Quantitative Energy Finance: Recent Trends and Developments 3031505964, 9783031505966

Table of contents :
Preface
Contents
List of Contributors
Part I Modelling of Energy Prices
Estimation of the Number of Factors in a Multi-Factorial Heath-Jarrow-Morton Model in Power Markets
1 Introduction
2 Model Description
3 Estimation
3.1 Distributions of the Changes in Futures and Spot Prices
3.2 State-Space System of Equations
3.2.1 Elements of the Space Equation
3.2.2 Elements of the State Equation
3.3 Implementation of the Kalman Filter and Minimization Algorithm
3.4 Criteria: AIC and BIC
4 Estimation Results
4.1 Data and Preprocessing Description
4.1.1 Description of the Data
4.1.2 Preprocessing of Data
4.1.3 Seasonal Adjustment of Spot Prices
4.2 Results
5 Conclusion
Appendix: Estimation Results
Belgian Power Market Data
French Power Market Data
German Power Market Data
References
Hawkes Processes in Energy Markets: Modelling, Estimation and Derivatives Pricing
1 Introduction
2 Empirical Evidence of Jump Clusters
3 Literature Review: Hawkes Processes in Energy Markets
4 Hawkes Processes
4.1 Definition Based on the Conditional Intensity
4.2 Moments of λT and NT
5 Simulation of Hawkes Processes with Exponential Kernel
5.1 Euler Scheme
5.2 Ogata's Modified Thinning Algorithm and a Variant
5.3 Dassios and Zhao's Exact Simulation
5.4 Comparison of Exact Methodologies
5.5 Comparison Between Algorithm 4 and the Euler Scheme
6 An Asset Pricing Model with Self-exciting Jumps
6.1 Dynamics Under the Historical Measure
6.2 Dynamics Under the Risk–Neutral Measure
7 Model Estimation
8 Model Simulation and Exotic Derivatives Pricing
8.1 Asian Options
8.2 Exact Simulation of (18)–(19)
8.3 Pricing Exotic Derivatives via Monte Carlo Simulation
9 Concluding Remarks
References
Periodic Trawl Processes: Simulation, Statistical Inference and Applications in Energy Markets
1 Introduction
1.1 Outline and Main Contributions of the Article
2 Mixed Moving Average and Periodic Trawl Processes
2.1 Background
2.2 Definition of a Mixed Moving Average Process
2.3 Periodic Trawl Processes
2.3.1 Second-Order Properties of Periodic Trawl Processes
2.3.2 Examples
3 Simulation of Periodic Trawl Processes
3.1 Slice-Based Simulation for Trawl Processes
3.1.1 Computing the Matrix of Slices
3.1.2 Adding the Weighted Slices
3.1.3 Simulating a Periodic Trawl Process
3.2 A Note on Stochastic Versus Deterministic Seasonality
4 Asymptotic Theory for MMAPs
4.1 Asymptotic Normality of the Sample Mean
4.2 Asymptotic Normality of the Sample Autocovariance
5 Inference for Periodic Trawl Processes Using Methods of Moments
5.1 Exponential Trawl Function
5.2 SupGamma Trawl Function
6 Inference for Periodic Trawl Processes Using a Generalised-Method-of-Moments Approach
6.1 Weak Dependence
6.2 GMM Estimation for Periodic Trawl Processes
7 Empirical Illustration on Electricity Day-Ahead Prices
8 Conclusion and Outlook
Appendix
Proof of the Second Order Properties
Proofs of the Asymptotic Theory
Examples
Exponential Trawl
SupGamma Trawl
Verifying the Assumptions of Theorem 2 for Selected Periodic Trawl Processes
Verifying Condition (11) from Proposition 7
Verifying Assumption (15) from Theorem 2
References
Part II Energy Transition
Fuelling the Energy Transition: The Effect of German Wind and PV Electricity Infeed on TTF Gas Prices
1 Introduction
1.1 European Natural Gas Consumption
1.2 Gas Key Player in the Energy Transition
1.3 Gas Pricing Factors
2 Literature Review
3 Data
4 Methodology
4.1 Baseline Model OLS
4.2 Threshold Regression Model
5 Empirical Results and Discussion
5.1 OLS
5.2 Threshold Regression Models
5.2.1 Wind Infeed Variables as Thresholds
5.2.2 PV Infeed Variables as Thresholds
6 Conclusion
7 Further Research
Appendix
References
A Mean-Field Game Model of Electricity Market Dynamics
1 Introduction
1.1 Literature Review
1.1.1 Optimization Models
1.1.2 Market Equilibrium Models
1.1.3 Simulation Models
1.1.4 Entry and Exit Models for Electricity Market
1.1.5 Mean-Field Games and Their Applications to Economic and Financial Modeling
2 The Model
2.1 Power Plants
2.2 Conventional Producers
2.3 Renewable Producers
2.4 Price Formation
3 Entry and Exit Mean-Field Game
3.1 Definition of the Mean-Field Setting
3.2 Price Formation
3.3 Optimization Functionals
3.4 Definition and Properties of Nash Equilibrium
4 Numerical Resolution and Illustrations
4.1 Numerical Illustration
Appendix: Proof of the Main Results
Preliminary Lemmas
Proof of Theorem 1
Proof of Proposition 2
References
PPA Investments of Minimal Variability
1 Introduction
2 An Infinite-Dimensional Problem
3 A Finite-Dimensional Problem
4 A Case Study with Solar
References
Part III Climate Risk
Climate Risk in Structural Credit Models
1 Introduction
2 Climate Risk
2.1 Physical Risk
2.2 Transition Risk
3 Structural Credit Models
3.1 Merton Model
3.2 Extensions
3.2.1 Jump-Diffusion Models
3.2.2 Optimal Capital Structure Models
4 Climate-Adjusted Models of Credit Risk
4.1 Growth Adjustment
4.2 Shocking the Value Process
4.3 Discontinuous Climate Impacts
4.4 Climate Scenario Uncertainty
5 Conclusions
References

Citation preview

Fred Espen Benth Almut E. D. Veraart   Editors

Quantitative Energy Finance Recent Trends and Developments

Quantitative Energy Finance

Fred Espen Benth • Almut E. D. Veraart Editors

Quantitative Energy Finance Recent Trends and Developments

Editors Fred Espen Benth Department of Mathematics University of Oslo Oslo, Norway

Almut E. D. Veraart Department of Mathematics Imperial College London London, UK

ISBN 978-3-031-50596-6 ISBN 978-3-031-50597-3 https://doi.org/10.1007/978-3-031-50597-3

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

The European power systems undergo a huge transition towards renewable generation, having a significant impact on risk management and operations in the markets for electricity. Gas- and coal-fired power plants are one of the major sources of carbon emissions, and have to be substituted to reach the net-zero goals of Europe. This rapid transition creates new and challenging problems within quantitative energy finance, some of which we address in this volume. In the book Quantitative Energy Finance – Modeling, Pricing, and Hedging in Energy and Commodities Markets (F. E. Benth, V. A. Kholodnyi and P. Laurence (eds.), Springer Verlag 2014), the focus was on bridging risk management tools from financial theory over to energy and commodity markets, with a particular view on power. Nearly a decade later, the markets for electricity in Europe have experienced a huge development, with closer integration between regions, fast development of renewable power and regulatory changes such as the EU taxonomy on sustainable finance. The markets after 2020 have also been hit by unprecedented highs and lows of electricity prices, explained by longer periods of little to no wind over Europe, a high degree of intermittency in the generation, and the cut in the import of Russian gas. In the future, climate change is predicted to further impact the power markets, with changing weather patterns leading to more frequent extreme weather such as heat waves, "Dunkelflaute" and cold spells. The electrification of society (transport, households, industry) leads, on the other hand, to an increased demand for power. With the current volume Quantitative Energy Finance – Recent Trends and Developments, which is a stand-alone continuation of the book published in 2014, we have collected a set of scientific papers analysing important aspects and challenges that we see for the moment and on the way ahead towards a net-zero energy system. We have grouped the papers according to three broad topics: The first group of articles is concerned with the modelling of energy prices taking recent changes in energy generation into account, followed by articles on the energy transition, and we conclude the book with a recent survey on the topic of climate risk. We will now briefly summarise the main contributions of each chapter.

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Preface

Modelling of Energy Prices Estimation of the Number of Factors in a Multi-factorial Heath–Jarrow–Morton Model in Power Markets by Olivier Féron, and Pierre Gruet: This chapter advances calibration methods for multi-factorial Heath–Jarrow–Morton models in the context of power markets with a particular focus on determining the optimal number of Gaussian factors in the model. The authors calibrate the model jointly on both spot and futures prices using maximum-likelihood techniques combined with information criteria. In an empirical study of Belgian, French, and German power prices, they demonstrate close similarities between the three markets and the number of factors needed to model the prices well. Hawkes Processes in Energy Markets: Modelling, Estimation and Derivatives Pricing by Riccardo Brignone, Luca Gonzato and Carlo Sgarra: In this chapter, the authors first review recent developments in using Hawkes processes to model energy prices and carry out derivatives pricing, including a description of exact simulation methods for Hawkes processes. Next, they propose a stylised new model for energy spot prices, which is built on a Hawkes process. Since this model is formulated under the historical probability measure, they furthermore establish a structure-preserving change of measure to also describe the corresponding risk-neutral dynamics of the spot prices needed for derivatives pricing. Using particle filtering techniques, the model can be estimated and an application to pricing exotic derivatives concludes this work. Periodic Trawl Processes: Simulation, Statistical Inference and Applications in Energy Markets by Almut E. D. Veraart: This chapter introduces the new class of continuous-time periodic trawl processes, which can account for periodic behaviour in the serial correlation either in a short- or long-memory framework. It presents their probabilistic properties and establishes the asymptotic theory for (generalised) method of moments estimators for the model parameters and proposes efficient simulation schemes for such processes. The methodology is applied to electricity spot prices from the German electricity market. Energy Transition Fuelling the Energy Transition: The Effect of German Wind and PV Electricity Infeed on TTF Gas Prices by Christoph Halser and Florentina Paraschiv: In order to facilitate the energy transition, low-carbon and flexible balancing tools are needed to deal with the intermittency of renewable energy generation. Gas has been playing an important role in the current energy transition and hence the authors study the substitution effect between gas and renewable energies wind and PV. They carry out their analysis in the context of threshold regression models applied to recent daily Dutch natural gas prices. They find a negative marginal effect of the day-ahead wind and PV infeed forecasts on day-ahead natural gas prices and a positive association between the day-ahead gas price and CO2 prices, coal prices, heating demand and supplier concentration. A Mean-Field Game Model of Electricity Market Dynamics by Alicia Bassiére, Roxana Dumitrescu and Peter Tankov: This chapter develops a mean-field game

Preface

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model for the long-term dynamics of electricity markets. This new model includes various refinements over existing models, such as that an arbitrary number of technologies with endogenous fuel prices can be considered, agents can both invest and divest, and various temporal aspects of the plant construction and age can be incorporated. The new model aims to describe the impacts of energy transition on electricity markets with a particular focus on the role gas plays, in the medium term, as a substitute for coal. The authors illustrate the properties of the new model through numerical computations and a stylised example. PPA Investments of Minimal Variability by Fred Espen Benth: A power purchase agreement (PPA) is a long-term financial contract between an electricity generator and a customer. In this chapter, the focus is on a PPA, where one can virtually operate a solar or wind power park. Since renewable energy generation is highly volatile, such a PPA could be used as a spatial hedge, where production is spread out geographically. This chapter proposes a model for the capacity factors of solar and wind generation by a square-integrable random field in a Hilbert space with an associated covariance operator and analyses how the variability of a portfolio of power plants spread out over various spatial locations can be minimised. In a case study of a PPA of a portfolio of solar power plants in Germany, it is shown that the variability of the difference between solar power production and electricity demand can be reduced significantly by a spatial hedge. Climate Risk Climate Risk in Structural Credit Models by Alexander Blasberg and Rüdiger Kiesel: The book concludes with a timely survey on how the impact of climate risk on financial markets can be described by structural credit risk models. Physical and transition risks, often considered the key components of climate risk, can be captured by the classical Merton model and its extensions, and the authors carefully describe the advantages and shortcomings of the existing models and outline possible improvements. All chapters have been refereed by peers, to whom we are grateful for their (anonymous) contribution to the scientific quality of this book. We thank Remi Lodh and Ute McCrory at Springer Verlag for creating the opportunity for publishing a second volume on quantitative energy finance and for their support and assistance throughout the preparations. F. E. Benth acknowledges financial support from Spatus, a thematic research group funded by UiO:Energy and Environment at the University of Oslo. Oslo, Norway London, UK October 2023

Fred Espen Benth Almut E. D. Veraart

Contents

Part I Modelling of Energy Prices Estimation of the Number of Factors in a Multi-Factorial Heath-Jarrow-Morton Model in Power Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Olivier Féron and Pierre Gruet

3

Hawkes Processes in Energy Markets: Modelling, Estimation and Derivatives Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Riccardo Brignone, Luca Gonzato, and Carlo Sgarra

41

Periodic Trawl Processes: Simulation, Statistical Inference and Applications in Energy Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Almut E. D. Veraart

73

Part II Energy Transition Fuelling the Energy Transition: The Effect of German Wind and PV Electricity Infeed on TTF Gas Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Christoph Halser and Florentina Paraschiv A Mean-Field Game Model of Electricity Market Dynamics . . . . . . . . . . . . . . . 181 Alicia Bassière, Roxana Dumitrescu, and Peter Tankov PPA Investments of Minimal Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Fred Espen Benth Part III Climate Risk Climate Risk in Structural Credit Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Alexander Blasberg and Rüdiger Kiesel

ix

List of Contributors

Alicia Bassière CREST, ENSAE, Institut Polytechnique de Paris, Palaiseau, France Fred Espen Benth Department of Mathematics, University of Oslo, Oslo, Norway Alexander Blasberg University of Duisburg-Essen, Essen, Germany Riccardo Brignone Department of Quantitative Finance, Institute for Economic Research, University of Freiburg, Freiburg im Breisgau, Germany Roxana Dumitrescu Department of Mathematics, King’s College London, London, UK Olivier Féron EDF R&D and Fime Lab, Palaiseau, France Luca Gonzato Department of Statistics and Operations Research, University of Vienna, Vienna, Austria Pierre Gruet EDF R&D and Fime Lab, Palaiseau, France Christoph Halser Business School, Norwegian University of Science and Technology, Trondheim, Norway Rüdiger Kiesel University of Duisburg-Essen, Essen, Germany Florentina Paraschiv Chair of Finance, Zeppelin University, Friedrichshafen, Germany Carlo Sgarra Department of Mathematics, Politecnico di Milano, Milan, Italy Peter Tankov CREST, ENSAE, Institut Polytechnique de Paris, Palaiseau, France Almut E. D. Veraart Department of Mathematics, Imperial College London, London, UK

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Part I

Modelling of Energy Prices

Estimation of the Number of Factors in a Multi-Factorial Heath-Jarrow-Morton Model in Power Markets Olivier Féron and Pierre Gruet

Abstract We study the calibration of specific multi-factorial Heath-Jarrow-Morton models to power market prices, with a focus on the estimation of the optimal number of Gaussian factors. We describe a common statistical procedure based on likelihood maximisation and Akaike/Bayesian information criteria, in the case of a joint calibration on both spot and futures prices. We perform a detailed analysis on three national markets within Europe: Belgium, France, and Germany. The results show a lot of similarities among all the markets we consider, especially on the optimal number of factors and on the behaviour of the different factors.

1 Introduction Electricity generation and supply have been widely liberalised in a large set of countries over the last decades. Although their precise organisation varies across places, power markets share a common structure linked to the specificities of electricity: it is not storable and therefore has to be produced exactly when it is consumed. For instance, in Western Europe, the spot market takes place everyday and allows one to define the amounts of electricity that will be produced (and consumed) during each of the hours in the next day, based on quite accurate forecasts of consumption needs and production capacities. However, as prices are very volatile on the spot market, utilities may casually want to avoid having their full production exposed to the spot price, and they can mitigate their financial exposure on the spot market by trading derived products on the financial futures market. On this futures market, standardised contracts can be exchanged continuously for the next weeks, months, quarters and years or seasons. Grasping the characteristics of the evolution of prices on the futures market is essential to be able to use it efficiently by computing relevant hedging strategies and risk indicators.

O. Féron · P. Gruet (O) EDF R&D and Fime Lab, Palaiseau, France e-mail: [email protected]; [email protected]; https://www.fime-lab.org © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. E. Benth, A. E. D. Veraart (eds.), Quantitative Energy Finance, https://doi.org/10.1007/978-3-031-50597-3_1

3

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O. Féron and P. Gruet

We address the statistical estimation of a family of models for electricity prices and propose a methodology to select one of those models using information criteria. We consider Heath-Jarrow-Morton (HJM) models, introduced in [18] to represent the dynamics of the forward rates. In that work devoted to the term structure of interest rates, the forward rates processes were led by a sum of N Brownian motions and a drift. Being common to all maturities, this set of stochastic factors was driving the whole forward rate curve. Reference [18] focuses on the use of their model for valuing contingent claims, and some examples are given. Using such models to represent electricity prices has been done by many authors. Reference [5] defined a 1- and a 3-factor HJM models to represent the dynamics of instantaneous delivery futures in power markets. As they acknowledged that the actually traded contracts are flow futures, meaning there is some delivery period, they used an approximation studied in [23] to derive a valuation formula for such flow futures contracts. In contrast, [7] applied HJM models to oil prices: they designed a methodology to search for the best number of factors by doing a Principal Component Analysis (PCA) on their data. Reference [25] followed the same approach on electricity prices, while accounting for the existence of the delivery period. Those last two articles let the volatility functions be totally unspecified: the PCA leads to nonparametric volatility functions. As they did, many authors have also looked for the best number of stochastic factors and for the shape of the volatility coefficients: keeping them simple ensures the models can be used operationally for risk management purposes and can lead to simple formulas for prices of derivatives. Reference [27] designed 1- and 2-factor models for the electricity spot price only. Reference [22] designed 1- to 4-factor models for electricity prices, accounting for the delivery period, and designed an extended Kalman filter to estimate the models on data from Nordpool market. They suggested to use 2- or 3factor models, but had no mathematical criterion to argue. Their model incorporates a noise process, which features market imperfections. Reference [28] proposed a sum of two Ornstein-Uhlenbeck mean-reverting processes to represent gas futures prices, which they estimated on Henry Hub price data. Reference [10] studied the risk premia in power markets with a 3-factor model that they estimated using a twostep procedure. Reference [24] introduced a 2-factor model for electricity prices, which they calibrated on German market to implied volatilities. The same model was studied in [13] where the calibration results show instability of the parameters, depending on the data that are considered. Reference [1] discussed the use of HJM models for futures contracts in power markets. They explained the implications of various modelling choices from a practical viewpoint. Reference [11] proposed a 2factor model similar to the one in [24]. Their model does not allow for delivery periods, but it can account for more commodities to be correlated. It is applied on oil prices data after many estimation methods are described. References [2, 26] recently proposed HJM-type additive models to jointly represent instantaneous and flow futures. The former article lists conditions so that such models do not allow for arbitrage. It gives examples of simple models suiting their frame. The latter article performs estimation in such an additive model with 2 factors, by minimising the difference between the theoretical and empirical covariations of processes.

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

5

Reference [14] performed the efficient estimation of a 2-factor model with stochastic volatility, working within a market model representing the dynamics of futures contracts with a delivery duration of one month. For a thorough introduction to factor models as well as a review of articles using them to model electricity prices, one may refer to [9]. In the context of interest rates, [3] estimated HJM models on Australian interest rates data. In their model, the volatility is a function of the level of the process. Reference [4] performed maximum likelihood estimation of a 1-factor model on American short term interest rates data. Reference [19] discussed the estimation of HJM models on German bond data by performing a PCA, and then by using nonlinear regression to estimate parametrically four specific models. Reference [20] worked on HJM models for interest rates dynamics where the volatility is an unspecified function of the rate level. They acknowledged that there are less Brownian motions than yield curves to be represented in their model, which implies stochastic singularity: some deterministic relationships between yield curves hold using the model, although they do not hold on real-world data. In order to be able to select a relevant model, one of the main stakes is to represent the volatility structure of prices. Among all possible models, we have to find an equilibrium between a good quality of representation and a simple and parsimonious form. We are focusing on models in which the dynamics of prices is driven by a sum of correlated Brownian motions, with deterministic volatility coefficients which decrease exponentially as the remaining time to delivery increases. This class of models is very well known and used in practice for its tractability to deal with option pricing and hedging purposes. It encompasses the set of seminal commodity models [15, 27, 30], which are used for pricing derivatives, for example, in [8] and more recently in [12]. Our aim is not to find the best price model, but rather to find differences and similarities on different power markets by means of a quite simple class of HJM models. We focus our study on the number of needed Brownian motions as a function of the market and the data used in order to be parsimonious while reaching a good quality of representation. To this end, we compute the classical Akaike information Criterion (AIC) and Bayesian information criterion (BIC), and we also propose some additional indicators to help the user choose an efficient number of factors. Selecting the appropriate number of stochastic components to describe electricity prices was also the focus of [16]. Starting with a Gaussian Ornstein-Uhlenbeck stochastic factor, they added Lévy-driven Ornstein-Uhlenbeck processes to the dynamics to model either positive or negative jumps until the predictive p-values of their models are satisfactory. Based on deseasonalised daily electricity spot prices without the weekends from the UK and Europe on various historical periods, they found that (depending on the datasets) one or two jump components give a satisfactory quality of representation given their metrics. Here we shall only consider Gaussian processes but the estimation will be performed on spot and futures contracts jointly. The rest of the chapter is organised as follows. In Sect. 2 we present the model and recall the corresponding dynamics of the prices for spot and futures contracts

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O. Féron and P. Gruet

with delivery periods. The estimation procedure is described in Sect. 3. In Sect. 4 we precisely describe the data used for the estimation, and we show and analyse the estimation results. Section 5 concludes this work and outlines some future research directions.

2 Model Description In this section we introduce the model for electricity price futures and we derive the equation for spot prices. In the sequel we use the following notation: • .Ft (T ) denotes the unitary power futures price at date t for the delivery of one megawatt-hour (hereafter MWh) of electricity at date T . Such a unitary futures is not traded on the markets, but is used as a modelling brick to write the spot price and the prices of quoted futures, see [9] for a thorough discussion on this approach; • .St = Ft (t) is the power spot price; • .Ft (T , θ ) denotes the power futures price quoted at t, delivering 1 MWh during all the hours between times T and .T + θ , where .θ is the length, in years, of the delivery period. Let .(o, F, F = (Ft )t≥0 , P) be a filtered probability space, and let us assume the absence of arbitrage. There exists a unique risk-neutral measure .Q equivalent to .P, and under this risk-neutral measure we consider the classical Heath-JarrowMorton [18] model written on the unitary futures price: dFt (T ) E −αk (T −t) = e σk dWtk , Ft (T ) N

.

(1)

k=1

where .N ≥ 1 is the number of stochastic factors, .(W k )k=1,...,N is a N -dimensional ' .F-adapted .Q-Brownian motion, of which components k and .k have correlation .ρk,k ' , .σk > 0 for .1 ≤ k ≤ N, and .0 < α1 < · · · < αN in order to guarantee identifiability of the model. By integrating Eq. (1) and letting .T = t, we deduce the expression of the spot price .St as a function of .Ft0 (t), for .t0 ≤ t: { St = Ft0 (t) exp

1 − e−(αk +αk' )(t−t0 ) 1EE ρk,k ' σk σk ' 2 αk + αk ' ' N

−

N

k=1 k =1

.

+

N f t E k=1 t0

}

σk e−αk (t−s) dWsk .

(2)

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

7

Applying Itô formula and assuming that .Ft0 is differentiable, we get the dynamics of the spot price, namely dSt = . St

(

) n N N ) ) E 1EE −(α +α )(t−t ) ' 0 + σk σk ' ρk,k ' 1 − e k k dZtk , dt + Ft0 (t) 2 '

Ft'0 (t)

k=1 k =1

k=1

where we introduced the auxiliary processes .Z k , .k = 1, . . . , N, defined by dZtk = −αk Ztk dt + σk dWtk , .

Ztk0 = 0 ,

(3)

for .1 ≤ k ≤ n. It is worth emphasising that the dynamics of the spot price can be written as being led by a sum of Ornstein-Uhlenbeck processes. Concerning the futures prices, as in [13], we can consider the no-arbitrage (discrete) relationship between futures contract prices and unitary futures prices in the form Ft (T , θ ) =

.

θ/ h−1 h E Ft (T + ih) , θ i=0

where h is a the timestep (1 hour or 1 day for example) considered in the discretization of the forward curve. Combining this equation with Eq. (1), we can deduce the dynamics of the futures contract prices: θ/ h−1 N E h E 1 dFt (T , θ ) = Ft (T + ih) e−αk (T +ih−t) σk dWtk Ft (T , θ ) θ Ft (T , θ ) i=0 k=1 ⎛ ⎞ . θ/E h−1 N E h Ft (T + ih) −αk ih ⎠ e = e−αk (T −t) σk ⎝ dWtk . θ Ft (T , θ ) k=1

(4)

i=0

3 Estimation In this section we describe the estimation methodology based on Kalman filtering and the maximum likelihood principle, and we explain how we compute the classical AIC and BIC in order to study the optimal number of factors. The likelihood function is obtained from a state-space equation system and computed via a Kalman filter: the likelihood function is the one of the residuals of the filter, which are multivariate Gaussian at each time step and are independent from one time step to the other. Also, as already described in [17], we will introduce a Gaussian error model to face the stochastic singularity (see [20]) when the number of model factors is lower than the number of observed futures contracts.

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In the sequel we consider .n + 1 quotation dates .t0 , . . . , tn and we use the notation Ati = ti − ti−1 and .Ani X = Xti − Xti−1 for any process X. Thus the index n stands for the number of updates in the Kalman filtering step, and thus for the number of Kalman residuals that will be computed. We will assume that the historical measure .P and the risk-neutral measure .Q coincide, so that the .Q-Brownian motions are also .P-Brownian motions and the dynamics (1) also holds under .P. Our motivation to do so is twofold:

.

• Assuming the dynamics (1) holds under .Q, we could consider a dynamics with a drift term under .P, for example

.

E dFt (T ) E -tk , = bk e−αk (T −t) dt + e−αk (T −t) σk d W Ft (T ) N

N

k=1

k=1

- k are .P-Brownian motions and .b1 , . . . , bN are real numbers. There where the .W is no technical difficulty in making inference about those N real numbers along with the volatility parameters by following the methodology described hereafter. Still this increases the dimension of the optimization problem while our goal is to discuss the volatility structure; • As [10] reported in 2006, there is empirical evidence that the absolute values of risk premia decrease as markets become more mature and attract speculators.

3.1 Distributions of the Changes in Futures and Spot Prices We consider the approximation of the futures prices dynamics, as in [13], stating +ih) that the shaping factors . FFt (T are all equal to 1. This assumption boils down to t (T ,θ) asserting the contribution of each hour in a given delivery period to the price of the whole delivery is the same, and then that, for each delivery period .[T , T + θ ], all prices .Ft (s) for .s ∈ [T , T + θ ], are identical and equal to the observed futures price .Ft (T , θ ). In order to avoid inconsistency, it is necessary to consider futures with disjoint delivery periods: for any couple .(Ft (T1 , θ1 ), Ft (T2 , θ2 )) of futures contracts considered in the sequel, we have .[T1 ; T1 + θ1 ] ∩ [T2 ; T2 + θ2 ] = ∅. In practice, this means that a preprocess (described hereafter in Sect. 4.1.2) is needed to remove any overlap in the futures’ delivery periods. With these assumptions, one can compute the dynamics (4) as dFt (T , θ ) E −αk (T −t) = e σk ψh (αk , θ )dWtk , Ft (T , θ ) N

.

k=1

where .ψh (α, θ ) =

h 1−e−αθ θ 1−e−αh .

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

9

By denoting .XtT ,θ = log(Ft (T , θ )) and applying Itô’s lemma we get Ani XT ,θ =

N f E

ti

k=1 ti−1

.

1EE 2 ' N

−

ψh (αk , θ )e−αk (T −t) σk dWtk N

f

ti

e−(αk +αk' )(T −t) ρk,k ' σk σk ' ψh (αk , θ )ψh (αk ' , θ )dt .

k=1 k =1 ti−1

(5) At each time .ti we assume observing .Li prices of futures contracts1 which are denoted by .Fti (Tl , θl ), for .l = 1, . . . , Li . When .Li > N, the model presents a stochastic singularity (already studied in [20] and in [17] in the case of a two-factor model applied to electricity prices). In order to face this problem we introduce a model error term and assume observing noisy returns. Precisely, we do not observe the increments .Ani XTl ,θl of the price process, but instead we observe Ani Y Tl ,θl = Ani XTl ,θl + εiTl ,θl ,

.

where .εiTl ,θl are identically distributed according to a Gaussian distribution 2 2 is unknown. Moreover, for all .i = 1, . . . , n, the random .N(0, v ), where .v T ' ,θ ' Tl ,θl variables .εi and .εi l l are independent for .1 ≤ l < l' ≤ Li , as well as the T ' ,θ

'

random variables .εiTl ,θl and .εj l l for .1 ≤ i < j ≤ n and .1 ≤ l ≤ Li , 1 ≤ l' ≤ Lj . ) )' Therefore, the vector . Ani Y T1 ,θ1 · · · Ani Y TLi ,θLi which stacks up the observed noise-perturbed increments at time step i is Gaussian .N(Mi , E i ) with .Mi = ' (Mil )1≤l≤Li , .E i = (Eill )1≤l,l' ≤Li and: ) ) Mil = E Ani Y Tl ,θl

.

N N ( 1EE ρk,k ' σk σk ' ψh (αk , θl )ψh (αk ' , θl ) =− 2 ' k=1 k =1

× e−(αk +αk' )(Tl −ti )

1 − e−(αk +αk' )Ati αk + αk '

(6) )

1 In practice the number of observed prices may vary, even after removing the redundant products (e.g. a quarterly contract when all the corresponding monthly contracts are observed), see Sect. 4.1.

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and ) ) ' Eill =Cov Ani Y Tl ,θl , Ani Y Tl' ,θl'

.

N ( N E E ρk,k ' σk σk ' ψh (αk , θl )ψh (αk ' , θl' ) +

=v 1l=l' 2

k=1 k ' =1

× e−αk (Tl −ti ) e−αk' (Tl' −ti )

1 − e−(αk +αk' )Ati αk + αk '

) ,

where .1l=l' = 1 if .l = l' and 0 otherwise. Let us introduce the auxiliary processes .Z k , for .k = 1, . . . , N , defined by .Ztk0 = 0 and dZtk = −αk Ztk dt + σk dWtk .

.

By integrating the above dynamics between dates .ti−1 and .ti , we have Ztki = Ztki−1 e−αk Ati +

f

ti

.

ti−1

σk e−αk (ti −t) dWtk

so that we can rewrite Eq. (5) with the auxiliary processes .Z k as: Ani XT ,θ =

N E

) ) ψh (αk , θ )e−αk (T −ti ) Ztki − Ztki−1 e−αk Ati

k=1 N N ( 1EE ρk,k ' σk σk ' ψh (αk , θ )ψh (αk ' , θ ) − 2 '

.

k=1 k =1

× e−(αk +αk' )(T −ti )

1 − e−(αk +αk' )Ati αk + αk '

Concerning the spot price, we use expression (2) and we write .St = seasonality adjusted spot price in order to obtain

(7) ) . St Ft0 (t)

for the

Ani X = log Sti − log Sti−1 = .

N ) ) E Ztki − Ztki−1 k=1

1 − e−(αk +αk' )Ati 1EE . ρk,k ' σk σk ' e−(αk +αk' )(ti−1 −t0 ) αk + αk ' 2 ' N

−

N

k=1 k =1

(8)

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

11

As for the dynamics of the futures prices, we consider a model (or measurement) error in the observed spot prices. We assume observing .Ani Y defined as Ani Y = Ani X + εi ,

.

where the random variable .εi is distributed according to a Gaussian distribution N(0, v 2 ) and is independent of all random variables .εjTl ,θl , .j = 1, . . . , n, .l = 1, . . . , Lj . We may notice that .Ani X cannot be considered as a price “return” because the underlying is the spot price, which stands for a different delivery period each day. We propose to consider this element in the calibration process in order to work with differences in price logarithms for all observed (futures and spot) prices.

.

3.2 State-Space System of Equations Having derived the distributions of price changes at each time step in the previous subsection, the multi-factor model can be written in a state-space model formulation: let us denote ⎛ 1 ⎞ Zti ⎛ n T1 ,θ1 ⎞ Ai Y ⎜Z 1 ⎟ ⎜ t ⎟ ⎜ ⎟ .. ⎜ .i−1 ⎟ ⎜ ⎟ n . ⎜ Zi = ⎜ .. ⎟ .Ai Y = ⎜ ⎟ , ⎟ . ⎝An Y TLi ,θLi ⎠ ⎜ N ⎟ i ⎝ Zti ⎠ Ani Y ZtNi−1 The multi-factor model can be written as follows: Ani Y = Mi + Fi Zi + ε i , .

Zi = Ai Zi−1 + ηi ,

(9)

with elements .Mi , .Fi , .Ai and the covariance matrices of .ε i and .ηi given in the upcoming subsections, in which we describe the components of the state-space system (9).

3.2.1

Elements of the Space Equation

Ani Y is a vector of size Li + 1, corresponding to the number of observed futures returns at date ti and the deseasonalized spot price change. The first Li components correspond to the futures prices returns and the last component corresponds to the spot. The mean vector Mi then has its Li first components defined by Eq. (6) and its last component defined, accordingly, by the last term in Eq. (7).

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Concerning the matrix Fi , we propose to write it as the stack of two different f linear forms Fi = ((Fi )' , (Fsi )' )' corresponding to the futures contracts and the spot, respectively. Using Eq. (7) we have f

f

f

Fi = Gi Hi

.

f

with Gi a (Li × N) matrix: ⎤ ψh (α1 , θ1 )e−α1 (T1 −ti ) . . . ψh (αN , θ1 )e−αN (T1 −ti ) ⎥ ⎢ f .. .. .G = ⎣ ⎦ . ... . i ψh (α1 , θLi )e−α1 (TLi −ti ) . . . ψh (αN , θLi )e−αN (TLi −ti ) ⎡

f

and Hi a (N × 2N) matrix: ⎡

1 −e−α1 Ati ⎢0 0 ⎢ ⎢ .. .. f ⎢ .H . i = ⎢. ⎢. .. ⎣ .. . 0 ...

0 ... 1 −e−α2 Ati .. . 0 .. .. . . ... ...

... ... 0 ...

0 0 .. .

... ... .. . 0 0 0 1 −e−αN Ati

⎤ ⎥ ⎥ ⎥ ⎥ . ⎥ ⎥ ⎦

f

The matrix Hi may not depend on time if the time step Ati is constant. However, f Gi depends on time because of the maturity terms Tl − ti . Concerning the part Fsi dedicated to the spot prices, we can use the same decomposition using Eq. (8): Fsi = Gsi His

.

with Gsi = 1'N the transpose of a N-dimensional vector composed of ones, and His a (N × 2N) matrix: ⎡

1 ⎢0 ⎢ ⎢ .. s .Hi = ⎢ . ⎢ ⎢. ⎣ .. 0

−1 0 .. . .. . ...

0 1 .. . .. . ...

⎤ ... ... ... 0 −1 0 . . . 0 ⎥ ⎥ .. ⎥ 0 ... ... . ⎥ ⎥ . ⎥ .. .. . . 0 0⎦ . . . 0 1 −1

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

13

In the calibration tests, as we explained above, we assume that the variance of the model errors is identical for all observed contracts, i.e. Qεi = v 2 ILi +1

.

where Im is the m × m identity matrix.

3.2.2

Elements of the State Equation

Using the solution of the Ornstein-Uhlenbeck processes on the factors .Z k defined in Eq. (3), we get the block-diagonal matrix ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ .Ai = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

e−α1 Ati 0 1 .. . .. . .. . 0 0

... .. .

0 .. −α At . e 2 i .. . 1 .. .. . . ... ... ... ...

... ...

...

... ...

...

0 ...

...

0 .. . ... ...

... ... .. . ... 0 e−αN Ati ... 1

0

⎤

⎥ 0⎥ ⎥ .. ⎥ .⎥ ⎥ .. ⎥ ⎥. .⎥ .. ⎥ ⎥ .⎥ ⎥ 0⎦ 0

If the time step .Ati is constant, then the matrix .Ai = A is also constant. The .(2N × 2N ) covariance matrix .Qηi is also deduced from the dynamics of the Ornstein-Uhlenbeck processes, and it is made from .2 × 2 blocks of which only the upper-left component is not zero. Precisely, for .1 ≤ k, k ' ≤ N, '

−1 Qη2k−1,2k = ρk,k ' σk σk ' i . ' Q2k−1,2k ηi

=

' −1 Q2k,2k ηi

=

' Q2k,2k ηi

1 − e−(αk +αk' )Ati , αk + αk '

(10)

=0.

3.3 Implementation of the Kalman Filter and Minimization Algorithm Now we explain how the Kalman filter is implemented and how the log-likelihood of the Kalman residuals is computed. Kalman filtering is named after Kalman [21] and is aimed at addressing problems in which one tries to get information about some state process that is shaded within noisy measurements. This is done by making, at each time step, some prediction of the next state and then updating the internal variables of the filter by using the comparison of this prediction to the realized state as a feedback. Namely, in the state-space equation (9), .Zi is the hidden state at

14

O. Féron and P. Gruet

step i and it is driven by the evolution of the stochastic factors on which one is willing to make inference. .Ani Y is the observation, it is made of combinations of the components of the hidden state vector .Zi , which is added to the noise vector .εi . Now we describe the filtering equations. At each time step .i = 1, . . . , n, we start with the a priori variables .Z i|i−1 and .Pi|i−1 , featuring the estimates of the mean and the variance of .Zi given the observations at time step i and prior to it. For the initialization at step 1, one needs initial values and we choose .Z 1|0 = 0 and .P1|0 = Qη1 , which is defined by (10). Yet the choice of the initial conditions never precludes the convergence of the parameters. Then we compute the Kalman gain .Ki , given by Ki = Pi|i−1 Fi' G−1 i ,

.

Gi = Fi Pi|i−1 Fi' + v 2 ILi +1

where for any matrix M, .M ' denotes its transpose. The estimate of .Ani Y conditionally to the past is given by Ani Y = Mi + Fi Z i|i−1 ,

.

then the Kalman residual .ri is given by .ri = Ani Y − Ani Y: this stands for the difference between the actual observed value and the expectation within the filter. Finally, one can compute the a posteriori variables Z i = Z i|i−1 + Ki ri , ( ) Pi = I2N − Ki Fi Pi|i−1 .

.

And one can prepare the a priori variables of the next step .i + 1 by computing Z i+1|i = Ai+1 Z i ,

.

Pi+1|i = Ai+1 Pi A'i+1 + Qηi+1 . It turns out that the Kalman residuals at two different steps are independent from each other, and that .ri ∼ N(0, Gi ). After iterating over all time steps, one can compute the likelihood of the sample .(r1 , . . . , rn ) of residuals as n | |

1

i=1

(2π )(Li +1)/2 det(Gi )1/2

.

) ( 1 . exp − ri' G−1 r i i 2

Therefore the negative log-likelihood is given by .Ln + c, where 1E log(det(Gi )) + ri' G−1 i ri 2 n

Ln =

.

i=1

and c is a real number which does not depend on the parameters.

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

15

As we are able to compute the negative log-likelihood for any set of parameters by going through the n filtering steps we described above, we use an optimization routine to minimize .Ln . However, leading the optimization over the pristine parameters v, .α1 < · · · < αN , .σ1 > 0, . . . , σN > 0 and .−1 < ρk,k ' < 1 for ' .1 ≤ k, k ≤ N that we have introduced in Sect. 2 is difficult as one has to handle constraints about the .α coefficients being ordered, the .σ coefficients being positive and the correlation ones being bounded above and below. Instead, we let a1 = log(α1 ) and ak = log(αk − αk−1 ) for 2 ≤ k ≤ N ,

.

and also we introduce the lower triangular matrix .S = (sk,k ' )1≤k,k ' ≤N such that .SS ' is the Cholesky decomposition of the positive-definite matrix ⎛

⎞ ρ1,2 σ1 σ2 . . . ρ1,N σ1 σN ⎜ ⎟ .. ⎜ ρ1,2 σ1 σ2 ⎟ σ22 ... . ⎜ ⎟. . ⎜ ⎟ .. .. . . . . ⎝ ⎠ . . . . ... ... σN2 ρ1,N σ1 σN σ12

We thus run an unconstrained optimization algorithm over the real numbers v, a1 , . . . , aN and .sk,k ' , .1 ≤ k ' ≤ k ≤ N . While this problem still has dimension (N +1)(N +2) . , it is easier to solve numerically as it embeds no constraint at all. We use 2 the method of Nelder and Mead [29], which is a standard simplex method without derivatives. .

3.4 Criteria: AIC and BIC The number of degrees of freedom (to account for in the AIC and BIC) is a function of the number N of factors in the model. For a definition and a discussion of AIC and BIC, we refer the reader to [6]. Given a fixed N, the parameters are as follows: N degrees of freedom corresponding to the parameters .ak , 1 degree of freedom corresponding to parameter v, and . N (N2+1) degrees of freedom corresponding to the coefficients in the lower triangular matrix S. The total number of degrees of freedom is then .(N + 1)(N + 2)/2. Therefore, in the case of the factorial models described above, the AIC and BIC are given by: -n , AI C = (N + 1)(N + 2) + 2L

.

BI C =

1 -n , (N + 1)(N + 2) log(n) + 2L 2

-n denoting the minimized negative log-likelihood of the Kalman residuals with .L that one obtains when applying Kalman filtering to the state-space equation (9).

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4 Estimation Results 4.1 Data and Preprocessing Description In this section, we describe the datasets that we used and we explain how we preprocessed them before running our estimation procedures. We also detail the process of seasonal adjustement of spot prices.

4.1.1

Description of the Data

We have used data of prices in several European power markets, namely Belgium, France, and Germany. Those prices are available on the websites https://eex.com (for the futures prices) and https://epexspot.com (for the spot prices). We collected the closing futures prices every business day, from 2018–01–01 to 2022–11–30, for various contracts, delivering 1 MWh of electricity over standardised periods. Those periods can be: • the nearest (or 2nd nearest, or 3rd nearest. . . ) week (from Monday to Sunday) that has not begun yet. The underlying contracts are named 1 week-ahead (hereafter 1WAH), 2WAH, 3WAH, .. . . ; • the nearest months that have not begun yet, corresponding to month-ahead contracts (hereafter MAH); • the nearest quarters (January–March, April–June, July–September, October– December) that have not begun yet, corresponding to quarter-ahead (QAH) contracts; • the nearest calendar years that have not begun yet, featuring year-ahead (YAH) contracts. For each of the previous time spans, a given number of contracts are traded. For every market we collected the following futures contracts: 1 to 4WAH, 1 to 6MAH, 1 to 4QAH and 1 to 2YAH. This leads to 16 futures contracts considered for the estimation. Concerning spot prices, 24 prices are issued every day, related to deliveries of 1 MWh over each of the 24 hours of the day after. We computed the average of those 24 prices each day, featuring the price of the delivery of 1 MWh over the following day. In total we thus considered 17 daily prices for the estimation. We note that some data are missing, but the maximisation of the likelihood with the Kalman filter as described in Sect. 3.3 can easily deal with a set of missing data. Indeed, in both cases one only has to consider a varying vector size .Li of available prices at each date .ti .

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

4.1.2

17

Preprocessing of Data

As noticed in Sect. 3.1, we preprocess the data in order to remove all overlaps in the futures’ delivery periods. To do so, we use the structure of futures contracts in power markets and the no arbitrage principle: consider 2 futures contracts .Ft (T1 , θ1 ) and .Ft (T2 , θ2 ) oberved at date t such that .[T1 , T1 + θ1 ] ∩ [T2 , T2 + θ2 ] /= ∅ and .θ1 < θ2 ; we consider that futures contracts with shorter delivery periods give more precise information. In the case .T1 ≤ T2 , we therefore replace .Ft (T2 , θ2 ) by the futures contract .Ft (T1 + θ1 , T2 + θ2 − T1 − θ1 ) whose delivery period is disjoint from .[T1 , T1 + θ1 ] and whose value is obtained by no arbitrage: .

T1 + θ1 − T2 T2 + θ2 − T1 − θ1 Ft (T1 + θ1 , T2 + θ2 − T1 − θ1 ) Ft (T1 , θ1 ) + θ2 θ2 = Ft (T2 , θ2 ) .

In the case .T1 > T2 we replace .Ft (T2 , θ2 ) by two futures contracts .Ft (T2 , T1 − T2 ) and .Ft (T1 +θ1 , T2 +θ2 −T1 −θ1 ) of same value deduced from the same no arbitrage principle. In particular, this preprocessing allows also to face some observed complete redundancy, as it happens that • three monthly contracts exactly cover a quarter contract; • four quarterly contracts exactly cover a calendar contract. The rule previously described removes the contract with longest delivery period (respectively, in the two previous cases, the quarter and the calendar contracts) on each of those days. Also, we computed the returns at dates of changing products, caring for the specific change. For example, at a date of a month change, e.g. 2018–02–01, we compute the returns between the .(n + 1)MAH and the nMAH, both corresponding to the same futures contract, namely March 2018 (.n = 1) to July 2018 (.n = 6). By doing so, we optimise the quantity of information available in the data, but we do not have the same number .Li of returns each day. As explained earlier, this is acknowledged and easily dealt with in the computation of the joint distribution of the price changes.

4.1.3

Seasonal Adjustment of Spot Prices

In order to use the estimation procedure described in Sect. 3.1, we have to remove the seasonality from spot prices. To do so, we assume that the daily spot price .St at time t is given by .St = St Ft0 (t), where .St is the residual that is modelled in Sect. 3.1 and .Ft0 (t) is the seasonality term, which we represent with dummy variables as Ft0 (t) = yyear(t) mmonth(t) dweekday(t) ,

.

18

O. Féron and P. Gruet

where .year(t) refers to the year to which t belongs, .month(t) ∈ {1, . . . , 12} is the number of its month, .weekday(t) ∈ {1, . . . , 7} is the number of its day. Furthermore, we let m12 = 12 −

11 E

.

j =1

mj and d7 = 7 −

6 E

dj ,

j =1

so that estimation bears only on the first 11 monthly dummies and on the first 6 daily dummies. All the coefficients are estimated with a Least Square procedure. The spot returns are then computed on the residual .St .

4.2 Results For each of the three markets (Belgium, France and Germany), we split the global set of data into three subsets: 2018–2019 (before the Covid crisis), 2020 (Covid period) and 2020–12–01 to 2022–11–30 (hereafter referred to as “2021–2022”, crisis period). For each of these nine datasets, we run the likelihood maximisation on power prices, as described in Sect. 3.1, starting with .N = 2 factors. Then we compute the AIC and BIC, and we keep increasing N until the AIC and the BIC start increasing, which means one has reached the balance between the number of parameters and the quality of representation of the prices. The estimation results are given in Appendix. Figures 1, 2 and 3 show the computed AIC and BIC as a function of the number of factors. Table 1 shows the optimal number N of factors and Tables 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28 show the estimated parameters (mean-reverting .αi ’s, volatility .σi ’s and correlations .ρij ’s. As a global remark on all three markets, we can observe that the BIC allows us to discriminate and find the optimal number of factors more easily than the AIC, which starts increasing quite later than the BIC due to the lower penalty it puts on the number of parameters in the model. Hereafter we will discuss the optimal number of factors according to the BIC, rather than the AIC, in order to stress the penalty linked to the number of parameters and enforce parsimony. In Table 1, in the second row, we show the optimal number of factors according to the BIC. This optimal number is very similar amongst all countries and historical periods, between 8 and 9. In the third row of Table 1 we show the greatest number N of factors where all the correlations are not close to 1 or -1 (.|ρij | < 0.98, for all .i < j , .i = 1, . . . , N − 1). These N (between 2 and 4) are significantly smaller than the BIC-optimal ones, which suggests it is possible to derive some parsimonious, low dimensional models that account quite well for the behaviour of the prices. The additional factors only allow one to adjust the volatility function in order to compensate the difficulty of the factorial model (with only exponential volatility functions) to represent the observed volatility which may incorporate non-monotonic behaviours.

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

19

Table 1 Optimal number N of factors according to the AIC (first row) or the BIC (second row) and greatest N for which all the correlations between two factors are smaller than 0.98 Belgium 18–19 20 10 AIC-optimal N 9 BIC-optimal N 9 8 2 Greatest N , .max |ρkk ' | < 0.98 3

21–22 9 8 3

France 18–19 20 10 9 9 8 3 3

21–22 9 8 4

Germany 18–19 20 10 9 9 8 2 3

21–22 10 8 3

Within a given market and a given time period, the values taken by a given estimated parameter are rather stable over the number of factors; see for instance Table 3 and choose a given column i: the values of the estimator of .αi remain quite the same as soon as the number of factors in the estimation is greater than i. One can make the same observation with the estimates of the .σi . It also turns out that for a fixed time period, the values of the estimated parameters have quite the same order of magnitude from one country to another, especially for .α and at the BIC-optimal number of factors. Also, we can observe similar characteristics of the first factor (i.e. the factor with the smallest value of .α), mainly driving the long term volatility), for all countries. In particular, for N close to the BIC-optimal number of factors, one can observe quite stable and similar estimated values of .α1 (between 0.58 in Germany and 1.39 in France) and a slight increase for the period 2021–2022. The estimated value of .σ1 is, for all countries, around 30 and 40% for periods 2018–2019 and 2020, respectively, whereas it explodes for period 2021–2022, highlighting the impact of the European crisis in the energy markets and the fact that the markets are interconnected and share some fundamentals. Let us discuss a bit the correlation matrices. For all countries and periods, the correlation matrices seem stable along the number of factors. Also, they feature a first factor which is generally weakly correlated to the other factors. When one, for instance, looks at the first column in the correlation matrices in Tables 8, 9, 17, 18, 26 and 27, it appears that very few of the noninitial elements have an absolute value higher than .0.20 for periods 2019–2019 and 2020. Interestingly, we are meeting a different situation in 2021–2022: the first factor is highly anticorrelated to the second factor, see .N = 8 in Belgium, in France or in Germany. We emphasize another very noticeable pattern, which is that most of the time (at least when .N > 5) factors with different parities would have a negative correlation, and have a positive one if they share the same parity. As an example, in Table 28, the cases .N = 5 and .N = 8 stand for perfect examples of this pattern, which the case .N = 7 is also quite close to doing so. We may compare our results to the ones available in the literature. A two-factor model has been calibrated to German option prices on futures contracts in [24] (from the year 2005). In [13] the calibration of the same two-factor model is done on marginal volatilities of futures contracts and on spot prices in the French and UK markets, using data from 2013 to 2015. In both articles the authors fixed .α1 and the correlation between the two factors to 0. The results obtained in the present paper

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O. Féron and P. Gruet

are significantly different. In particular, our estimation results for a two-factor model in the German market, period 2018–2019, lead to a non-zero .σ1 (.2.11 yr.−1/2 ) and values of .α2 and .σ2 are incomparably higher than the ones obtained in [24], Table 2. This can be explained by the fact that we are considering the spot prices and short term (weekly) futures contracts in the calibration, which puts a high emphasis on the short-term behaviours of prices, and then noteworthy results in a high value of .α2 . The same observation can be done when comparing our results in the French market to the ones obtained in [13]: the estimated parameters in [14] on the French market and in the present paper in France in 2018–2019 with .N = 2 feature parameter values that are between 7 and 45 times higher in our setting. Whatever the number of factors considered, our estimation results lead to a non-zero .α1 . Indeeed, the smallest estimated values of .α1 for all N are .0.78 for the French market and .0.58 for the German one, leading to a half-time duration associated to the exponential decay of 0.89 and 1.2 yr, respectively, and the values of the associated exponential function for a maturity of 2 years are approximately 1/5 and 1/3, respectively. This would argue for considering a non-zero .α1 in the electricity price models, contrarily to the other implementations of HJM models for electricity prices proposed in [24] or [13]. It is also possible to draw a comparison between [10] and our results. Reference [10] estimates a 3-factor model on French, German and Dutch data from 2001 to 2005. Their estimation setting is somewhat similar to ours, as they use Kalman filtering as well. Yet they use only very short-term contracts, namely day-ahead ones. We compare their parameters, presented in their Table 3, to the ones we got for .N = 3 for France in 2018–2019 (Tables 11 and 14) and in Germany in 2018– 2019 (Tables 20 and 23). The values of the .α and .σ coefficients that we estimate are 1 to 5 times higher than the ones in [10], and approximately 15 times higher for .α3 in France and in Germany. Globally the comparison between our results and the ones of the literature show significant differences. This can be explained by the difference of contracts used for the calibration and also the difference in the period of historical data: for instance, the dataset of [10] is 13 years older than ours. At that time in Western Europe, prices were lower and less volatile, there were less actors in the market, and also the share of renewables was quite limited compared to the one of today.

5 Conclusion In this paper, we studied the number of factors to be considered within the specific class of factorial HJM models, widely used in practice and in the literature of pricing and risk management in the power markets. We focused the study on three power markets in Europe and applied the classical methods of likelihood maximisation to estimate the parameters, from which we derived the AIC and BIC for selecting the optimal number of factors. We also proposed to look at the estimated correlations between the Brownian motions to get some additional

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

21

guidance on the choice of the number and the form of the volatility functions. We then formulated some observations that are common to all the considered markets. The main result is that the optimal number of factors, according to the classical BIC, varies from 8 to 10 depending on the market and time periods. This number is a bit lower than the number of prices considered in the calibration (being 17 most of the time). The optimal number of factors can also be discussed, especially when observing the characteristics of the last factors (with the highest values of the mean-reverting parameter). Indeed the last factors present very high correlations, and seem to be only useful to compensate the difficulties of the model (with only exponential volatility functions) to fit the observed volatilities: the number of significant “random factors” is smaller than 6. Therefore, in the case of this specific class of factorial HJM models, an acceptable number of factors to take into account might be between 3 and 5. Contrary to previous (and old) results in the literature, the first factor should be considered with a non-constant volatility (.α1 /= 0) for all the markets and historical periods considered. Obviously the considered model class is restrictive, although widely used. In particular, the Gaussian restriction of the random processes may be inconsistent with the presence of spikes and jumps (at least in spot prices). This inconsistency affects the analysis on the number of factors. A natural improvement would be to consider more general Lévy processes in an HJM model, although this implies models more difficult to use in practice. A simpler, but still restrictive, way to improve this study is to consider other forms of volatility functions in order to better fit the observed data. Also, one future objective is to consider an HJM factor model jointly on several markets, and study, by this way, the optimal number of factors to represent prices on several markets simultaneously. We expect, because of interconnections and market price convergence, to get a smaller optimal number of needed factors than the sum of optimal numbers computed independently for each market.

Appendix: Estimation Results See Tables 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, and 28.

22

O. Féron and P. Gruet

Belgian Power Market Data

2021−2022

2020

−42000

−25000 2

4

6

8

10

AIC BIC

−40000

AIC BIC

−22000

AIC BIC

−47000 −50000

AIC−BIC criteria

2018−2019

2

4

Number of factors

6

8

10

2

4

6

8

10

Number of factors

Number of factors

Fig. 1 AIC-BIC on Belgian Power market data Table 2 Estimated αi ’s (yr−1 ) in the Belgian Power Market, period 2018–2019. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 1 4.51 3.02 1.57 1.27 1.28 1.26 1.23 1.22 1.20

f. 2 54.28 18.37 14.69 11.29 12.79 12.78 12.48 12.40 12.32

f. 3

f. 4

f. 5

f. 6

f. 7

f. 8

f. 9

f. 10

62.10 19.10 12.74 17.67 18.24 18.60 18.59 18.55

339.11 39.67 29.40 29.36 28.36 27.88 27.54

46.66 40.77 41.54 40.24 39.66 39.05

52.49 57.97 55.28 53.84 52.60

75.81 74.68 73.29 71.89

96.69 100.61 100.00

133.79 141.46

187.17

Table 3 Estimated αi ’s (yr−1 ) in the Belgian Power Market, period 2020. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 1 14.67 2.57 1.50 1.34 1.29 1.05 1.02 1.02 1.02

f. 2 59.24 32.80 10.91 8.66 10.16 6.13 6.12 5.97 5.94

f. 3

f. 4

f. 5

f. 6

f. 7

f. 8

f. 9

f. 10

37.03 39.14 28.61 23.99 18.07 17.18 17.43 17.48

46.41 36.51 31.26 23.43 22.21 22.40 22.50

49.73 43.52 32.52 30.51 31.44 31.67

56.66 44.57 42.45 43.76 44.26

60.55 57.67 59.62 60.48

73.07 80.67 82.16

104.06 113.99

151.92

f. 11

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

23

Table 4 Estimated αi ’s (yr−1 ) in the Belgian Power Market, period 2021–2022. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 1 1.29 1.29 1.45 2.02 2.03 1.95 1.93 1.95 1.96

f. 2 1341.72 199.71 5.46 2.51 2.49 2.77 2.74 2.67 2.65

f. 3

f. 4

f. 5

f. 6

f. 7

f. 8

f. 9

f. 10

255.35 39.63 12.53 19.59 18.77 14.46 14.46 14.29

48.73 45.48 25.59 23.62 17.82 17.88 17.63

51.13 33.47 30.05 21.72 21.80 21.44

60.34 61.04 49.14 48.92 49.22

80.62 67.46 67.33 68.18

88.33 94.60 98.14

125.60 140.05

192.50

Table 5 Estimated σi ’s (yr−1/2 ) in the Belgian Power Market, period 2018–2019. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 1 0.79 0.58 0.35 0.31 0.31 0.31 0.31 0.31 0.30

f. 2 26.42 3.67 36.44 134.13 96.54 90.34 77.09 78.00 77.38

f. 3

f. 4

f. 5

f. 6

f. 7

f. 8

f. 9

f. 10

25.55 40.34 137.07 164.47 181.12 203.69 221.11 230.08

142.75 351.98 272.23 302.07 347.14 367.41 384.20

351.06 416.62 426.44 488.81 522.00 552.41

213.97 376.39 525.24 557.68 589.25

169.46 408.22 433.74 461.81

153.75 277.46 333.33

103.38 227.06

98.45

Table 6 Estimated σi ’s (yr−1/2 ) in the Belgian Power Market, period 2020. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 3 f. 4 f. 5 f. 6 f. 7 f. 8 f. 9 f. 10 f. 11 f. 1 f. 2 25.68 4.25 0.66 1140.90 1144.28 1.71 437.73 441.13 0.46 1.35 322.63 519.69 201.31 0.43 0.41 3.71 266.39 519.80 492.81 239.69 1.12 274.64 555.64 530.27 386.45 141.21 0.36 1.19 262.71 543.95 517.58 456.24 435.41 222.91 0.35 1.07 277.14 537.09 537.39 580.13 608.71 507.20 204.66 0.35 0.35 1.04 281.26 545.96 552.00 617.51 677.42 560.11 319.17 111.20

24

O. Féron and P. Gruet

Table 7 Estimated σi ’s (yr−1/2 ) in the Belgian Power Market, period 2021–2022. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 1 f. 2 f. 3 f. 4 f. 5 1.13 303.33 1.13 66.53 129.09 1.34 1.07 474.57 479.38 27.26 26.93 2.15 1245.89 1249.72 31.92 31.55 428.66 842.79 456.99 10.02 9.66 530.91 966.55 458.40 9.96 9.69 458.49 921.53 475.18 12.47 12.20 455.89 929.35 486.50 13.74 13.47 458.22 928.25 481.48

f. 6

f. 7

f. 8

f. 9

f. 10

47.99 159.34 252.14 269.98 265.43

151.33 499.34 278.94 599.59 638.17 302.84 619.41 757.25 630.14 257.32

Table 8 Estimated correlation matrices in the French Power Market, period 2018–2019. N is the number of factors N 2

3

4

Estimated correlation matrix ] [ 1.00 −0.05 −0.05 1.00 ⎤ ⎡ 1.00 −0.11 0.04 ⎥ ⎢ ⎣−0.11 1.00 −0.34⎦ 0.04 −0.34 1.00 ⎤ ⎡ 1.00 0.11 −0.12 0.52 ⎥ ⎢ ⎢ 0.11 1.00 −1.00 0.68 ⎥ ⎥ ⎢ ⎣−0.12 −1.00 1.00 −0.72⎦ ⎡

5

6

0.52

1.00 ⎢ 0.06 ⎢ ⎢ ⎢−0.05 ⎢ ⎣−0.01 0.01 ⎡ 1.00 ⎢ ⎢−0.04 ⎢ ⎢ 0.05 ⎢ ⎢−0.06 ⎢ ⎢ ⎣ 0.05 −0.05

0.68 −0.72 1.00 0.06 1.00 −1.00 0.59 −0.58

−0.05 −1.00 1.00 −0.59 0.59

−0.01 0.59 −0.59 1.00 −1.00

−0.04 1.00 −1.00 0.94 −0.89 0.86

0.05 −1.00 1.00 −0.96 0.92 −0.89

−0.06 0.94 −0.96 1.00 −0.99 0.98

⎤ 0.01 ⎥ −0.58⎥ ⎥ 0.59 ⎥ ⎥ −1.00⎦ 1.00 0.05 −0.89 0.92 −0.99 1.00 −1.00

⎤ −0.05 ⎥ 0.86 ⎥ ⎥ −0.89⎥ ⎥ 0.98 ⎥ ⎥ ⎥ −1.00⎦ 1.00 (continued)

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

25

Table 8 (continued) N

7

8

Estimated correlation matrix ⎡ 1.00 0.02 −0.02 0.04 ⎢ 0.02 1.00 −1.00 0.95 ⎢ ⎢ ⎢−0.02 −1.00 1.00 −0.98 ⎢ ⎢ 0.04 0.95 −0.98 1.00 ⎢ ⎢ ⎢−0.06 −0.91 0.94 −0.99 ⎢ ⎣ 0.09 0.82 −0.86 0.93 −0.11 −0.73 0.76 −0.84 ⎡ 1.00 0.03 −0.04 0.07 ⎢ ⎢ 0.03 1.00 −0.99 0.95 ⎢ ⎢−0.04 −0.99 1.00 −0.98 ⎢ ⎢ ⎢ 0.07 0.95 −0.98 1.00 ⎢ ⎢−0.10 −0.84 0.89 −0.96 ⎢ ⎢ 0.12 0.65 −0.70 0.82 ⎢ ⎢ ⎣−0.11 −0.43 0.48 −0.62 0.09 0.29 −0.34 0.47

−0.06 −0.91 0.94 −0.99 1.00 −0.97 0.89

0.09 0.82 −0.86 0.93 −0.97 1.00 −0.98

−0.10 −0.84 0.89 −0.96 1.00 −0.94 0.79 −0.66

0.12 0.65 −0.70 0.82 −0.94 1.00 −0.95 0.86

⎤ −0.11 −0.73⎥ ⎥ ⎥ 0.76 ⎥ ⎥ −0.84⎥ ⎥ ⎥ 0.89 ⎥ ⎥ −0.98⎦ 1.00 −0.11 −0.43 0.48 −0.62 0.79 −0.95 1.00 −0.97

⎤ 0.09 ⎥ 0.29 ⎥ ⎥ −0.34⎥ ⎥ ⎥ 0.47 ⎥ ⎥ −0.66⎥ ⎥ 0.86 ⎥ ⎥ ⎥ −0.97⎦ 1.00

Table 9 Estimated correlation matrices in the French Power Market, period 2020. N is the number of factors N 2

3

4

5

Estimated correlation matrix [ ] 1.00 −0.36 −0.36 1.00 ⎤ ⎡ 1.00 −0.20 0.20 ⎥ ⎢ ⎣−0.20 1.00 −1.00⎦ 0.20 −1.00 1.00 ⎤ ⎡ 1.00 0.06 −0.22 0.22 ⎥ ⎢ ⎢ 0.06 1.00 −0.54 0.54 ⎥ ⎥ ⎢ ⎣−0.22 −0.54 1.00 −1.00⎦ 0.22 0.54 −1.00 1.00 ⎤ ⎡ 1.00 0.05 0.03 −0.04 0.04 ⎢ 0.05 1.00 −0.45 0.43 −0.39⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 0.03 −0.45 1.00 −1.00 0.99 ⎥ ⎥ ⎢ ⎣−0.04 0.43 −1.00 1.00 −1.00⎦ 0.04 −0.39 0.99 −1.00 1.00 (continued)

26

O. Féron and P. Gruet

Table 9 (continued) N

6

7

8

Estimated correlation matrix ⎡ 1.00 0.15 −0.14 0.14 ⎢ ⎢ 0.15 1.00 −0.69 0.63 ⎢ ⎢−0.14 −0.69 1.00 −0.99 ⎢ ⎢ 0.14 0.63 −0.99 1.00 ⎢ ⎢ ⎣−0.14 −0.50 0.95 −0.98 0.14 0.43 −0.90 0.95 ⎡ 1.00 0.26 −0.10 0.09 ⎢ 0.26 1.00 −0.61 0.58 ⎢ ⎢ ⎢−0.10 −0.61 1.00 −1.00 ⎢ ⎢ 0.09 0.58 −1.00 1.00 ⎢ ⎢ ⎢−0.06 −0.50 0.97 −0.98 ⎢ ⎣ 0.03 0.39 −0.89 0.92 −0.00 −0.31 0.81 −0.85 ⎡ 1.00 0.23 −0.06 0.06 ⎢ ⎢ 0.23 1.00 −0.58 0.54 ⎢ ⎢−0.06 −0.58 1.00 −1.00 ⎢ ⎢ ⎢ 0.06 0.54 −1.00 1.00 ⎢ ⎢−0.05 −0.44 0.95 −0.98 ⎢ ⎢ 0.07 0.32 −0.83 0.87 ⎢ ⎢ ⎣−0.13 −0.24 0.62 −0.67 0.16 0.21 −0.49 0.54

−0.14 −0.50 0.95 −0.98 1.00 −0.99

⎤ 0.14 ⎥ 0.43 ⎥ ⎥ −0.90⎥ ⎥ 0.95 ⎥ ⎥ ⎥ −0.99⎦ 1.00

−0.06 −0.50 0.97 −0.98 1.00 −0.98 0.93

0.03 0.39 −0.89 0.92 −0.98 1.00 −0.98

−0.05 −0.44 0.95 −0.98 1.00 −0.95 0.78 −0.65

0.07 0.32 −0.83 0.87 −0.95 1.00 −0.93 0.83

⎤ −0.00 −0.31⎥ ⎥ ⎥ 0.81 ⎥ ⎥ −0.85⎥ ⎥ ⎥ 0.93 ⎥ ⎥ −0.98⎦ 1.00 −0.13 −0.24 0.62 −0.67 0.78 −0.93 1.00 −0.98

⎤ 0.16 ⎥ 0.21 ⎥ ⎥ −0.49⎥ ⎥ ⎥ 0.54 ⎥ ⎥ −0.65⎥ ⎥ 0.83 ⎥ ⎥ ⎥ −0.98⎦ 1.00

Table 10 Estimated correlation matrices in the Belgian Power Market, period 2021–2022. N is the number of factors N 2

3

4

Estimated correlation matrix [ ] 1.00 −0.54 −0.54 1.00 ⎤ ⎡ 1.00 −0.91 0.17 ⎥ ⎢ ⎣−0.91 1.00 0.18⎦ 0.17 0.18 1.00 ⎤ ⎡ 1.00 −0.50 −0.10 0.10 ⎥ ⎢ ⎢−0.50 1.00 −0.04 0.03 ⎥ ⎥ ⎢ ⎣−0.10 −0.04 1.00 −1.00⎦ ⎡

5

0.10

1.00 ⎢−1.00 ⎢ ⎢ ⎢ 0.07 ⎢ ⎣−0.06 0.06

0.03 −1.00 1.00 −1.00 1.00 −0.09 0.07 −0.07

0.07 −0.09 1.00 −0.57 0.57

−0.06 0.07 −0.57 1.00 −1.00

⎤ 0.06 −0.07⎥ ⎥ ⎥ 0.57 ⎥ ⎥ −1.00⎦ 1.00 (continued)

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

27

Table 10 (continued) Estimated correlation matrix ⎡ 1.00 −1.00 0.09 −0.09 ⎢ ⎢−1.00 1.00 −0.10 0.10 ⎢ ⎢ 0.09 −0.10 1.00 −1.00 ⎢ ⎢−0.09 0.10 −1.00 1.00 ⎢ ⎢ ⎣ 0.08 −0.09 0.98 −0.99 −0.08 0.08 −0.86 0.89 ⎡ 1.00 −0.99 0.03 −0.02 ⎢−0.99 1.00 −0.05 0.04 ⎢ ⎢ ⎢ 0.03 −0.05 1.00 −1.00 ⎢ ⎢−0.02 0.04 −1.00 1.00 ⎢ ⎢ ⎢ 0.00 −0.02 0.99 −1.00 ⎢ ⎣ 0.29 −0.27 −0.33 0.34 −0.33 0.30 0.19 −0.20 ⎡ 1.00 −0.99 0.08 −0.07 ⎢ ⎢−0.99 1.00 −0.10 0.09 ⎢ ⎢ 0.08 −0.10 1.00 −1.00 ⎢ ⎢ ⎢−0.07 0.09 −1.00 1.00 ⎢ ⎢ 0.06 −0.08 0.99 −1.00 ⎢ ⎢−0.01 0.01 −0.66 0.69 ⎢ ⎢ ⎣ 0.01 −0.01 0.56 −0.59 −0.01 0.00 −0.45 0.48

N

6

7

8

0.08 −0.09 0.98 −0.99 1.00 −0.92

⎤ −0.08 ⎥ 0.08 ⎥ ⎥ −0.86⎥ ⎥ 0.89 ⎥ ⎥ ⎥ −0.92⎦ 1.00

0.00 −0.02 0.99 −1.00 1.00 −0.36 0.22

0.29 −0.27 −0.33 0.34 −0.36 1.00 −0.99

0.06 −0.08 0.99 −1.00 1.00 −0.72 0.62 −0.50

−0.01 0.01 −0.66 0.69 −0.72 1.00 −0.96 0.87

⎤ −0.33 0.30 ⎥ ⎥ ⎥ 0.19 ⎥ ⎥ −0.20⎥ ⎥ ⎥ 0.22 ⎥ ⎥ −0.99⎦ 1.00 0.01 −0.01 0.56 −0.59 0.62 −0.96 1.00 −0.97

⎤ −0.01 ⎥ 0.00 ⎥ ⎥ −0.45⎥ ⎥ ⎥ 0.48 ⎥ ⎥ −0.50⎥ ⎥ 0.87 ⎥ ⎥ ⎥ −0.97⎦ 1.00

French Power Market Data

2020

2021−2022

2

4

6

8

Number of factors

10

2

4

6

8

Number of factors

Fig. 2 AIC-BIC on French Power market data

10

AIC BIC

−38000 −41000

−21000

AIC BIC

−24000

−49000

AIC BIC

−53000

AIC−BIC criteria

2018−2019

2

4

6

8

Number of factors

10

28

O. Féron and P. Gruet

Table 11 Estimated αi ’s (yr−1 ) in the French Power Market, period 2018–2019. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 1 4.89 1.54 1.21 0.93 0.95 0.95 0.89 0.81 0.79

f. 2 66.64 14.90 13.42 6.55 7.02 7.63 8.66 6.25 5.50

f. 3

f. 4

f. 5

f. 6

f. 7

f. 8

f. 9

f. 10

325.81 37.09 24.77 22.67 22.34 16.96 16.57 16.30

45.10 29.20 30.34 30.02 22.85 22.32 22.05

51.01 42.30 42.95 32.64 31.93 31.52

56.39 59.23 43.20 43.09 42.94

78.50 59.46 59.47 59.04

75.25 80.82 78.54

104.72 107.35

141.37

f. 11

Table 12 Estimated αi ’s (yr−1 ) in the French Power Market, period 2020. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 1 49.20 2.48 1.58 1.51 1.48 1.40 1.39 1.39 1.39

f. 2 49.20 20.21 7.97 7.45 8.19 7.29 7.13 7.05 7.10

f. 3

f. 4

f. 5

f. 6

f. 7

f. 8

f. 9

f. 10

64.13 35.94 26.52 24.24 19.12 18.65 18.71 18.72

37.12 27.46 25.20 19.85 19.36 19.44 19.45

46.08 39.69 29.91 29.45 29.67 29.67

56.58 44.79 43.29 44.08 43.95

62.48 66.07 65.58 65.62

93.30 99.08 97.71

137.99 145.98

220.85

Table 13 Estimated αi ’s (yr−1 ) in the French Power Market, period 2021–2022. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 1 1.67 1.57 1.46 1.45 1.49 1.82 1.82 1.78 1.80

f. 2 213.22 6.59 8.82 7.75 6.67 3.61 3.60 3.71 3.67

f. 3

f. 4

f. 5

f. 6

f. 7

f. 8

f. 9

f. 10

258.71 14.68 18.56 20.40 12.68 12.55 12.54 12.48

880.27 44.61 31.67 14.17 13.97 13.92 13.85

54.40 42.23 32.88 34.12 34.46 34.31

56.55 39.23 40.98 41.04 40.78

51.77 60.08 62.99 63.56

85.11 92.71 93.58

126.82 138.59

187.60

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

29

Table 14 Estimated σi ’s (yr−1/2 ) in the French Power Market, period 2018–2019. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 1 f. 2 0.79 34.09 0.35 2.61 0.31 3.24 0.28 0.65 0.28 0.84 0.28 1.16 0.28 5.84 0.27 1.10 0.27 0.74

f. 3

f. 4

f. 5

f. 6

f. 7

f. 8

f. 9

f. 10

f. 11

146.77 186.62 258.26 145.64 145.99 142.58 133.80 131.49

188.25 286.08 318.19 315.28 301.52 308.97 312.88

32.34 315.76 372.54 432.08 446.15 453.17

144.69 345.13 430.73 478.85 518.54

151.49 322.60 166.91 409.52 331.69 133.38 517.10 427.81 261.57 105.37

Table 15 Estimated σi ’s (yr−1/2 ) in the French Power Market, period 2020. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 2 f. 1 11.93 16.83 0.70 5.57 0.53 1.07 0.51 1.18 0.49 1.67 0.48 1.69 0.47 1.62 0.47 1.54 0.47 1.58

f. 3

f. 4

f. 5

f. 6

f. 7

f. 8

f. 9

f. 10

25.67 12332.60 12310.80 12298.70 12306.80 12306.40 12308.40 12308.20

12336.00 12375.70 12423.60 12453.00 12448.40 12451.70 12451.20

69.03 237.63 244.06 254.68 260.13 261.28

115.57 186.63 94.53 191.19 137.13 64.48 215.85 164.48 116.92 52.29 216.15 170.24 120.12 68.20 26.74

Table 16 Estimated σi ’s (yr−1/2 ) in the French Power Market, period 2021–2022. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 3 f. 4 f. 5 f. 6 f. 1 f. 2 1.21 157.05 1.37 181.54 1.26 6.37 7.53 288.99 1.16 2.60 6.60 274.45 272.93 1.16 1.20 2.01 57.67 308.13 433.29 184.59 2.53 300.46 310.03 931.82 1247.70 2.56 2.54 299.04 307.13 944.57 1206.82 2.56 2.27 301.63 309.01 973.81 1190.98 2.27 2.39 2.39 300.90 308.20 968.46 1171.96

f. 7

f. 8

f. 9

f. 10

329.12 386.80 122.14 402.94 309.19 131.82 429.68 407.30 328.97 155.04

30

O. Féron and P. Gruet

Table 17 Estimated correlation matrices in the French Power Market, period 2018–2019. N is the number of factors N 2

3

4

Estimated correlation matrix ] [ 1.00 −0.03 −0.03 1.00 ⎤ ⎡ 1.00 0.07 0.13 ⎥ ⎢ ⎣0.07 1.00 −0.90⎦ 0.13 −0.90 1.00 ⎤ ⎡ 1.00 0.11 −0.12 0.12 ⎥ ⎢ ⎢ 0.11 1.00 −0.79 0.78 ⎥ ⎥ ⎢ ⎣−0.12 −0.79 1.00 −1.00⎦ ⎡

5

6

7

8

0.12

1.00 ⎢−0.04 ⎢ ⎢ ⎢ 0.09 ⎢ ⎣−0.09 0.08 ⎡ 1.00 ⎢ ⎢−0.05 ⎢ ⎢ 0.02 ⎢ ⎢−0.01 ⎢ ⎢ ⎣−0.02 0.04 ⎡ 1.00 ⎢−0.03 ⎢ ⎢ ⎢−0.02 ⎢ ⎢ 0.05 ⎢ ⎢ ⎢−0.09 ⎢ ⎣ 0.11 −0.12 ⎡ 1.00 ⎢ ⎢ 0.10 ⎢ ⎢−0.09 ⎢ ⎢ ⎢ 0.08 ⎢ ⎢−0.05 ⎢ ⎢ 0.02 ⎢ ⎢ ⎣ 0.02 −0.04

0.78 −1.00 1.00

⎤ 0.08 ⎥ −0.11⎥ ⎥ 0.93 ⎥ ⎥ −0.93⎦

−0.04 1.00 −0.19 0.19 −0.11

0.09 −0.19 1.00 −1.00 0.93

−0.09 0.19 −1.00 1.00 −0.93

−0.05 1.00 −0.57 0.55 −0.51 0.48

0.02 −0.57 1.00 −0.99 0.95 −0.91

−0.01 0.55 −0.99 1.00 −0.98 0.95

−0.02 −0.51 0.95 −0.98 1.00 −0.99

⎤ 0.04 ⎥ 0.48 ⎥ ⎥ −0.91⎥ ⎥ 0.95 ⎥ ⎥ ⎥ −0.99⎦ 1.00

−0.03 1.00 −0.70 0.67 −0.61 0.53 −0.44

−0.02 −0.70 1.00 −0.99 0.95 −0.87 0.76

0.05 0.67 −0.99 1.00 −0.98 0.91 −0.80

−0.09 −0.61 0.95 −0.98 1.00 −0.97 0.88

0.11 0.53 −0.87 0.91 −0.97 1.00 −0.97

0.10 1.00 −0.86 0.79 −0.65 0.52 −0.31 0.19

−0.09 −0.86 1.00 −0.99 0.94 −0.86 0.64 −0.49

0.08 0.79 −0.99 1.00 −0.97 0.91 −0.71 0.56

−0.05 −0.65 0.94 −0.97 1.00 −0.98 0.83 −0.70

0.02 0.52 −0.86 0.91 −0.98 1.00 −0.92 0.81

1.00

⎤ −0.12 −0.44⎥ ⎥ ⎥ 0.76 ⎥ ⎥ −0.80⎥ ⎥ ⎥ 0.88 ⎥ ⎥ −0.97⎦ 1.00 0.02 −0.31 0.64 −0.71 0.83 −0.92 1.00 −0.97

⎤ −0.04 ⎥ 0.19 ⎥ ⎥ −0.49⎥ ⎥ ⎥ 0.56 ⎥ ⎥ −0.70⎥ ⎥ 0.81 ⎥ ⎥ ⎥ −0.97⎦ 1.00

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

31

Table 18 Estimated correlation matrices in the French Power Market, period 2020. N is the number of factors N 2

3

4

Estimated correlation matrix ] [ 1.00 0.53 0.53 1.00 ⎤ ⎡ 1.00 −0.12 −0.10 ⎥ ⎢ ⎣−0.12 1.00 −0.40⎦ −0.10 −0.40 1.00 ⎤ ⎡ 1.00 0.12 −0.52 0.52 ⎥ ⎢ ⎢ 0.12 1.00 −0.32 0.32 ⎥ ⎥ ⎢ ⎣−0.52 −0.32 1.00 −1.00⎦ ⎡

5

6

7

8

0.52

1.00 ⎢ 0.13 ⎢ ⎢ ⎢−0.52 ⎢ ⎣ 0.52 −0.53 ⎡ 1.00 ⎢ ⎢ 0.23 ⎢ ⎢−0.52 ⎢ ⎢ 0.52 ⎢ ⎢ ⎣−0.50 0.47 ⎡ 1.00 ⎢ 0.36 ⎢ ⎢ ⎢−0.52 ⎢ ⎢ 0.52 ⎢ ⎢ ⎢−0.46 ⎢ ⎣ 0.34 −0.27 ⎡ 1.00 ⎢ ⎢ 0.38 ⎢ ⎢−0.52 ⎢ ⎢ ⎢ 0.52 ⎢ ⎢−0.46 ⎢ ⎢ 0.37 ⎢ ⎢ ⎣−0.31 0.29

0.32 −1.00 1.00

⎤ −0.53 ⎥ −0.28⎥ ⎥ 0.98 ⎥ ⎥ −0.98⎦

0.13 1.00 −0.33 0.33 −0.28

−0.52 −0.33 1.00 −1.00 0.98

0.52 0.33 −1.00 1.00 −0.98

0.23 1.00 −0.38 0.38 −0.23 0.17

−0.52 −0.38 1.00 −1.00 0.95 −0.89

0.52 0.38 −1.00 1.00 −0.95 0.89

−0.50 −0.23 0.95 −0.95 1.00 −0.99

⎤ 0.47 ⎥ 0.17 ⎥ ⎥ −0.89⎥ ⎥ 0.89 ⎥ ⎥ ⎥ −0.99⎦ 1.00

0.36 1.00 −0.43 0.43 −0.29 0.11 −0.01

−0.52 −0.43 1.00 −1.00 0.96 −0.81 0.69

0.52 0.43 −1.00 1.00 −0.97 0.81 −0.69

−0.46 −0.29 0.96 −0.97 1.00 −0.93 0.84

0.34 0.11 −0.81 0.81 −0.93 1.00 −0.98

⎤ −0.27 −0.01⎥ ⎥ ⎥ 0.69 ⎥ ⎥ −0.69⎥ ⎥ ⎥ 0.84 ⎥ ⎥ −0.98⎦ 1.00

0.38 1.00 −0.44 0.44 −0.31 0.15 −0.02 −0.04

−0.52 −0.44 1.00 −1.00 0.96 −0.84 0.65 −0.50

0.52 0.44 −1.00 1.00 −0.96 0.84 −0.65 0.50

−0.46 −0.31 0.96 −0.96 1.00 −0.95 0.81 −0.66

0.37 0.15 −0.84 0.84 −0.95 1.00 −0.94 0.81

−0.31 −0.02 0.65 −0.65 0.81 −0.94 1.00 −0.96

1.00

⎤ 0.29 ⎥ −0.04⎥ ⎥ −0.50⎥ ⎥ ⎥ 0.50 ⎥ ⎥ −0.66⎥ ⎥ 0.81 ⎥ ⎥ ⎥ −0.96⎦ 1.00

32

O. Féron and P. Gruet

Table 19 Estimated correlation matrices in the French Power Market, period 2021–2022. N is the number of factors N 2

3

4

Estimated correlation matrix ] [ 1.00 −0.35 −0.35 1.00 ⎤ ⎡ 1.00 −0.36 −0.26 ⎥ ⎢ ⎣−0.36 1.00 −0.41⎦ −0.26 −0.41 1.00 ⎤ ⎡ 1.00 −0.08 0.01 −0.37 ⎥ ⎢ ⎢−0.08 1.00 −0.97 −0.22⎥ ⎥ ⎢ ⎣ 0.01 −0.97 1.00 0.06 ⎦ ⎡

5

6

7

8

−0.37 −0.22 0.06

1.00

−0.11 1.00 −0.76 0.29 −0.28

−0.10 −0.76 1.00 −0.68 0.68

0.12 0.29 −0.68 1.00 −1.00

−0.23 1.00 −0.47 0.35 −0.30 0.26

0.08 −0.47 1.00 −0.96 0.90 −0.84

−0.11 0.35 −0.96 1.00 −0.99 0.96

0.12 −0.30 0.90 −0.99 1.00 −0.99

⎤ −0.14 ⎥ 0.26 ⎥ ⎥ −0.84⎥ ⎥ 0.96 ⎥ ⎥ ⎥ −0.99⎦ 1.00

−0.87 1.00 −0.36 0.36 −0.14 0.14 −0.11

0.25 −0.36 1.00 −1.00 0.64 −0.61 0.55

−0.25 0.36 −1.00 1.00 −0.65 0.62 −0.56

0.06 −0.14 0.64 −0.65 1.00 −1.00 0.99

−0.05 0.14 −0.61 0.62 −1.00 1.00 −0.99

⎤ 0.03 −0.11⎥ ⎥ ⎥ 0.55 ⎥ ⎥ −0.56⎥ ⎥ ⎥ 0.99 ⎥ ⎥ −0.99⎦ 1.00

−0.87 1.00 −0.37 0.37 −0.13 0.12 −0.08 0.04

0.25 −0.37 1.00 −1.00 0.64 −0.62 0.54 −0.49

−0.25 0.37 −1.00 1.00 −0.65 0.63 −0.55 0.50

0.05 −0.13 0.64 −0.65 1.00 −1.00 0.98 −0.93

−0.04 0.12 −0.62 0.63 −1.00 1.00 −0.99 0.94

−0.00 −0.08 0.54 −0.55 0.98 −0.99 1.00 −0.98

1.00 ⎢−0.11 ⎢ ⎢ ⎢−0.10 ⎢ ⎣ 0.12 −0.13 ⎡ 1.00 ⎢ ⎢−0.23 ⎢ ⎢ 0.08 ⎢ ⎢−0.11 ⎢ ⎢ ⎣ 0.12 −0.14 ⎡ 1.00 ⎢−0.87 ⎢ ⎢ ⎢ 0.25 ⎢ ⎢−0.25 ⎢ ⎢ ⎢ 0.06 ⎢ ⎣−0.05 0.03 ⎡ 1.00 ⎢ ⎢−0.87 ⎢ ⎢ 0.25 ⎢ ⎢ ⎢−0.25 ⎢ ⎢ 0.05 ⎢ ⎢−0.04 ⎢ ⎢ ⎣−0.00 0.04

⎤ −0.13 ⎥ −0.28⎥ ⎥ 0.68 ⎥ ⎥ −1.00⎦ 1.00

⎤ 0.04 ⎥ 0.04 ⎥ ⎥ −0.49⎥ ⎥ ⎥ 0.50 ⎥ ⎥ −0.93⎥ ⎥ 0.94 ⎥ ⎥ ⎥ −0.98⎦ 1.00

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

33

German Power Market Data

2020

6

8

10

2

4

6

8

AIC BIC

−40000

AIC BIC

10

−43000

4

−26000

2

2021−2022

−23000

−50000

AIC BIC

−54000

AIC−BIC criteria

2018−2019

2

Number of factors

Number of factors

4

6

8

10

Number of factors

Fig. 3 AIC-BIC on German Power market data

Table 20 Estimated αi ’s (yr−1 ) in the German Power Market, period 2018-2019. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 3 f. 1 f. 2 2.11 603.81 1.04 15.27 360.10 0.73 15.02 16.69 0.68 11.53 12.50 0.69 13.20 16.17 0.68 13.31 17.28 0.62 10.92 14.05 0.61 10.95 14.13 0.60 10.96 14.17

f. 4

f. 5

f. 6

f. 7

f. 8

f. 9

f. 10

76.17 41.12 29.01 28.10 21.87 22.09 22.25

50.52 39.39 40.05 31.42 31.93 32.04

52.84 55.11 42.27 43.55 44.06

71.43 54.00 72.05 57.69 77.13 99.09 58.59 78.12 105.83 138.69

f. 11

Table 21 Estimated αi ’s (yr−1 ) in the German Power Market, period 2020. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 1 15.85 1.93 1.44 1.23 1.17 1.16 1.16 1.16 1.16

f. 2 60.49 23.07 19.22 13.70 13.66 13.46 13.54 13.57 13.64

f. 3

f. 4

f. 5

f. 6

f. 7

f. 8

f. 9

f. 10

66.89 22.00 15.12 17.08 17.22 17.62 17.82 18.11

60.51 39.59 28.01 27.02 27.24 27.41 27.61

48.55 40.45 39.61 39.93 40.01 40.12

54.55 55.43 55.89 55.42 55.58

72.37 77.81 77.53 77.31

102.05 107.74 108.73

141.05 151.41

199.78

34

O. Féron and P. Gruet

Table 22 Estimated αi ’s (yr−1 ) in the German Power Market, period 2021–2022. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 3 f. 4 f. 5 f. 6 f. 7 f. 8 f. 9 f. 10 f. 11 f. 1 f. 2 1.44 228.78 8.47 1320.73 1.45 6.39 37.89 44.91 1.47 4.76 26.89 34.76 45.46 1.58 3.58 20.82 26.38 35.30 58.57 1.85 2.80 14.97 18.70 23.47 45.37 58.29 2.02 2.81 14.56 18.07 22.33 47.91 67.88 91.73 2.02 2.78 14.71 18.34 22.94 47.47 66.16 93.16 125.70 2.02 2.88 14.73 18.54 23.64 43.43 59.68 84.84 118.38 152.36 2.01

Table 23 Estimated σi ’s (yr−1/2 ) in the German Power Market, period 2018-2019. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 1 0.40 0.30 0.28 0.27 0.28 0.28 0.27 0.27 0.27

f. 2 178.91 2.52 247.00 244.95 250.21 226.68 232.59 229.82 230.59

f. 3

f. 4

f. 5

f. 6

f. 7

f. 8

f. 9

f. 10

f. 11

152.37 250.92 247.36 301.77 332.67 360.10 357.98 359.72

34.46 240.20 272.51 317.54 343.53 349.22 364.39

241.02 375.36 443.91 543.95 554.86 572.69

156.81 408.57 663.81 680.28 715.36

183.60 448.70 118.79 579.83 372.58 146.27 661.16 469.21 312.44 129.29

Table 24 Estimated σi ’s (yr−1/2 ) in the German Power Market, period 2020. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 1 5.62 0.54 0.45 0.41 0.40 0.40 0.40 0.40 0.39

f. 2 25.56 13.95 434.08 416.00 330.31 312.99 300.52 293.64 285.24

f. 3

f. 4

f. 5

f. 6

f. 7

f. 8

f. 9

f. 10

34.08 451.23 424.11 436.66 456.72 471.71 480.42 494.33

30.42 298.32 290.48 334.36 390.04 432.36 481.61

294.23 354.88 407.32 486.86 566.84 621.28

175.77 397.63 510.24 603.28 668.06

192.14 418.03 519.86 590.19

178.53 398.62 496.65

165.95 371.95

153.13

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

35

Table 25 Estimated σi ’s (yr−1/2 ) in the German Power Market, period 2021–2022. N is the number of factors N 2 3 4 5 6 7 8 9 10

f. 1 f. 2 f. 3 f. 4 f. 5 f. 6 f. 7 f. 8 f. 9 f. 10 f. 11 1.19 162.96 1.25 1.42 297.61 1.29 1.19 608.02 612.04 1.48 1.21 526.35 959.78 436.87 2.83 2.48 552.65 935.47 445.87 66.46 11.44 11.15 485.63 941.58 483.64 214.86 191.22 10.92 10.65 479.07 937.50 475.67 218.83 362.19 168.54 12.18 11.91 486.49 935.60 472.17 264.24 497.38 438.88 180.52 9.47 9.21 484.36 930.73 486.20 260.72 500.67 556.86 489.00 219.77

Table 26 Estimated correlation matrices in the German Power Market, period 2018–2019. N is the number of factors N 2

3

4

Estimated correlation matrix ] [ 1.00 0.16 0.16 1.00 ⎤ ⎡ 1.00 0.03 0.22 ⎥ ⎢ ⎣0.03 1.00 −0.91⎦ 0.22 −0.91 1.00 ⎤ ⎡ 1.00 0.05 −0.05 0.09 ⎥ ⎢ ⎢ 0.05 1.00 −1.00 0.26 ⎥ ⎥ ⎢ ⎣−0.05 −1.00 1.00 −0.27⎦ ⎡

5

6

0.09

1.00 ⎢−0.04 ⎢ ⎢ ⎢ 0.04 ⎢ ⎣−0.04 0.04 ⎡ 1.00 ⎢ ⎢−0.06 ⎢ ⎢ 0.06 ⎢ ⎢−0.03 ⎢ ⎢ ⎣ 0.01 −0.00

0.26 −0.27 1.00 −0.04 1.00 −1.00 0.46 −0.46

0.04 −1.00 1.00 −0.47 0.46

−0.04 0.46 −0.47 1.00 −1.00

−0.06 1.00 −1.00 0.95 −0.91 0.87

0.06 −1.00 1.00 −0.96 0.92 −0.88

−0.03 0.95 −0.96 1.00 −0.99 0.97

⎤ 0.04 ⎥ −0.46⎥ ⎥ 0.46 ⎥ ⎥ −1.00⎦ 1.00 0.01 −0.91 0.92 −0.99 1.00 −0.99

⎤ −0.00 ⎥ 0.87 ⎥ ⎥ −0.88⎥ ⎥ 0.97 ⎥ ⎥ ⎥ −0.99⎦ 1.00 (continued)

36

O. Féron and P. Gruet

Table 26 (continued) N

7

8

Estimated correlation matrix ⎡ 1.00 −0.04 0.03 0.01 ⎢−0.04 1.00 −1.00 0.98 ⎢ ⎢ ⎢ 0.03 −1.00 1.00 −0.98 ⎢ ⎢ 0.01 0.98 −0.98 1.00 ⎢ ⎢ ⎢−0.05 −0.94 0.95 −0.99 ⎢ ⎣ 0.08 0.84 −0.85 0.91 −0.09 −0.74 0.75 −0.81 ⎡ 1.00 0.01 −0.01 0.01 ⎢ ⎢ 0.01 1.00 −1.00 0.95 ⎢ ⎢−0.01 −1.00 1.00 −0.96 ⎢ ⎢ ⎢ 0.01 0.95 −0.96 1.00 ⎢ ⎢ 0.00 −0.84 0.87 −0.97 ⎢ ⎢−0.02 0.67 −0.71 0.85 ⎢ ⎢ ⎣ 0.02 −0.46 0.50 −0.67 −0.00 0.17 −0.20 0.36

−0.05 −0.94 0.95 −0.99 1.00 −0.97 0.89

0.08 0.84 −0.85 0.91 −0.97 1.00 −0.98

0.00 −0.84 0.87 −0.97 1.00 −0.96 0.82 −0.55

−0.02 0.67 −0.71 0.85 −0.96 1.00 −0.95 0.75

⎤ −0.09 −0.74⎥ ⎥ ⎥ 0.75 ⎥ ⎥ −0.81⎥ ⎥ ⎥ 0.89 ⎥ ⎥ −0.98⎦ 1.00 0.02 −0.46 0.50 −0.67 0.82 −0.95 1.00 −0.92

⎤ −0.00 ⎥ 0.17 ⎥ ⎥ −0.20⎥ ⎥ ⎥ 0.36 ⎥ ⎥ −0.55⎥ ⎥ 0.75 ⎥ ⎥ ⎥ −0.92⎦ 1.00

Table 27 Estimated correlation matrices in the German Power Market, period 2020. N is the number of factors N 2

3

4

Estimated correlation matrix ] [ 1.00 −0.41 −0.41 1.00 ⎤ ⎡ 1.00 −0.01 −0.12 ⎥ ⎢ ⎣−0.01 1.00 −0.70⎦ −0.12 −0.70 1.00 ⎤ ⎡ 1.00 0.17 −0.17 0.22 ⎥ ⎢ ⎢ 0.17 1.00 −1.00 0.81 ⎥ ⎥ ⎢ ⎣−0.17 −1.00 1.00 −0.82⎦ ⎡

5

6

0.22

1.00 ⎢ 0.04 ⎢ ⎢ ⎢−0.04 ⎢ ⎣−0.14 0.14 ⎡ 1.00 ⎢ ⎢ 0.06 ⎢ ⎢−0.06 ⎢ ⎢ 0.07 ⎢ ⎢ ⎣−0.10 0.12

0.81 −0.82 1.00 0.04 1.00 −1.00 0.73 −0.72

−0.04 −1.00 1.00 −0.73 0.72

−0.14 0.73 −0.73 1.00 −1.00

0.06 1.00 −1.00 0.96 −0.90 0.85

−0.06 −1.00 1.00 −0.97 0.91 −0.87

0.07 0.96 −0.97 1.00 −0.98 0.95

⎤ 0.14 −0.72⎥ ⎥ ⎥ 0.72 ⎥ ⎥ −1.00⎦ 1.00 −0.10 −0.90 0.91 −0.98 1.00 −0.99

⎤ 0.12 ⎥ 0.85 ⎥ ⎥ −0.87⎥ ⎥ 0.95 ⎥ ⎥ ⎥ −0.99⎦ 1.00 (continued)

Estimation of the Number of Factors in a Multi-Factorial HJM Model in Power Markets

37

Table 27 (continued) N

7

8

Estimated correlation matrix ⎡ 1.00 0.07 −0.07 0.07 ⎢ 0.07 1.00 −1.00 0.96 ⎢ ⎢ ⎢−0.07 −1.00 1.00 −0.97 ⎢ ⎢ 0.07 0.96 −0.97 1.00 ⎢ ⎢ ⎢−0.08 −0.88 0.90 −0.97 ⎢ ⎣ 0.06 0.72 −0.74 0.85 −0.04 −0.57 0.59 −0.70 ⎡ 1.00 0.11 −0.11 0.10 ⎢ ⎢ 0.11 1.00 −1.00 0.97 ⎢ ⎢−0.11 −1.00 1.00 −0.98 ⎢ ⎢ ⎢ 0.10 0.97 −0.98 1.00 ⎢ ⎢−0.09 −0.90 0.92 −0.97 ⎢ ⎢ 0.05 0.77 −0.79 0.86 ⎢ ⎢ ⎣−0.01 −0.61 0.62 −0.70 0.01 0.53 −0.54 0.60

−0.08 −0.88 0.90 −0.97 1.00 −0.94 0.84

0.06 0.72 −0.74 0.85 −0.94 1.00 −0.97

−0.09 −0.90 0.92 −0.97 1.00 −0.95 0.82 −0.72

0.05 0.77 −0.79 0.86 −0.95 1.00 −0.95 0.88

⎤ −0.04 −0.57⎥ ⎥ ⎥ 0.59 ⎥ ⎥ −0.70⎥ ⎥ ⎥ 0.84 ⎥ ⎥ −0.97⎦ 1.00 −0.01 −0.61 0.62 −0.70 0.82 −0.95 1.00 −0.98

⎤ 0.01 ⎥ 0.53 ⎥ ⎥ −0.54⎥ ⎥ ⎥ 0.60 ⎥ ⎥ −0.72⎥ ⎥ 0.88 ⎥ ⎥ ⎥ −0.98⎦ 1.00

Table 28 Estimated correlation matrices in the German Power Market, period 2021–2022. N is the number of factors N 2

3

4

5

Estimated correlation matrix [ ] 1.00 −0.29 −0.29 1.00 ⎤ ⎡ 1.00 −0.31 −0.55 ⎥ ⎢ ⎣−0.31 1.00 −0.21⎦ −0.55 −0.21 1.00 ⎤ ⎡ 1.00 −0.36 −0.05 0.04 ⎥ ⎢ ⎢−0.36 1.00 −0.07 0.07 ⎥ ⎥ ⎢ ⎣−0.05 −0.07 1.00 −1.00⎦ 0.04 0.07 −1.00 1.00 ⎤ ⎡ 1.00 −0.56 −0.03 0.03 −0.03 ⎢−0.56 1.00 −0.05 0.05 −0.05⎥ ⎥ ⎢ ⎥ ⎢ ⎢−0.03 −0.05 1.00 −1.00 0.99 ⎥ ⎥ ⎢ ⎣ 0.03 0.05 −1.00 1.00 −1.00⎦ −0.03 −0.05 0.99 −1.00 1.00 (continued)

38

O. Féron and P. Gruet

Table 28 (continued) N

6

7

8

Estimated correlation matrix ⎡ 1.00 −0.90 0.03 −0.02 ⎢ ⎢−0.90 1.00 −0.09 0.09 ⎢ ⎢ 0.03 −0.09 1.00 −1.00 ⎢ ⎢−0.02 0.09 −1.00 1.00 ⎢ ⎢ ⎣ 0.02 −0.08 0.98 −0.99 −0.03 0.08 −0.88 0.91 ⎡ 1.00 −0.99 0.06 −0.05 ⎢−0.99 1.00 −0.07 0.07 ⎢ ⎢ ⎢ 0.06 −0.07 1.00 −1.00 ⎢ ⎢−0.05 0.07 −1.00 1.00 ⎢ ⎢ ⎢ 0.04 −0.06 0.99 −1.00 ⎢ ⎣−0.01 0.01 −0.80 0.82 0.00 −0.01 0.77 −0.80 ⎡ 1.00 −0.99 0.07 −0.06 ⎢ ⎢−0.99 1.00 −0.09 0.08 ⎢ ⎢ 0.07 −0.09 1.00 −1.00 ⎢ ⎢ ⎢−0.06 0.08 −1.00 1.00 ⎢ ⎢ 0.06 −0.07 0.99 −1.00 ⎢ ⎢−0.02 0.03 −0.75 0.77 ⎢ ⎢ ⎣ 0.02 −0.02 0.72 −0.74 −0.02 0.02 −0.68 0.70

0.02 −0.08 0.98 −0.99 1.00 −0.95

⎤ −0.03 ⎥ 0.08 ⎥ ⎥ −0.88⎥ ⎥ 0.91 ⎥ ⎥ ⎥ −0.95⎦ 1.00

0.04 −0.06 0.99 −1.00 1.00 −0.84 0.83

−0.01 0.01 −0.80 0.82 −0.84 1.00 −1.00

0.06 −0.07 0.99 −1.00 1.00 −0.79 0.76 −0.72

−0.02 0.03 −0.75 0.77 −0.79 1.00 −0.99 0.94

⎤ 0.00 −0.01⎥ ⎥ ⎥ 0.77 ⎥ ⎥ −0.80⎥ ⎥ ⎥ 0.83 ⎥ ⎥ −1.00⎦ 1.00 0.02 −0.02 0.72 −0.74 0.76 −0.99 1.00 −0.98

⎤ −0.02 ⎥ 0.02 ⎥ ⎥ −0.68⎥ ⎥ ⎥ 0.70 ⎥ ⎥ −0.72⎥ ⎥ 0.94 ⎥ ⎥ ⎥ −0.98⎦ 1.00

References 1. Benth, F.E., Koekebakker, S.: Stochastic modeling of financial electricity contracts. Energy Econ. 30(3), 1116–1157 (2008) 2. Benth, F.E., Piccirilli, M., Vargiolu, T.: Mean-reverting additive energy forward curves in a Heath–Jarrow–Morton framework. Math. Financ. Econ. 13(4), 543–577 (2019) 3. Bhar, R., Chiarella, C.: The estimation of the Heath-Jarrow-Morton model by use of Kalman filtering techniques. In: Computational Approaches to Economic Problems, pp. 113–126 (1997) 4. Bhar, R., Chiarella, C., To, T.D.: A maximum likelihood approach to estimation of HeathJarrow-Morton models. Research Paper Series 80, Quantitative Finance Research Centre, University of Technology, Sydney (2002) 5. Bjerksund, P., Rasmussen, H., Stensland, G.: Valuation and risk management in the Norwegian electricity market. In: Energy, Natural Ressources And Environmental Economics, pp. 167– 185. Springer, Berlin (2000) 6. Burnham, K.P., Anderson, D.R.: Multimodel inference. Soc. Methods Res. 33(2), 261–304 (2004) 7. Clewlow, L., Strickland, C.: Energy Derivatives: Pricing and Risk Management. Lacima Group, Sydney (2000) 8. Dahlgren, M.: A continuous time model to price commodity-based swing options. Rev. Deriv. Res. 8(1), 27–47 (2005)

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9. Deschatre, T., Feron, O., Gruet, P.: A survey of electricity spot and futures price models for risk management applications. Energy Econ. 102, 105504 (2021) 10. Diko, P., Lawford, S., Limpens, V.: Risk premia in electricity forward prices. Stud. Nonlinear Dynam. Econom. 10(3), 1–24 (2006) 11. Edoli, E., Tasinato, D., Vargiolu, T.: Calibration of a multifactor model for the forward markets of several commodities. Optimization 62(11), 1553–1574 (2013) 12. Fabbiani, E., Marziali, A., De Nicolao, G.: Fast calibration of two-factor models for energy option pricing. Appl. Stoch. Models Bus. Ind. 37(3), 661–671 (2021) 13. Feron, O., Daboussi, E.: Calibration of electricity price models. Fields Inst. Commun. 74, 183– 210 (2015) 14. Feron, O., Gruet, P., Hoffmann, M.: Efficient volatility estimation in a two-factor model. Scand. J. Stat. 47(3), 862–898 (2020) 15. Gibson, R., Schwartz, E.S.: Stochastic convenience yield and the pricing of oil contingent claims. J. Financ. 45(3), 959–976 (1990) 16. Gonzalez, J., Moriarty, J., Palczewski, J.: Bayesian calibration and number of jump components in electricity spot price models. Energy Econ. 65, 375–388 (2017) 17. Gruet, P.: Some problems of statistics and optimal control for stochastic processes in the field of electricity markets prices modeling. Ph.D. Thesis, Université Paris Diderot (2015) 18. Heath, D., Jarrow, R., Morton, A.: Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica. 60(1), 77–105 (1992) 19. Heitmann, F., Trautmann, S.: Gaussian multi-factor interest rate models: theory, estimation and implications for option pricing. Lehrstuhl f. Finanzwirtschaft, University of Mainz (1995) 20. Jeffrey, A., Kristensen, D., Linton, O., Nguyen, T., Phillips, P.C.B.: Nonparametric estimation of a multifactor Heath-Jarrow-Morton model: an integrated approach. J. Financ. Econom. 2(2), 251–289 (2004) 21. Kalman, R.E.: A new approach to linear filtering and prediction problems. J. Basic Eng. 82(1), 35–45 (1960) 22. Karesen, K.F., Husby, E.: A joint state-space model for electricity spot and futures prices. Norsk Regnesentral (2000) 23. Kemna, A.G.Z., Vorst, A.C.F.: A pricing method for options based on average asset values. J. Bank. Financ. 14(1), 113–129 (1990) 24. Kiesel, R., Schindlmayr, G., Börger, R.H.: A two-factor model for the electricity forward market. Quant. Financ. 9(3), 279–287 (2009) 25. Koekebakker, S., Ollmar, F.: Forward curve dynamics in the Nordic electricity market. Manag. Financ. 31(6), 73–94 (2005) 26. Latini, L., Piccirilli, M., Vargiolu, T.: Mean-reverting no-arbitrage additive models for forward curves in energy markets. Energy Econ. 79, 157–170 (2019) 27. Lucia, J.J., Schwartz, E.S.: Electricity prices and power derivatives: evidence from the Nordic power exchange. Rev. Deriv. Res. 5(1), 5–50 (2002) 28. Manoliu, M., Tompaidis, S.: Energy futures prices: term structure models with Kalman filter estimation. Appl. Math. Financ. 9(1), 21–43 (2002) 29. Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J. 7(4), 308–313 (1965) 30. Schwartz, E.S.: The stochastic behavior of commodity prices: implications for valuation and hedging. J. Financ. 52(3), 923–973 (1997)

Hawkes Processes in Energy Markets: Modelling, Estimation and Derivatives Pricing Riccardo Brignone, Luca Gonzato, and Carlo Sgarra

Abstract The purpose of the present contribution is to illustrate the extensive use of Hawkes processes in modeling price dynamics in energy markets and to show how they can be applied for derivatives pricing. After a review of the literature devoted to the subject and on the exact simulation of Hawkes processes, we introduce a simple, yet useful, Hawkes-based model for energy spot prices. We present the model under the historical measure and illustrate a structure preserving change of measure, allowing to specify a risk-neutral dynamics. Then, we propose an effective estimation methodology based on particle filtering. Finally, we show how to perform exotic derivatives pricing both through exact simulation and characteristic function inversion techniques.

1 Introduction Hawkes processes are a particular class of (self-exciting) point processes where the occurrence of any event increases the probability of future events. They became popular modeling tools since their introduction in [40, 41]. Hawkes processes are ubiquitous across many fields of science. As a first application, they have been considered in earthquake modeling, where the aftershocks following a main event are described with an Hawkes process; a notable example is given by Ogata [63]. Moreover, self-exciting features proved to be useful to forecast infection

R. Brignone Department of Quantitative Finance, Institute for Economic Research, University of Freiburg, Freiburg im Breisgau, Germany e-mail: [email protected] L. Gonzato Department of Statistics and Operations Research, University of Vienna, Vienna, Austria e-mail: [email protected] C. Sgarra () Department of Mathematics, Università degli studi di Bari “Aldo Moro”, Bari, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. E. Benth, A. E. D. Veraart (eds.), Quantitative Energy Finance, https://doi.org/10.1007/978-3-031-50597-3_2

41

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diseases [57], crimes [58] and terrorist attacks [66]. The popularity of Hawkes processes have also been established in other fields: [71] propose applications in e-marketing, [30] for modeling events and social networks behaviour, [70] in mechanical twinning, [48] finds applications in neuron activity, [61] for modeling population dynamics. Nevertheless, an impressive spread of applications of Hawkes processes has recently taken place in economics and finance. We refer to [42, 43] for detailed surveys. A first strand of financial applications of Hawkes processes is related to high-frequency trading and limit order book. Some notable examples are given by Chavez-Demoulin et al. [17], Chavez-Demoulin and McGill [16], Bacry et al. [4] and Cartea et al. [15]. For insurance markets, [37, 38] proposes Hawkes processes to analyse the impact of volatility clustering on the evaluation of equity indexed annuities and the contagion between insurance and financial markets. Further applications in insurance are given by Stabile and Torrisi [69], Dassios and Zhao [20], Zhu [73], and Jang and Dassios [45]. From the point of view of credit risk, we mention the paper by Errais et al. [24], where the authors proposed to model credit default events in a portfolio of securities as correlated point processes, where the events dynamics are described by a mutually exciting Hawkes process. A more detailed review of the literature devoted to Hawkes-based models for commodities will be provided in Sect. 3. Hawkes processes have been thoroughly investigated from the mathematical viewpoint. In general, they are not affine and not even Markov processes, but in one case they are both, and this is the case in which the memory kernel is of exponential type (the definition will be provided in Sect. 4). This case is extremely relevant and attractive from the modeling viewpoint since for affine processes it is possible to compute explicitly the characteristic function, allowing for the usage of powerful transform-based pricing methods for derivatives valuation. Moreover, it also allows for more efficient simulation, as we illustrate later in this chapter. As a result, extensive literature is available on derivative pricing. We mention here the contributions by Hainaut and Moraux [39] on the pricing and hedging of options on stocks, by Brignone and Sgarra [10] and Brignone et al. [12] for Asian options, by Kokholm [51] for exchange options. In addition, we want to recall that Hawkes processes are not the only class of processes exhibiting self-exciting features: continuous time branching processes with immigration (CBPI) is another relevant class of processes sharing with Hawkes processes this remarkable property. CBPI processes have been used to describe populations dynamics by Pardoux [65]. In finance, [46] use CBPI to model short rates and illustrate their relationship with Hawkes processes. Another motivation for the popularity of Hawkes processes in financial markets modeling is their strict relationship with rough volatility models, i.e. models including in the driver one or more fractional Wiener processes with Hurst index .H < 1/2. Jaisson and Rosenbaum [44] proved that rough volatility models can be obtained as a proper scaled limit of a sequence of Hawkes processes with long memory kernels, and this property turns out to be relevant for computational purposes [23]. The purpose of this work is fourfold: (1) we provide some more evidence of self-exciting effects in the spot price of energy commodities; (2) we provide a

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brief literature review on the usage of Hawkes processes in energy finance; (3) we provide a literature review on the exact simulation of Hawkes process with exponential kernel, and compare the performances of the various methodologies available in the literature; (4) we illustrate a simple but realistic and practically relevant approach to model energy spot prices based on a Hawkes process with exponential kernel, which can be applied for derivatives pricing purposes. Indeed, we first illustrate a change of measure technique to move from the historical to the risk-neutral measure. Second, we propose an effective estimation methodology based on recent econometric literature. Third, we show how to price exactly some derivative instruments such as geometric Asian options under the suggested model. Fourth, we develop an exact simulation scheme for such models and show how to apply it for pricing path-dependent derivatives instruments, such as BermudanAsian options, which are popular in commodity markets. The plan of this chapter is the following: in Sect. 2 we discuss empirical evidence of jump clustering features in energy commodity price dynamics, in Sect. 3 we present a review of the literature available on Hawkes models in energy markets, in Sect. 4 we define Hawkes processes and illustrate their basic properties, in Sect. 5 we review existing simulation schemes adopted for Hawkes processes and compare their performances. In Sect. 6 we present a Hawkes-type model for a generic asset, and we discuss the relation between the historical and the risk-neutral dynamics. In Sect. 7 we propose an estimation methodology for the proposed model and in Sect. 8 we provide a pricing method for a class of exotic derivatives, the Asian options (with and without early exercise features). Finally, in the last section, we resume the main results and discuss possible developments.

2 Empirical Evidence of Jump Clusters Jump clustering in financial markets is the phenomenon for which, whenever a jump in the price process occurs, there is an increase in the probability of observing a new jump over the next period. This statistical feature is widely observed in financial markets, such as equity (see [31, 32], among others), and commodity (see Sect. 3 for a detailed review). For illustrative purposes, we provide some evidence of jump clusters for two energy commodities. We consider the time series of spot returns of West Texas Intermediate (WTI) and NY Harbor Ultra Low Sulphur Heating Oil (NYULSD) from 30-Jun-2006 to 15-Dec-2022 (4294 days, source: Datastream® ). These are reported in the top panel of Fig. 1. Next, we need to detect jumps in the price processes. To this aim, we suggest applying the iterated re-weighted least squares technique developed in [13]. This simple algorithm is used to extract jumps from a time series of spot prices and can be summarized as follows. Step 1: the mean and standard deviation of the daily simple returns are computed. Returns which are greater (respectively, smaller) than the mean plus (minus) three standard deviations are considered as jumps. Step 2: a new time series of simple returns is created by replacing the realizations corresponding to a jump with the sample

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0.5

0.2 0.1

0

0 -0.1

-0.5

2008 2010 2012 2014 2016 2018 2020 2022

2008 2010 2012 2014 2016 2018 2020 2022

-0.2

2008 2010 2012 2014 2016 2018 2020 2022

2008 2010 2012 2014 2016 2018 2020 2022

Fig. 1 Jump clusters in energy markets. Top subplots contain the time series of the returns of WTI and NYULSD; bottom subplots contain the time series of the jumps (indicated as vertical gray lines) detected using the methodology developed in [13]

mean. Steps 1 and 2 are repeated until the ratio between standard deviations of two consecutive iterations is greater than 0.99. By this way, the separation between the jumps and the Gaussian oscillations is done (see [6, 13] and [12] for more details). The algorithm detects 170 jumps in WTI and 147 jumps in NYULSD. Note that, based on this result, a model for energy commodity prices which omit jumps is likely misspecified. The gray lines in the bottom panel of Fig. 1 represent the jumps detected via the aforementioned algorithm. By an inspection of this figure, it is possible to see that jumps appear in clusters. We have periods with high jump activity (e.g. during the global financial crisis in 2008/2009), followed by calm periods where we do not observe any jump. This means that it is unlikely that jumps in the energy commodity spot prices exhibit a constant arrival rate. Following [6] and [12], we assess this fact with a simple statistical test. If the arrival rate is constant, then the jump process is a homogeneous Poisson process and the distribution of the interarrival times is exponential with mean .1/λ, where .λ is the average arrival rate computed as the ratio between the total number of jumps and the number of observations in the sample (i.e. .λ = 0.0396 for WTI spot price and .λ = 0.0342 for NYULSD). We then perform a two sample KolmogoroffSmirnov test where the null hypothesis is that the interarrival times are exponentially distributed with mean .1/λ. We strongly reject the null hypothesis, with p-values around 5.53E-06 for WTI spot price and 2.05E-04 for NYULSD. This analysis justifies the usage of stochastic processes for properly modeling the jumps intensity. In this work, we study the usage of Hawkes processes for this purpose.

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3 Literature Review: Hawkes Processes in Energy Markets A large amount of literature is devoted to model energy prices dynamics by Hawkes processes and in this section we provide a review on the main contributions available on the subject, and we apologize if the list presented will not be exhaustive. Filimonov et al. [29] investigate endogeneity of jumps in several asset classes, including commodities, and, in particular, crude oil and natural gas. By estimating from real data the branching ratio they provide evidence of endogeneity of jumps in prices, meaning that a large amount of jumps is self-exciting, or, put in a different way, they appear in clusters. Several authors aim at modeling intraday electricity prices by Hawkes processes: [27] investigates the European intraday market while [36] investigate the German intraday market, providing evidence of jumps clustering. Blasberg et al. [7] consider the question whether there is a temporal structure prevalent in the parameters of Hawkes processes estimated for adjacent delivery hours. They model a daily seasonality pattern found in the data and provide a suitable economic interpretation. In order to model the other components of the time series, they propose simple (vector) autoregressive models. They conduct a forecasting study and find superior performances of their proposed model against various benchmarks. Callegaro et al. [14] propose and investigate two model classes for forward power price dynamics, based on continuous branching processes with immigration, and on Hawkes processes with exponential kernel, respectively. Both the proposed models exhibit jumps clustering features. They adopt a Heath–Jarrow–Morton approach to capture the whole forward curve evolution. By examining daily data in the French power market, and by performing a goodness-of-fit test, they find evidence of adequacy of Hawkes processes in describing the forward price dynamics. Clements et al. [18] are the first to develop a multivariate self-exciting point process model for dealing with price spikes across connected regions in the Australian National Electricity Market. They examine the importance of the physical infrastructure connecting the regions on the transmission of spikes, and find that: (1) spikes are transmitted between the regions; (2) the size of spikes is influenced by the available transmission capacity. Giordano and Morale [34] propose a new model based on a Hawkes process to model the Italian power market. They find evidence of the presence of jumps in the Italian electricity market, many of which appear clustered over short time periods. The two-factor model they propose is driven by both Hawkes and fractional Brownian processes. They perform a calibration procedure, discussing the seasonality, spikes and estimate the Hurst coefficient. After calibrating and validating the model, they examine its forecasting performance via a class of adequate evaluation metrics. Eyjolfsson and Tjøstheim [25] suggest a model for spot prices in power markets based on Hawkes processes. They introduce a class of jump processes with selfexciting jump intensity. In particular, the paper focuses on two main aspects: (1) stochastic analysis of the proposed processes; (2) empirical application using daily quotes of UK electricity prices.

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Jiao et al. [47] propose a model for spot power prices based on a CBPI process and, after introducing a suitable relation between the historical and the risk-neutral spot price dynamics, they compute the forward and futures prices and provide a sound description of the sign change in the term structure of risk premium often observed in power markets. Gonzato and Sgarra [35] present a Hawkes type model for crude oil prices, including a stochastic dynamics for volatility. Their model specification is completed by a stochastic convenience yield. For model estimation, they apply the two-stage Sequential Monte Carlo (SMC) sampler [31] to both spot and futures quotations. From the estimation results, they find evidence of self-excitation in the oil market, which leads to an improved fit and a better out of sample forecasting performance with respect to jump-diffusion models with constant intensity. Brignone et al. [12] propose a mean-reverting model for commodity prices with Hawkes-type jumps, stochastic volatility and a diffusion-type dynamics for the convenience yield. This model is calibrated on real derivatives data, including energy related products, and is used to price exotic derivatives instruments. The model we are going to propose as a working example in Sect. 6 is somehow similar to the model proposed in [12], where a different estimation procedure has been used. We point out that the simulation scheme used in the last mentioned paper could be successfully applied to price options with early exercise features, and this is the subject of ongoing work.

4 Hawkes Processes We give here the definition of Hawkes process (with exponential kernel) and outline some basic properties.

4.1 Definition Based on the Conditional Intensity A Hawkes process is a special kind of point process whose conditional intensity depends on the history of the events. First, let us define the conditional intensity function and the univariate Hawkes process. Definition 1 Let N be a point process and let .FN t be the natural filtration generated by N itself. Then, the left continuous process defined by:

λt := λ(t | FN t− ) = lim

.

h→0+

  P Nt+h − Nt > 0 | FN t− h

is called the conditional intensity function of the point process.

(1)

Hawkes Processes in Energy Markets: Modelling, Estimation and Derivatives Pricing

Definition 2 The univariate Hawkes process be defined for all .t > 0 and .h → 0+ as: ⎧   N ⎪ P N − N = 1 | F ⎪ t+h t t− ⎪ ⎪ ⎨   . P Nt+h − Nt > 1 | FN t− ⎪ ⎪   ⎪ ⎪ ⎩P Nt+h − Nt = 0 | FN t−

47

N with conditional intensity .λt , can = λt h + o(h) = o(h)

(2)

= 1 − λt h + o(h),

where .o(h) denotes infinitesimals of higher order with respect to h, or, more qualitatively speaking, terms going to 0 faster than h. Furthermore, the dynamics of conditional intensity of a Hawkes process with exponentially decay function can be represented by the following stochastic differential equation (SDE): dλt = k(θ − λt )dt + βdNt ,

(3)

.

where, k is the constant rate of decay, .θ is the background intensity, .β is the magnitude of self-excited jump and N is a univariate Hawkes process. For sake of illustration, we report in Fig. 2 a simulated trajectory of a Hawkes process. The vertical gray dashed lines represent jump occurrences. Whenever the jump occurs, the intensity increases. As a result, we have a higher probability of observing a new jump. When there are no jumps, the intensity reverts to its long run mean. Given the initial datum .λ0 the solution of (3), can be written as follows: λt = θ + (λ0 − θ )e−kt +

Nt 

.

βe−k(Tk −t) ,

t ≥ 0.

(4)

k=1

80

60

40

20

0 0

0.2

0.4

0.6

0.8

1

1.2

Fig. 2 Simulated sample path of the intensity of a Hawkes process

1.4

1.6

1.8

2

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Then, (4) can be written again in the following way: λt = θ + (λ0 − θ )e−kt + β



.

t

e−k(t−s) dNs ,

(5)

0

which provides the meaning of the expression “exponential memory kernel” sometimes used to denote this class of processes. This definition can be extended to more general memory kernels .K(s), when .λt is given by:

t

λt = λ0 + β

.

K(t − s)dNs .

0

When .K(s) = s γ , the Hawkes process with conditional intensity .λt is called a Hawkes process with power (memory) kernel. The Markov property of the couple .(λt , Nt ) in this case is lost. Definition 3 A D-variate Hawkes process is a D-variate counting process N with intensity vector given by: λit = λi0 +

D 

.

t

j

K i,j (t − s)dNs ,

j =1 0

where the non-negative and causal (i.e. .K i,j (s) = 0, .s < 0) .L1 functions (see [5, Section 2]) .K i,j (s) denote the matrix kernel. D-variate Hawkes processes provide a description of mutual contagion or mutual-exciting behavior effects among different assets. A notable example is given by Ait-Sahalia et al. [2], where the authors propose a mutually-exciting jump-diffusion process to describe the dynamics of market indices in six different countries. Ait-Sahalia and Hurd [1] investigate the optimal investment problem for a financial market with mutual contagion described by a D-variate Hawkes process, while [51] presents an evaluation algorithm for spread options based on the explicit computation of the characteristic function of a bi-variate Hawkes process with exponential kernel. In the present chapter, we shall focus on the one-dimensional case.

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4.2 Moments of λT and NT We provide, next, the expressions for the conditional expectation and variance of .λt , given as in (3), and for the conditional expected number of jumps: 

θ k −k1 T θk e + λ0 − , .E[λT ] = k1 k1

 

θk β2 θ k −k1 T θk −2k1 T e , − λ0 e + λ0 − + Var[λT ] = k1 2k1 k1 2k1

 1 θk θk λ0 − (1 − e−k1 T ), E[NT ] = T + k1 k1 k1

(6)

where, .k1 = k − β > 0. The analytical expression in (6) will be used, following [21], to evaluate the correctness and accuracy of the simulations in Sect. 5.

5 Simulation of Hawkes Processes with Exponential Kernel In this section, we illustrate several approaches that have been proposed in the literature to simulate a Hawkes process, which do not require any numerical method for their implementation.1 Together with the description of the algorithms, we also provide pseudocodes. These are presented in the easiest possible form to allow a straightforward implementation. Matlab® codes are available at https:// www.mathworks.com/matlabcentral/fileexchange/74738-hawkes-simulation. Note that these codes can be slightly different from pseudocodes provided below, since they are more efficient.

5.1 Euler Scheme The Euler scheme is used to simulate the exact solution of a discrete time process approximating the system dynamics. In our framework, we can discretize (3) by approximating the jump times .Nt with a Bernoulli random variable, as in [31]. Therefore, λti = λti−1 + k(θ − λti−1 )Δt + βΔNti ,

.

(7)

1 In other words, we exclude from our review those methods which are expectantly slow since based on repeated implementation of computationally intensive numerical techniques such as, for example, root finding algorithms (as in [64]), or numerical integration (as in [59]).

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where, .ΔNti = Nti − Nti−1 ∼ Ber(λti−1 Δt). Despite affected by time discretization bias, this method is widely used in practice due to its simplicity. Moreover, it is expected to be fast in general, with a lower dependence on the expected number of jumps than competing methods. Pseudocode is given in Algorithm 1. Algorithm 1 Euler scheme Input: λ0 , k, θ , β, T , n Output: {λtj }nj =1 , {Ntj }nj =1 1: Set Δt = Tn and N0 = 0 2: for j = 1 : n do 3: Set ΔNtj = 0 and draw u ∼ U(0, 1) % draw from uniform 4: if u < λtj −1 Δt then 5: ΔNtj = 1, Ntj = Ntj −1 + 1 end 6: Set λtj = λtj −1 + k(θ − λtj −1 )Δt + βΔN (tj ) 7: end

5.2 Ogata’s Modified Thinning Algorithm and a Variant The simulation algorithm for Hawkes processes proposed by Ogata [62] is the most popular in the literature. It is based on the thinning simulation method for non-homogeneous Poisson processes proposed in [53]. The idea is to simulate a homogeneous Poisson process and then remove excess points, such that the remaining points satisfy the conditional intensity .λt . This algorithm requires the ¯ ∀t. A conditional intensity to be upper bounded, i.e. .∃λ¯ < ∞ such that .λt ≤ λ, generalization introduced by Ogata [62] (typically called Ogata’s modified thinning) is outlined in Algorithm 2. This generalization requires only the local boundedness of conditional intensity. Indeed, if .λt is a non-increasing function (i.e. .k > 0 in (3)) in the interval between two adjacent occurrences, we have that .λt ≤ λt + for i + ¯ .t ∈ (ti , ti+1 ), where .t i is the time just after .ti . Therefore, a local bound .λt could be set equal to .λt + in the interval .(ti , ti+1 ) and it has to be updated after each i occurrence. Pseudocode is given in Algorithm 2. Afterwards, [19] modified Ogata’s algorithm by setting .λ¯ t = λt . In this way we are not interested if t is a point of the process or not; whereas we add a function ¯ t + , for arbitrary .k¯ (the authors suggest to set of time interval of length .Lt = kλ ¯ ¯ .k = 0.5). Then, .λt is updated if a new point of the process occurs or if the time frame .Lt has elapsed. In Algorithm 3 we describe how to implement Ogata’s thinning algorithm as in [19].

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Algorithm 2 Ogata’s modified thinning algorithm Input: λ0 , θ , k, β, T NT T Output: NT , {Tj }N j =1 , {λTj }j =1

1: Set N0 = 0, ϵ = 10−6 , j = 0, t = 0, T0 = 0 2: while t < T do 3: Compute M = λt+ϵ according to (5) 4: u1 ∼ U(0, 1), u2 ∼ U(0, 1) % draw from two independent uniforms 1) % transform in an exponential with mean 1/M 5: R = − ln(u M 6: Compute H = λt+R /M according to (5) 7: if u2 > H then 8: t =t +R 9: else 10: j = j + 1, t = t + R , Tj = t, Ntj = Ntj −1 + 1 end end T 11: Compute {λTj }N j =1 using formula (5)

Algorithm 3 Ogata’s modified thinning algorithm as in [19] Input: λ0 , θ , k, β, T NT T Output: NT , {Tj }N j =1 , {λTj }j =1 1: Set k¯ = 0.5, N0 = 0, ϵ = 10−6 , j = 0, t = 0, T0 = 0 2: while t < T do ¯ t+ϵ 3: Compute M = λt according to (5) and L = kλ 4: u1 ∼ U(0, 1) % draw from uniform 1) % transform in an exponential with mean 1/M 5: R = − ln(u M 6: if R > L then 7: t =t +L 8: else 9: u2 ∼ U(0, 1) % draw from uniform 10: Compute H = λt+R /M according to (5) 11: if u2 > H then 12: t =t +R 13: else 14: j = j + 1, t = t + R, Tj = t, Ntj = Ntj −1 + 1 end end end T 15: Compute {λTj }N j =1 using formula (5)

5.3 Dassios and Zhao’s Exact Simulation Dassios and Zhao [21] proposed an efficient sampling algorithm for Hawkes process that does not rely on the accept-reject method. Moreover, this algorithm can generate jump times with either stationary or non-stationary intensity. The starting point is the cluster-based representation of a Hawkes process (see [21] for more details). The authors decompose the interarrival times into two independent random variables: the first random variable .S1 represents the interarrival time of the next event, if it is coming from the background intensity .θ . The second random variable .S2 represents

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the interarrival time of the next event if it comes from either the exponential immigrant kernel − θ )e−k t or the Hawkes self-exciting kernel from each of the  .(λ0−k(t−t j ) . Then, the sampled interarrival time is the minimum of past events . j :tj 0 then

7: 8:

S = min(S1 , S2 ) else

9:

S = S2 end

10:

Tj +1 = Tj + S and t = Tj +1

11:

λj +1 = (λj − θ)e−k(Tj +1 −Tj ) + θ + β

12:

Ntj +1 = Ntj + 1 j =j +1

13: end

5.4 Comparison of Exact Methodologies We start by comparing Algorithms 2, 3, and 4. When evaluating the efficiency of a methodology, three considerations are important: computing time, bias, and variance. All these methods are unbiased and present the same variance. Hence, we just need to evaluate the running time. To this aim, we implement the following experiment. We generate 100 parameters combinations respecting the following conditions: .k ∈ [0.5, 35], .θ ∈ [0.1, 3], .β ∈ [1, 30], .λ0 ∈ [0.1, 5], .T ∈ {1, 3, 5, 7, 10}, .k > β, .λ0 > θ. With these choices we aim to incorporate standard parameter settings used in the literature, in particular [8, 31, 35, 51] which calibrated the Hawkes model on real financial data. For each parameter set we run the three algorithms with .106 simulations and register the running time. We choose such

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40 30 20 10 0 0

20

40

60

80

100

120

140

160

180

200

Fig. 3 Time in seconds of each simulation scheme with respect to the expected number of jumps

a high number of simulations in order to minimize the risk that the comparison between the different methodologies is influenced by the effective number of jumps over all the simulations. Indeed, with only few simulations, we could observe that a method is faster than another one only because the (random) number of jumps is much smaller and not because the method is effectively faster. In Fig. 3 we report the running time in seconds and the expected number of jumps for each model parameter set. Results show that Algorithm 4 is the best performing overall. For the same accuracy it results slightly faster than Algorithm 2. This difference is anyway small and basically negligible for small .E[NT ]. The superior performance is due to the fact that random sampling is necessary only when the jump effectively occurs, while in thinning algorithms those steps run also when the jump is discarded, resulting in a higher running time. Moreover, in Algorithm 2, formula (5) must be implemented to compute the upper bound for the intensity M, the acceptance probability H and to update the jump intensity, while, in Algorithm 4, this is evaluated only to update the jump intensity (i.e. only when the jump effectively occurs). We compare now Algorithms 2 and 3. Let’s start by an inspection of Fig. 3. We note the strange results for the running time of the method of [19] for small values of the expected number of jumps. The issue comes exactly from the adjustment they propose. Indeed, a small value for (6) is usually due to small values of the intensity parameters. Therefore, in Algorithm 3, we get stuck at the first if statement, since the variable L will be always smaller than R, causing a very small update of t=t+L.2 To the best of our knowledge, this unsatisfactory behavior of the algorithm was not previously noted in the literature. However, the proposed modification is expected to give a reasonable compromise between setting the bound M too high, and so generating excessive trial points, and setting it too low, thus requiring too many iterations of steps 5 and 6 in Algorithm 3. In other words, the idea is to avoid running the accept-reject step whenever the possibility partial solution to this problem is given by increasing the value of the parameter .k¯ in Algorithm 3, which controls the function L.

2A

54 Table 1 Parameter sets taken from financial literature

R. Brignone et al. Set A B C D

k 20.8 18.16 30.05 2.56

.θ

.β

.λ0

.E[N1 ]

1.13 0.32 1.39 2.14

13.5 16.62 23.46 2

1.3 0.5 1.5 2.3

2.9569 2.1036 5.6051 4.0532

Literature [51] [8] [35] [31]

of observing a jump is very low, saving the CPU time needed for random sampling. This allows in practice to reduce the number of discarded points, but our numerical results show that this benefit is not commensurate with the extra labour involved. Indeed, with respect to Algorithm 2 some extra operations are necessary, i.e. an if statement and the computation of L. As a result, Algorithm 3 results slower than Algorithm 2.

5.5 Comparison Between Algorithm 4 and the Euler Scheme We compare, next, the performances of Algorithms 1 and 4. To this aim, we need first to estimate the bias of the Euler scheme. We select some parameter sets taken from financial literature (see Table 1). For each parameter set, we compute (6) analytically and through Monte Carlo simulation using Algorithm 1 (.109 simulations). The absolute difference between the true value and the Monte Carlo estimate is the bias. Then, we run Algorithms 1 and 4 for different number of simulations .M and compute the Root Mean Squared Error (RMSE) and register the running time in seconds.3 For the implementation of the Euler scheme we must also specify the number of time discretization steps √ (denoted with n in Algorithm 1). We follow the suggestion of [22] to select .n = M. The numerical results are reported in Table 2. Few comments are in order. The Euler scheme is, as expected, less accurate with a non-negligible average error in computing the expected number of jumps. When we use less than .M = 4 × 104 simulations, the Euler scheme is faster than Algorithm 4. Indeed, the most time consuming steps in Algorithm 4 (but also Algorithms 2 and 3) are: (1) random sampling from the uniform distribution; (2) updating the vectors of jump times and intensity. Indeed, the total number of jumps is unknown at the beginning of each simulation. This fact precludes more efficient vectorizations and pre-allocations. In other words, it is not possible to sample all the needed random quantities in one time at the beginning of the simulation. Contrarily, the Euler scheme does not suffer of this problem as the number of uniforms to be sampled is .n × √T and does not depend on the effective number of jumps. Hence, for small .n = M, the Euler scheme is faster than the competitor. Despite that, in practical

3 See

e.g. [52] and the references therein for more details on the design of this experiment and definition of RMSE.

.

4 × 104 4 . 16 × 10 4 . 64 × 10 4 .256 × 10

.

4 × 104 4 . 16 × 10 4 . 64 × 10 4 .256 × 10

.M

Algorithm 1 n Bias Set A 0.0157 200 400 0.0087 0.0047 800 0.0013 1600 Set C 200 0.0321 0.0165 400 0.0080 800 0.0037 1600

0.0538 0.0281 0.0141 0.0070

0.0268 0.0140 0.0073 0.0031

RMSE

0.27 2.32 20.43 224.39

0.36 2.27 19.59 204.11

Time

0.0477 0.0239 0.0120 0.0060

0.0223 0.0112 0.0056 0.0028 0.58 2.06 8.22 31.75

0.48 1.59 6.24 24.87

Algorithm 4 RMSE Time

Algorithm 1 n Bias Set B 200 0.0178 400 0.0083 800 0.0059 1600 0.0018 Set D 200 0.0332 400 0.0172 800 0.0083 1600 0.0043 0.0376 0.0193 0.0094 0.0048

0.0425 0.0223 0.0120 0.0057

RMSE

0.24 2.33 20.41 210.39

0.25 2.29 19.95 206.86

Time

0.0183 0.0090 0.0045 0.0023

0.0441 0.0223 0.0109 0.0054

0.51 1.78 7.09 27.24

0.44 1.54 6.19 23.17

Algorithm 4 RMSE Time

Table 2 Speed–accuracy comparisons of Algorithms 1 and 4 for different parameter sets: the case of .E[N1 ]. All computing times are in seconds

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-1.5

-1.4 -1.6 -1.8

-2

-2 -2.2

-2.5 -0.5

0

0.5

1

1.5

2

-1.2

2.5

-1

0

1

2

0

1

2

-1.5

-1.4 -1.6

-2

-1.8 -2

-2.5

-2.2 -2.4 -1

0

1

2

-1

Fig. 4 Speed–accuracy comparisons of Algorithms 1 (blue line with circles) and 4 (red line with diamonds) for different parameter sets: the case of .E[N1 ]. All computing times are in seconds

applications n is not a small number since the accuracy of this simulation scheme depends on the number of time discretization points, as it is clear from the evolution of the bias in Table 2. For this reason the Euler scheme is slower than the other methods for higher n (or, equivalently in our experiment, higher .M). Therefore, a good environment for its implementation is when the expected number of jumps is very high, but the simulation horizon is small (e.g. .T < 1). However, when high accuracy is required, Algorithm 4 is much faster and more accurate than the Euler scheme, with a higher convergence rate of the RMSE, as evident from Fig. 4, where we report speed–accuracy portrayals on a log-log scale for the competing methods. Based on these numerical results, we suggest to use Algorithm 4 to simulate Hawkes processes with an exponential kernel.

6 An Asset Pricing Model with Self-exciting Jumps In this section we suggest a simple model for spot returns where the jump intensity evolves according to a Hawkes process with exponential kernel. We first define the model under the historical measure, then we propose a structure preserving change of measure which allows the specification of the risk-neutral dynamics.

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57

6.1 Dynamics Under the Historical Measure Under the historical measure, asset price and jump intensity are given by:

  σ2 J − λt E e − 1 dt + σ dWt + J dNt . .dXt = d ln(St ) = μ−δ− 2 dλt = k(θ − λt )dt + βdNt

(8) (9)

This model can be seen as an extension of [56]. J is the jumps size, which is normally distributed with given mean .μJ and standard deviation .σJ . .β is constant and is responsible for the self-excitation property of .λt . .dWt is a standard Brownian motion independent from the jump component .J dNt . Further, .δ represents a constant convenience yield. We further assume that the following non-explosion condition for the Hawkes process .Nt in (9) holds: k > β.

.

(10)

The following shows that, despite .σ is assumed to be constant, the variance of simple returns still changes randomly as function of .λ: Vt dt = Var

.

dSt St

= σ 2 dt + (μ2J + σJ2 )λt dt.

6.2 Dynamics Under the Risk–Neutral Measure In order to deal with derivatives pricing we need to introduce a risk-neutral measure. The purpose of the present subsection is to illustrate briefly the basic technical tools required in order to perform a structure-preserving probability measure change in a Hawkes jump-diffusion framework. To this end, we extend the results in [72] and [39] by introducing an Esscher–type measure change. This is defined as follows.  If we denote by .Lt := J dNt the jump term in (8), and by .ψ P (z) := E ezJ the moment generating function of the jump size density, we can define the following family of exponential martingales:

t 1 t 2 Mt (ξ, φ) := exp κ1 (ξ )λt + ξ Lt + κ2 (ξ )t − φ (u)du − φ(u)dWu 2 0 0

.

where .κ1 (ξ ), κ2 (ξ ) (with .κ1 (ξ ), κ2 (ξ ) functions of .ξ ) denote the risk premium related to the jumps and .φ (a stochastic process adapted to the reference filtration .Ft ) denotes the risk premium related to the diffusion component of log-returns. We state the following result. We remark that in the present case, where the logreturn probability density is Gaussian, .ψ P (z) exists for all values of z, while for different

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distribution it will exist only for some values of the parameters characterizing the probability density of the logreturn jump size. Proposition 1 If, for any .ξ , there exist functions .κ1 (ξ ), κ2 (ξ ) solutions of the following system of algebraic equations:  .

κ1 k − [exp (κ1 (ξ )β)ψ(ξ ) − 1] = 0

(11)

κ2 + κ1 kθ = 0,

the non-explosion condition (10) then .Mt (ξ, φ) is a local martingale. If, moreover,   T holds for .Nt and the Novikov condition .(E exp 0 φ 2 (u)du < ∞) holds for .φ, .Mt is a true martingale. Proof We extend to the present modelling framework the results in [72] (in our case the process .J dNt driving the log-returns dynamics is marked). Let’s define .mt := ln Mt ; then 1 dmt = κ1 (ξ )k(θ − λt )dt + κ2 (ξ )dt + κ1 (ξ )βdNt + ξ J dNt − φ 2 (t)dt − φ(t)dWt 2

.

By applying Itô’s lemma to .Mt we obtain: 1 dMt = Mt dmt + Mt d[m, m]ct 2 +∞ + Mt [exp (κ1 (ξ )β + ξ z) − 1 − (κ1 (ξ )β + ξ z)]F (dz)dNt ,

.

−∞

where .[m, m]ct denotes the quadratic variation of the continuous component of .mt and .F (·) denotes the random measure of the jumps size J and .λt ν(dz)dt denotes the predictable compensator of the process .J dNt . We can write the last equation in the following form: dMt = Mt

.

κ1 (ξ )k(θ − λt )dt + κ2 (ξ )dt + κ1 (ξ )β + ξ



+∞ −∞

zF (z)dz dNt



1 1 + Mt − φ 2 (t)dt − φ(t)dWt + Mt d[m, m]ct 2 2 +∞ + Mt [exp (κ1 (ξ )β + ξ z) − 1 − (κ1 (ξ )β + ξ z)]F (dz)dNt , −∞

Hawkes Processes in Energy Markets: Modelling, Estimation and Derivatives Pricing

59

which, after the introduction of the compensator of the jump process, can be written as dMt = Mt κ1 (ξ )kθ dt + Mt κ2 (ξ )dt − Mt λt κ1 (ξ )k +∞ − Mt φdWt − Mt λt [exp (κ1 (ξ )β + ξ z) − 1]ν(dz)dt+

.

+ Mt

−∞

+∞

−∞

[exp (κ1 (ξ )β + ξ z) − 1][F (dz)dNt − λt ν(dz)dt],

since . 12 Mt d[m, m]ct = 12 φ 2 (t)Mt dt. The integral with respect to the compensated jump measure .F (dz)dNt − λt ν(dz)dt is a local martingale as the integrals with respect to the Wiener process .dWt , so .Mt is a local martingale if and only if   = Mt (ξ,φ) is a true (11) hold. In order to prove that the likelihood process . dQ dP Ft M0 (ξ,φ)     = 1, we need to apply a uniform integrability criterion martingale with .E dQ dP Ft separately for the diffusion and the jump part in the dynamics (8) and (9), since they are independent. The Novikov condition grants that the diffusion part of .Mt is a true martingale, while the criterion proposed by [68], which in this case holds thanks to the non-explosion condition (10), grants that the jump part of .Mt is a true martingale as well. A more detailed proof of the martingale property for the jump part of .Mt can be found in [9, Prop. .2.6]. ⨆ ⨅ In strict analogy with [39], by defining the moment-generating function under exp (κ1 (ξ )β)ψ(z+ξ ) ψ(z+ξ ) Q Q .Q in the following way: .ψ (z) = exp (κ1 (ξ )β)ψ(ξ ) , i.e. .ψ (z) = ψ(xi) , it is possible to prove that the measure change just introduced by the likelihood process  Mt (ξ,φ) dQ  . = dP Ft M0 (ξ,φ) preserves the model structure. We have the following theoretical result. Proposition 2 The dynamics under the risk-neutral measure .Q of .XtQ , λQ t is given by the following system of stochastic differential equations: Q .dXt



 Q  1 2 Q J = r − σ − E e − 1 λt − δ dt + σ dWtQ + J Q dNtQ , . 2

Q Q Q Q Q dλQ t = k (θ − λt )dt + β dNt ,

(12) (13)

where .J Q NtQ denotes the jump process with respect to .Q and .J Q the jump size with respect to .Q. The relations between the relevant parameters under .P and .Q are the following: dWtQ = dWtP + φ(t)dt,

.

k Q = k, θ Q = θ eκ1 (ξ )β ψ(ξ ), β Q = eκ1 (ξ )β ψ(ξ )β.

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Moreover, under .Q the jump size .J Q still have a normal distribution with mean Q Q Q .μ J and volatility .σJ , with moment generating function given by .ψ (z) = ψ(z + ξ )/ψ(ξ ). Proof The conditional moment generating function of .λ under .Q is   P κ1 (ξ )β P exp (m . EQ [exp (uλQ )|F ] = exp (−m )E + ue ψ(ξ )λ )|F t t T t T T

.

  If we denote by .f (t, u, λ, m) = EP exp (mT + ueκ1 (ξ )β ψ(ξ )λPT )|Ft the conditional expectation, which is a martingale by definition, we can apply Itô’s lemma and impose a vanishing condition on the drift. In this way we obtain the following PIDE: .

∂f 1 ∂ 2f ∂f ∂f + [κ1 (ξ )k(θ − λ) + κ1 (ξ )] + k(θ − λ) − φ2 2 ∂t ∂m ∂λ 2 ∂m +∞ +λ [f (t, λ + β, m + κ1 β + ξ z) − f (t, λ, m)]ν(dz)] = 0.

(14)

−∞

If we guess a solution of the following form: f (t, λt , mt ) = exp [A(t, T ) + eκ1 (ξ )β ψ(ξ )B(t, T )λt + C(t, T )mt ],

.

with terminal conditions .A(T , T ) = 0, B(T , T ) = u, C(T , T ) = 1, and denote by ϵ = eκ1 (ξ )β ψ(ξ ), we can write:

.

.



∂A ∂f ∂ 2f ∂B ∂C ∂f ∂f = + ϵλt + mt f, = Cf, = ϵBf. = C 2 f, 2 ∂t ∂t ∂t ∂t ∂m ∂λ ∂m

By inserting these expressions into (14) we obtain the following equation, that must be satisfied for every value of .λ, m: .

∂C ∂A + [ϵkθ B + kθ κ1 (ξ )C + κ2 (ξ )C] + mt ∂t ∂t

+∞ ∂B − κ1 (ξ )kC − ϵkB + + λt ϵ [eCκ1 (ξ )β+Bϵβ+ξ z − 1]dν(z) ∂t −∞ 1 1 − φ 2 C + φ 2 C 2 = 0. 2 2

We obtain .C(t, T ) = 1 and  .

∂A ∂t + ϵkθ B + kθ κ1 (ξ ) + κ2 (ξ ) = 0  +∞ κ (ξ )β+ϵBβ+ξ z 1 ϵ ∂B ∂t − κ1 (ξ )k − ϵkB + −∞ [e

− 1]dν(z) = 0.

Hawkes Processes in Energy Markets: Modelling, Estimation and Derivatives Pricing

61

Finally, by using conditions (11), we get:  .

∂A ∂t = −ϵkθ B ϵβB ϵ ∂B ∂t = kϵB − e

+ 1.

(15)

We can repeat the same computation of the conditional moment generating function of .λ under .P without the introduction of the terms .mt and .mT and we get the following system of ordinary differential equations:  .

∂A ∂t ∂B ∂t

= −kθ B = kB − eβB + 1.

(16)

By comparing (15) written under .Q with (16), written under .P, the result follows, since they are the same, but under .Q the unknown B appears multiplied by .ϵ = ⨆ ⨅ eκ1 (ξ )β ψ(ξ ). Remark 1 We shall assume in the following that the risk premium term .φ is such that the following equality is satisfied for all .t ∈ [0, T ]: P

μ − r + λt [(eκ1 (ξ )β ψ(ξ ) − 1)(eμJ +σJ /2 − 1)] = σ φ(t), 2

.

(17)

in such a way that the dynamics of log-returns under .Q can be written as in (12). Remark 2 We remark that, on the basis of the present construction, the relations between the mean and the variance of the jump size with respect to .Q and .P are the Q P 2 following: .μQ J = μJ + ξ σJ and .σJ = σJ . The dynamics of the present model under .Q can then be written as follows:

 σ2 J − λt E[e − 1] dt + σ dWt + J dNt . .d ln(St ) = r −δ− 2 dλt = k(θ − λt )dt + βdNt ,

(18) (19)

where we dropped the superscript .Q on .λt , J, k, θ, β for simplicity of notation.

7 Model Estimation We perform an empirical estimation of the model (12)–(13) using WTI spot prices ranging from 05 October 2006 to 19 April 2020, for a total of 3531 daily observations. The model under consideration is highly non-linear and nonGaussian, therefore standard Kalman filtering techniques cannot be applied in this context. Hence, in order to conduct inference, we rely on the Continuous

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Table 3 Priors specification .Θ .μ .μJ

k .β

Dist Tr. Normal Tr. Normal Tr. Normal Tr. Normal

Support .[−0.20, 0.20]

.(μ0 , σ0 )

(0.00, 0.10) .[−0.20, 0.20] (-0.03, 0.20) .[0.00, 20.0] (2.50, 4.00) .[0.00, 20.0] (2.00, 4.00)

.Θ .σ .σJ .θ .δ

Dist Tr. Normal Tr. Normal Tr. Normal Tr. Normal

Support

.(μ0 , σ0 )

.[0.00, 1.00]

(0.20, 2.00) (0.05, 0.20) .[0.00, 20.0] (2.10, 4.00) .[−1.50, 1.50] (0.00, 1.00) .[0.00, 0.30]

Sequential Importance Resampling (CSIR) algorithm of [55]. This technique allows to perform Simulated Maximum Likelihood Estimation (SMLE) in non-linear and non-Gaussian state-space models using particle filtering methods. With respect to standard SIR algorithms, the CSIR replaces the resampling step with a linear interpolation. By this way the simulated log-likelihood obtained with the particle filter will be smooth and prone to numerical optimization. To implement the CSIR we need to cast model (12)–(13) in a state-space form. To this end, we consider a first order Euler approximation to the continuous-time system. Denoting by .Δt a small time interval, we can define the measurement and transition equations, under the historical measure .P, as follows  √ σ2 J .Xt = Xt−1 + μ − δ − − λt−1 E[e − 1] Δt + σ Δt Zt + J ΔNt , . 2

(20) λt = λt−1 + k(θ − λt−1 )Δt + βΔNt ,

(21)

where .Zt ∼ N(0, 1), .Jt ∼ N(μJ , σJ2 ) and .ΔNt = Nt −Nt−1 ∼ Bernoulli(λt−1 Δt). Given the state-space model (20)–(21), the SMLE procedure is performed in two steps: first, we sample .M = 104 random model parameters from the prior distributions in Table 3. The hyperparameters of the prior distributions are chosen to be consistent with those estimated in the recent literature (see e.g. [12, 31]). We use a truncated Normal prior to respect the domain of the parameters and to incorporate some upper and lower bounds used later in the optimization step. Given the M set of parameters, we evaluate the log-likelihood using our CSIR algorithm with .N = 100 particles4 for each of them and take the 8 random model parameter combinations corresponding to the highest log-likelihood. Second, we run 8 optimizations using those model parameters as starting points and take as MLE the parameters with the highest (maximized) log-likelihood. For this second step we use .N = 3000 particles which is approximately equal to the number of total observations and ensures that

4 We performed several experiments and we found that, for the log–likelihood computation in step 1, .N = 100 and .M = 104 are enough to fully represent the spectrum of reasonable starting points. A higher number of particles N is only needed to control the Monte Carlo variance of the likelihood estimator, but not to increase its level. Therefore, we increase N only for the subsequent optimization in order to stabilize the inferential procedure.

Hawkes Processes in Energy Markets: Modelling, Estimation and Derivatives Pricing Table 4 Estimated parameters for the model in (20)–(21) on WTI log-returns. The log-likelihood at the estimated parameters is equal to .8.51E + 03

.Θ .μ .δ .σ

k

Estimate −0.0726 −0.7693 0.2809 11.2570

Std error 0.1163 0.1382 0.0039 0.2966

.Θ .θ .β .μJ .σJ

Estimate 3.8132 10.1608 0.0165 0.0987

63 Std error 0.3361 0.3599 0.0037 0.0048

200 180 160 140 120 100 80 60 40 20 0

2008

2010

2012

2014

2016

2018

2020

Fig. 5 Filtered jump intensity (green line) and 5% and 95% quantiles (black line)

the simulated likelihood estimator is not too noisy (see e.g. [3]). During both steps we enforce the non-explosion condition (10): .k > β. In Table 4 we present the parameters estimated through the CSIR algorithm. Few comments are in order. First, all parameters are statistically significant (except .μ), which confirms that our estimation strategy can deliver reliable parameter estimates. The result for .μ is somehow expected since the sample mean of log-returns is almost equal to zero. Nevertheless, the convenience yield .δ is negative and highly significant. Second, in line with the existing literature, we confirm that jumps are self-exciting in the oil market. Third, the size of jumps is on average positive and the associated parameters are identified with high accuracy. Next, we plot in Fig. 5 the filtered jump intensity. From the figure we can appreciate the tightness of confidence intervals, which is another evidence in favor of our estimation approach. In addition, from Fig. 6 we note that the filtered jump intensity .λt is high during periods of

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0.3 0.2 0.1 0 -0.1 -0.2 -0.3

2008

2010

2012

2014

2016

2018

2020

2008

2010

2012

2014

2016

2018

2020

200 150 100 50 0

Fig. 6 WTI log-returns (top panel) and filtered jump intensity (bottom panel)

market stress with many jumps in close succession and reverts to its long run mean in normal periods.

8 Model Simulation and Exotic Derivatives Pricing In this section, we outline some results on affine option pricing models which allows to price quickly and exactly geometric (both with continuous and discrete monitoring) Asian options, which are very popular in commodity markets (see e.g. [12, 49, 67]). Then, we develop an exact simulation scheme for the model (18)– (19) which allows to price other derivative instruments with more involved payoff structure, such as arithmetic Asian, Bermudan and Bermudan-Asian options.

8.1 Asian Options The following result provides the joint moment generating function of the logreturns and jump intensity.

Hawkes Processes in Energy Markets: Modelling, Estimation and Derivatives Pricing

Proposition 3 Define .XT := log



ST St

65

 , the following holds

E[eu1 XT +u2 λT | Ft ] = eu1 Xt +A(u1 ,u2 ,τ )+B(u1 ,u2 ,τ )λt ,

.

where .τ = T − t and  ⎧ ∂A(u1 ,u2 ,τ ) ⎪ = u1 r − δ − ⎪ ∂τ ⎨ .

⎪ ⎪ ⎩

σ2 2



2

+ u21 σ2 + kθ B(u1 , u2 , τ ),   2 2 = −kB(u1 , u2 , τ ) + eu1 μJ + u1 σJ /2 + βB(u1 ,u2 ,τ ) − 1

∂B(u1 ,u2 ,τ ) ∂τ μJ + σJ2 /2

−(e

− 1)u1 ,

with initial conditions .A(u1 , u2 , 0) = 0, .B(u1 , u2 , 0) = u2 . ⨆ ⨅

Proof The proof strictly follows [12, Proposition 4]. Proposition 3 can be further extended. Proposition 4 The following holds M(u1 , u2 , u3 , u4 ) := E[eu1 XT +u2 λT +u3

.

T t

Xs ds+u4

T t

λs ds

| Ft ]

= e(u1 +u3 τ )Xt +A(u1 ,u2 ,u3 ,u4 ,τ )+B(u1 ,u2 ,u3 ,u4 ,τ )λt , where   ⎧ 2 2 2 ,u3 ,u4 ,τ ) ⎪ ∂A(u1 ,u∂τ = (u1 + u3 τ ) μ − δ − σ2 + (u1 + u3 τ )2 σ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ +kθ B(u1 , u2 , u3 , u4 , τ ), ∂B(u1 ,u2 ,u3 ,u4 ,τ ) . = −kB(u1 , u2 , u3 , u4 , τ )  ∂τ  ⎪ ⎪ 2 2 ⎪ (u +u τ )μ J +(u1 +u3 τ ) σJ /2+βB(u1 ,u2 ,u3 ,u4 ,τ ) − 1 1 3 ⎪ + e ⎪ ⎪ ⎪ ⎩ 2 −(eμJ +σJ /2 − 1)(u1 + u3 τ ) + u4 , ⨆ ⨅

Proof See [10, p. 106]. Finally, we state the following theoretical result which follows from [33].

Proposition 5 Suppose .n ∈ N and .Δ = Tn (we are assuming equally-spaced dates for simplicity), .0 := t0 < t1 < ... < tn := T , the following holds   n n u X Mn (u) := E e 1 j =0 tj | F0 = e j =1 Aj +Bn λ0

.

where A0 := 0,

.

Aj := A(j u, Bj −1 , Δ),

B0 := 0,

Bj := B(j u, Bj −1 , Δ).

66

R. Brignone et al.

Proof We provide a proof which is simpler than the one proposed in [33]. We assume for simplicity that .X0 = 0 and .tj − tj −1 = Δ .∀j ∈ {1, n}. By induction, for .n = 2 E[eu(Xt1 +Xt2 ) |F0 ] = E[euXt1 E[euXt2 |F1 ]|F0 ]  = exp A(u, 0, Δ) + A(2u, B(u, 0, Δ), Δ)+  B(2u, B(u, 0, Δ), Δ)λ0 = eA1 +A2 +B2 λ0 .

.

For .n = 3, E[eu(Xt1 +Xt2 +Xt3 ) |F0 ] = E[eu(Xt1 +Xt2 ) E[euXt3 |F2 ]|F0 ]

.

= eA1 E[euXt1 E[e2uXt2 +B1 λt2 |F1 ]|F0 ] eA1 +A2 E[e3Xt1 +B2 λt1 |F0 ] = eA1 +A2 +A3 +B3 λ0 . ⨆ ⨅

Proceeding in the same way for .n > 3 one gets the thesis. We will use these results to price geometric Asian options in Sect. 8.3.

8.2 Exact Simulation of (18)–(19) The simulation of the stochastic jump intensity model in (18)–(19) crucially depends on the simulation of the Hawkes process with exponential kernel. Based on our discussion/numerical results in Sect. 5 we will adopt the exact simulation scheme developed by [21] (see Algorithm 4) for this purpose. Given an initial date .t0 = 0, a final date T , an initial value for .λ0 and the parameters of the model,   the algorithm allows to obtain a sample of the triplet NT NT . NT , {τj } j =1 , {λτj }j =1 , where .τj is the .j −th jump time. Given the triplet, we can compute: λT = θ + (λτNT − θ )e−k(T −τNT ) ,



T

.

λs ds = −

0

λT − λ0 − kθ T − βNT . k

Then .

  XT λT ,

T 0

 T λs ds, {Jj }N j =1 , NT

∼ N(m, s 2 )

(22)

Hawkes Processes in Energy Markets: Modelling, Estimation and Derivatives Pricing

67

T where .{Jj }N j =1 is a sequence of i.i.d. normal (with mean .μJ and standard deviation .σJ ) random variables

 T 

NT σ2  σ2 μJ + 2J .m = r −δ− −1 λs ds + Jx,j , T − e 2 0

s2 = σ 2T .

j =1

(23) Hence, it is possible to simulate .XT given .λ0 exactly and efficiently. Naturally, it is possible to extend the methodology in order to generate paths for log-returns process observed at a discrete time grid, we summarize the procedure in Algorithm 5, which will be used later for pricing exotic derivatives. Algorithm 5 Exact simulation of the model in (18)–(19) Input: X0 , λ0 , r, δ, k, θ , β, σ , μJ , σJ , {t0 , t1 , ..., tn := T } Output: {Xtj }nj =1 1: Set k¯ = 0   NT T 2: Simulate the triplet NT , {τj }N j =1 , {λτj }j =1 using Algorithm 4 3: for j = 1 : n do 4:

Compute number of jumps in the interval: A = #{k¯ : tj −1 < τk¯ ≤ tj }

5:

if A = 0 then

6: 7:

L = 0 and λtj = θ + (λtj −1 − θ)e−k(tj −tj −1 ) else



8:

L=

9:

k¯ = k¯ + A

k¯

βItj −1 0, x ≥ 0 and the .τ -periodic function .p(x) = sin(2π x/τ ). Then 𝔼(Yt ) = 𝔼(L' )

.

2π τ , + 4π 2

λ2 τ 2

Var(Yt ) = Var(L' )

8π 2 , λ3 τ 2 + 16π 2 λ (continued)

Periodic Trawl Processes

81

Cov(Y0 , Yt ) = Var(L' )2π

     e−λt sin 2πτ t τ λ + 4π cos 2πτ t λ(λ2 τ 2 + 16π 2 )

,

Cor(Y0 , Yt ) = e−λt c(t),

with  

 1 2π t 2π t c(t) = sin τ λ + 4π cos , 4π τ τ

where c is a .τ -periodic function.

Next, we consider an example based on the supGamma trawl function, which can allow for both short and long memory.

Example −H  , for Consider the supGamma trawl function defined as .g(x) = 1 + αx .α > 0, H > 1, x ≥ 0. As before, let .p(x) = sin(2π x/τ ). Then Cov(Y0 , Yt ) = Var(L' )



∞

.

0

p(u)p(t + u)g(t + u)du

  2π u 2π(t + u) = Var(L ) sin sin τ τ 0

−H (t + u) 1+ du. α '



∞

−H

∞

∞ H We note that . 0 g(t + u)du = 0 1 + (t+u) du = Hα−1 (t + α)1−H . α According to Proposition 6, there exists a .τ -periodic function c such that

∞

 g(t + u)du t 1−H

= c(t) 1 + .Cor(Y0 , Yt ) = c(t) . ∞ α 0 g(u)du 0

For .H ∈ (1, 2], we are in the long-memory case and for .H > 2 in the shortmemory case.

82

A. E. D. Veraart

Example Let us briefly consider the general case of a superposition trawl. Let 

∞

g(x) =

.

e−λx fλ (λ)dλ,

0

for a density .fλ . If fλ (λ) =

.

1 α H λH −1 e−λα , 𝚪(H )

for .α > 0, H > 1, then 

∞

g(x) =

.

e−λx

0

 1 x −H , α H λH −1 e−λα dλ = 1 + 𝚪(H ) α

i.e. we are in the case of the previous example. Then 

'

∞

Cov(Y0 , Yt ) = Var(L )

.

p(u)p(t + u)g(t + u)du,

0

where 

∞

.

0

p(u)p(t + u)g(t + u)du



=

∞



0

 =

∞

e

−λt

0

 =

∞

p(u)p(t + u)

e−λ(t+u) fλ (λ)dλdu

0



∞

p(u)p(t + u)e−λu dufλ (λ)dλ

0

∞

2π 0

     e−λt sin 2πτ t τ λ + 4π cos 2πτ t λ(λ2 τ 2 + 16π 2 )

fλ (λ)dλ

 ∞ e−λt 2π t fλ (λ)dλ τ (λ2 τ 2 + 16π 2 ) 0

 ∞ e−λt 2π t fλ (λ)dλ. + 8π 2 cos τ λ(λ2 τ 2 + 16π 2 ) 0

= 2π τ sin

(continued)

Periodic Trawl Processes

83

From Proposition 6, we deduce that there exists a .τ -periodic function c such ∞ ∞ that .Cor(Y0 , Yt ) = c(t) 0 g(t + u)du/ 0 g(u)du. Note that  .

∞

∞ ∞

 g(t + u)du =

0

0



∞

=

e−λ(t+u) fλ (λ)dλdu

0

e−λt fλ (λ)

0



∞

e−λu dudλ

0 ∞

=



0

1 −λt e fλ (λ)dλ = 𝔼(e−Λt /Λ), λ

where .Λ is a continuous random variable with density .fλ . Also, 

∞

.

 g(u)du =

0

0

∞

1 fλ (λ)dλ = 𝔼(1/Λ). λ

That is, we have Cor(Y0 , Yt ) = c(t)

.

𝔼(e−Λt /Λ) . 𝔼(1/Λ)

3 Simulation of Periodic Trawl Processes We will now address the question of how a periodic trawl process can be simulated efficiently. A natural first choice might be to develop a grid-based method, using a suitable spatio-temporal grid. However, it turns out that we can adapt a slice-based simulation algorithm which was developed for trawl processes, see [40] and also [37], and extend it to allow for any additional temporal kernel function, which in our case, will be a .τ -periodic function p. The advantage of this slice-based approach is that we can have a coarser, but exact, approximation in the spatial domain, and the simulation error only appears through the discretisation in time, whereas a grid-based approach would result in a discretisation error both in the spatial and the temporal domain.

84

A. E. D. Veraart

Fig. 1 Slices for .n = 4

S11 S12 S13 S14 S15 S21 S22 S23 S24 S31 S32 S33 S41 S42 S51 0

Δn 2Δn 3Δn 4Δn

t n=4

Suppose that we would like to simulate Y on the grid .t0 , . . . , tn with .ti = iΔ for i = 0, . . . , n for .Δ > 0 and .tn = nΔ = T .

.

3.1 Slice-Based Simulation for Trawl Processes The idea behind the slicing, see [40] and [37], is that we consider the partition of ∪ni=0 Ati , where .Ati = {(x, s) : s ≤ t, 0 ≤ x ≤ g(t − s)}, which is obtained when considering all the disjoint sets obtained from the intersections of the various trawl sets. We illustrate this idea in Fig. 1 for the case when .n = 4 and spell out the mathematical details in Algorithm 1. We create an .(n + 1) × (n + 1)-dimensional slice matrix .S = (Sij ). Capital letters refer to slices themselves, which are Borel sets of finite Lebesgue measure, and small letters .s = (sij ) to the Lebesgue measures of the associated slices (.sij = Leb(Sij )). We will draw independent random variables .L(Si,j ) whose dis' tribution has cumulant

function .si,j C(θ ; L ) and simulate periodic trawl processes of the form .Yt = (−∞,t]×ℝ p(t − s)𝕀(0,g(t−s)) (x)L(dx, ds), using the following approximation, for .k = 0, . . . , n,

.

YkΔn =

k+1 

.

j =1

p((k + 2 − j )Δn )

n+2−j 

L(Si,j ).

i=k+2−j

We will now describe the simulation algorithm for periodic trawl processes in more detail by splitting the task at hand into three algorithms.

Periodic Trawl Processes

3.1.1

85

Computing the Matrix of Slices

We only consider slices .Sij (or .sij ) with .j ≤ n + 2 − i and store them in the matrix of slices of the form ⎛

S11 S21 .. .

⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Sn1 S(n+1)1

S12 . . . S22 . . . .. . .. . Sn2 ▭ . . . ▭ ...

⎞ S1n S1(n+1) S2n ▭ ⎟ ⎟ .. ⎟ , . .. . ⎟ ⎟ ... ▭ ⎠ ... ▭

see also Algorithm 1.

Program Code Algorithm 1 Computing the slices .(si,j ), for .i ≤ n + 1, j ≤ n + 2 − i, using the trawl function g, n, and the grid width .Δn as inputs 1: procedure SLICES(g, n, Δn ) 2: b ← numeric(n + 1) ⊳ Create four vectors of zeros 3: c ← numeric(n) 4: d ← numeric(n + 1) 5: e ← numeric(n) 6: for k in 1:(n+1) do

Δ 7: b[k] ← 0 n g(kΔn − x)dx

0 8: d[k] ← −∞ g(x − (k − 1)Δn )dx 9: end for 10: for k in 1:n do 11: c[k] ← b[k] − b[k + 1] 12: e[k] = d[k] − d[k + 1] 13: end for 14: s ← matrix(0, n + 1, n + 1) ⊳ Create slice matrix of zeros 15: for k in 1:n do 16: s[k, 1 : (n + 1 − k)] ← replicate(c[k], n + 1 − k) 17: s[k, (n + 1 − k + 1)] ← b[k] 18: end for 19: s[, 1] ← c(e, d[n + 1]) ⊳ Compute the column of first slices 20: return s ⊳ Return the slice matrix 21: end procedure

86

A. E. D. Veraart

3.1.2

Adding the Weighted Slices

Program Code Algorithm 2 Adding the weighted slices given a slicematrix s = (si,j ), for i ≤ n + 1, j ≤ n + 2 − i and an n-dimensional weight vector w 1: procedure ADDWEIGHTEDSLICES(s, w) 2: n ← nrow(s) − 1 3: x ← numeric(n + 1) ⊳ Create vector of zeros 4: tmp ← 0 5: for k in 0:n do 6: tmp ← 0 7: for j in 1:(k+1) do 8: tmp ← tmp + w[k + 2 − j ] · sum(s[(k + 2 − j ) : (n + 2 − j ), j ]) 9: end for 10: x[1 + k] ← tmp 11: end for 12: return x 13: end procedure

3.1.3

Simulating a Periodic Trawl Process

Program Code Algorithm 3 Simulating a periodic trawl process on the grid 0, Δn , . . . , nΔn given the functions p, g, the distribution of L' , and the grid width Δn and n 1: procedure SIMULATEPERIODICTRAWL(p, g, C(·; L' ), Δn , n) ⊳ Compute the slice-matrix (si,j ) 2: s ← SLICES(g, n, Δn ) 3: L ← matrix(n + 1, n + 1) ⊳ Create matrix of r.v.s L(Sij ) 4: for k in 1:n do 5: L[k, 1 : (n + 1 − k)] ← vector of (n + 1 − k) i.i.d. r.v.s ∼ s[k, 2] · C(·; L' ) 6: L[k, (n + 1 − k + 1)] ← r.v. ∼ s[k, (n + 1 − k + 1)] · C(·; L' ) 7: end for 8: for k in 1:(n+1) do 9: L[k, 1] ← r.v. ∼ s[k, 1] · C(·; L' ) 10: end for 11: w ← numeric(n + 1) ⊳ Create weight vector 12: for k in 1:(n+1) do 13: w[k] ← p(k · Δn ) 14: end for 15: y ← ADDWEIGHTEDSLICES(L, w) 16: return y ⊳ Return Y0 , YΔn , . . . , YnΔn 17: end procedure

Periodic Trawl Processes

87

We note that in the above algorithm, Algorithm 3, the notation “r.v.s .∼ s[k, (n + 1 − k + 1)] · C(·; L' )” is a short-hand notation for random variables whose distribution is characterised by the cumulant function .s[k, (n + 1 − k + 1)] · C(·; L' ). We note that the approximation for the first column of the slice matrix is rather rough. Hence, in practice, we should use a burn-in period in the simulation to minimise the effect of the initial approximation error. The simulation algorithm has been implemented in the R package ambit available on CRAN, see [53].

3.2 A Note on Stochastic Versus Deterministic Seasonality We note that periodic trawl processes are stationary with stochastic seasonality, which is reflected in the time-invariant mean, variance, and autocorrelation, where the autocorrelation function is a periodic function. One might wonder how such periodic trawl processes compare to trawl processes with deterministic seasonality. As before, let L denote a homogeneous Lévy basis with characteristic triplet .(ζ, a, 𝓁), .p : [0, ∞) → ℝ a periodic function with period .τ > 0 and .g : [0, ∞) → ℝ a continuous, monotonically decreasing function. Consider the trawl process  Xt =

𝕀(0,g(t−s)) (x)L(dx, ds)

.

(−∞,t]×ℝ

and the periodic trawl process  Yt =

p(t − s)𝕀(0,g(t−s)) (x)L(dx, ds).

.

(−∞,t]×ℝ

Let .q : [0, ∞) → ℝ a periodic function with period .τ > 0. We consider trawl processes with additive and multiplicative seasonality. Xa (t) := q(t) + Xt ,

Xm (t) := q(t)Xt .

.

Then .Xa , Xm are not stationary and their second-order properties are given by 𝔼(Xta ) = q(t) + 𝔼(L' )



∞

.

Var(Xta ) = Var(L' )



∞ 0

g(u)du = q(t) + 𝔼(X0 )

0

g(u)du = Var(X0 ),

88

A. E. D. Veraart

Cov(X0a , Xta )



'

g(t + u)du = Cov(X0 , Xt ),

0

∞ Cor(X0a , Xta )

∞

= Var(L )

g(t + u)du

∞ = Cor(X0 , Xt ), 0 g(u)du

=

0

for .t ≥ 0. Also, m .𝔼(Xt )

'



= q(t)𝔼(L )

∞

0

Var(Xtm ) = q 2 (t)Var(L' ) Cov(X0m , Xtm ) Cor(X0m , Xtm )

g(u)du = q(t)𝔼(X0 ),



∞

0 '

g(u)du = q 2 (t)Var(X0 ),



∞

= q(0)q(t)Var(L ) =

q(0)q(t) q 2 (t)

∞

0∞ 0

g(t + u)du = q(0)q(t)Cov(X0 , Xt ),

0

g(t + u)du g(u)du

=

q(0) Cor(X0 , Xt ). q(t)

Example Consider the case where .L' ∼ N(0, 1), .p(x) = q(x) = sin(2π x/τ ), for a m .τ = 3, .g(x) = exp(−0.5x). We simulate the processes X, .X , .X , Y and the function q on the grid .ti = iΔn , for .i = 0, . . . , n = 499 and .Δn = 0.1. We visually compare the sample paths and empirical autocorrelation functions of the stationary trawl process X, the nonstationary processes .Xa and .Xm , the stationary periodic trawl process Y and the seasonality function q in Fig. 2, see [54] for the corresponding R code. Here we consider Gaussian processes, and the stochastic noise terms in each simulation of the various paths were kept the same to simplify the comparison. We note that the sample paths and empirical autocorrelation functions of Y and .Xm look very similar, which suggests that it might be hard to distinguish between these two models in practice. Note however that the key probabilistic difference between the two processes is that Y is stationary whereas .Xm is not.

Periodic Trawl Processes

89 1.00

ACF of Xt

4

Xt

2 0 −2

0.75 0.50 0.25 0.00

0

100

200

300

400

0

500

10

1.00

2

0.75

Xt

a

ACF of Xt

a

4

0 −2 0

20

30

20

30

20

30

20

30

0.50 0.25

100

200

300

400

0

500

10

Lag

t

m

3

ACF of Xt

2 1 m

30

0.00

−4

Xt

20

Lag

t

0 −1

1.0 0.5 0.0

−2 −3 0

100

200

300

400

0

500

10

t

Lag

1.0

3

ACF of Yt

2

Yt

1 0 −1 −2

0.5 0.0 −0.5

0

100

200

300

400

500

0

10

Lag

1.0

1.0

0.5

0.5

ACF of q(t)

q(t)

t

0.0 −0.5 −1.0

0.0 −0.5 −1.0

0

100

200

300

t

400

500

0

10

Lag

Fig. 2 Comparing the sample paths and empirical autocorrelation functions of the stationary trawl process X, the non-stationary processes .Xa and .Xm , the stationary periodic trawl process Y and the seasonality function q. (a) Sample path of X. (b) Empirical ACF of X. (c) Sample path of .Xa . (d) Empirical ACF of .Xa . (e) Sample path of .Xm . (f) Empirical ACF of .Xm . (g) Sample path of Y . (h) Empirical ACF of Y . (i) Function .q(). (j) Empirical ACF of .q()

90

A. E. D. Veraart

4 Asymptotic Theory for the Sample Mean, the Sample Autocovariances and the Sample Autocorrelations of Mixed Moving Average Processes Since MMAPs and hence also periodic trawl processes are mixing and, hence, ergodic, see [23], we know that their corresponding moment estimators are consistent. In order to develop a suitable estimation theory for (periodic) trawl processes, we first derive the results in the more general setting of MMAPs. Consider an MMAP given by .Y = (Yt )t∈ℝ for  Yt = μ +

.

ℝ×ℝ

f (x, t − s)L(dx, ds),

for .μ ∈ ℝ, where f satisfies the integrability conditions stated in (7). We are interested in the asymptotic behaviour of the sample mean and sample autocovariances/-correlations for such processes. Note that for all .t ∈ ℝ,  ' .𝔼(Yt ) = μ + 𝔼(L ) f (x, u)dxdu, ℝ×ℝ

and the autocovariance function of Y is denoted by  γ (h) = γf (h) = Cov(Y0 , Yh ) = κ2

.

ℝ×ℝ

f (x, −s)f (x, h − s)dxds,

for any .h ∈ ℝ, where .κ2 = Var(L' ). Suppose that the process Y is sampled over a fixed-time grid of width .Δ > 0 at times .(nΔ)n∈ℕ . The proofs of the following results are presented in section “Proofs of the Asymptotic Theory” in the Appendix.

4.1 Asymptotic Normality of the Sample Mean We denote the sample mean by 1 YiΔ . := n n

Y n;Δ

.

i=1

By adapting [14, Theorem 2.1] to our more general setting, we get the following asymptotic result for the sample mean.

Periodic Trawl Processes

91

Theorem 1 Suppose that .𝔼(L' ) = 0, κ2 = Var(L' ) < ∞, μ ∈ ℝ and .Δ > 0. Further, assume that ⎛ ⎝FΔ : ℝ × [0, Δ] → [0, ∞], (x, u) I→ FΔ (x, u) =

.

∞ 

⎞ |f (x, u + j Δ)|⎠

j =−∞

∈ L2 (ℝ × [0, Δ]).

(9)

 Then . ∞ j =−∞ |γ (Δj )| < ∞, 

∞ 

VΔ :=

γ (Δj ) = κ2

.

j =−∞

ℝ×[0,Δ]

⎞2 ⎛ ∞  ⎝ f (x, u + j Δ)⎠ dxdu,

(10)

j =1

and the sample mean of .YΔi , for .i = 1, . . . , n, is asymptotically Gaussian as .n → ∞, i.e. d √  n Y n;Δ − μ → N (0, VΔ ) ,

.

as n → ∞.

Remark 5 As in [14], we remark that the assumption that .𝔼(L' ) = 0 can be removed if the kernel function f satisfies .f ∈ L1 (ℝ × ℝ) ∩ L2 (ℝ × ℝ).

4.2 Asymptotic Normality of the Sample Autocovariance We define the sample autocovariance function as  γn;Δ (Δh) := n−1

n−h 

(Yj Δ − Y n,Δ )(Y(j +h)Δ − Y n,Δ ),

.

h ∈ {0, . . . , n − 1},

j =1

and the sample autocorrelation function as ρ n;Δ (Δh) =

.

 γn;Δ (Δh) ,  γn;Δ (0)

h ∈ {0, . . . , n − 1}.

In the case when .μ = 0 (and we assumed that .𝔼(L' ) = 0), one could use the following simpler estimators ∗  γn;Δ (Δh) := n−1

n 

.

j =1

Yj Δ Y(j +h)Δ ,

h ∈ {0, . . . , n − 1},

92

A. E. D. Veraart

and ∗ (Δh)  γn;Δ

∗ ρ n;Δ (Δh) =

.

∗ (0)  γn;Δ

,

h ∈ {0, . . . , n − 1}.

Consider the case where .μ = 0, for which we will study the limiting behaviour ∗ (Δp),  ∗ (Δq)). For this, we need a formula for the fourth (joint) of .Cov( γn,Δ γn,Δ moments of the mixed moving average process, which we derive in the following lemma. Lemma 1 Let .L' be a non-zero Lévy seed with .𝔼(L' ) = 0 and .𝔼(L'4 ) < ∞. Let 2 2 '2 ' '4 .κ2 := 𝔼(L ) = Var(L ), .η := 𝔼(L )/κ and .κ4 := (η − 3)κ . Let .Δ > 0, and 2 2 assume that .f ∈ L2 (ℝ × ℝ) ∩ L4 (ℝ × ℝ). Then, for .t1 , t2 , t3 , t4 ∈ ℝ, we have 𝔼(Yt1 Yt2 Yt3 Yt4 )

.

= γ (t1 − t2 )γ (t3 − t4 ) + γ (t1 − t3 )γ (t2 − t4 ) + γ (t1 − t4 )γ (t2 − t3 )  f (x, t1 − t3 + s)f (x, t2 − t3 )f (x, s)f (x, t4 − t3 + s)dxds. + κ4 ℝ×ℝ

The following result is an extension of [14, Proposition 3.1]. Proposition 7 Let .μ = 0. Let .L' be a non-zero Lévy seed with .𝔼(L' ) = 0 and 2 '4 '2 ' '4 .𝔼(L ) < ∞. Let .κ2 := 𝔼(L ) = Var(L ) and .η := 𝔼(L )/κ . Let .Δ > 0, and 2 2 4 assume that .f ∈ L (ℝ × ℝ) ∩ L (ℝ × ℝ) and ⎛ .

⎝ℝ × [0, Δ] → ℝ, (x, u) I→

⎞

∞ 

f 2 (x, u + j Δ)⎠ ∈ L2 (ℝ × [0, Δ]).

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j =−∞

For .q ∈ ℤ, define the function

.

Gq;Δ : ℝ × [0, Δ] → ℝ, (x, u) I→ Gq;Δ (x, u) =

∞ 

 f (x, u + j Δ)f (x, u + (j + q)Δ) ,

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j =−∞

which is in .L2 (ℝ × [0, Δ]) due to (11). Further, assume that ∞  .

j =−∞

|γ (Δj )|2 < ∞.

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Periodic Trawl Processes

93

Then, for each .p, q ∈ ℕ, we have ∗ ∗ lim nCov( γn;Δ (Δp),  γn;Δ (Δq)) = vpq;Δ ,  := (η − 3)κ22 Gp;Δ (x, u)Gq;Δ (x, u)dxdu

n→∞

vpq;Δ .

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ℝ×[0,Δ]

∞ 

+

[γ (lΔ)γ ((l + p − q)Δ) + γ ((l − q)Δ)γ ((l + p)Δ)].

l=−∞

We now state a joint central limit theorem for the sample autocovariance, autocorrelations and their counterparts when .μ = 0. Theorem 2 1. Suppose that the assumptions of Proposition 7 hold and that, in addition, ∞   .

j =−∞

ℝ×ℝ

2 |f (x, u)||f (x, u + j Δ)|dxdu < ∞.

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Then, for each .h ∈ ℕ, we have √ d ∗ ∗ n(γn;Δ (0) − γ (0), . . . , γn;Δ (hΔ) − γ (hΔ))⏉ → N(0, VΔ ),

.

n → ∞,

where the asymptotic covariance matrix is given by .VΔ = (vpq,Δ )p,q=0,...,h ∈ ℝh+1,h+1 with .vpq;Δ defined as in (14). 2. Suppose that the same  assumptions as in 1. hold2and further assume that the function .(x, u) I→ ∞ j =−∞ |f (x, u + j Δ)| is in .L (ℝ × [0, Δ]). Then, for each .h ∈ ℕ, we have √ d n( γn;Δ (0) − γ (0), . . . ,  γn;Δ (hΔ) − γ (hΔ))⏉ → N(0, VΔ ),

.

n → ∞,

where the asymptotic covariance matrix is given by .VΔ = (vpq;Δ )p,q=0,...,h ∈ ℝh+1,h+1 with .vpq;Δ defined as in (14). 3. Suppose that the same assumptions as in 1. hold and that f is not almost everywhere equal to zero. Then, for each .h ∈ ℕ, we have √ d ∗ ∗ n(ρn;Δ (Δ) − ρ(Δ), . . . , ρn;Δ (hΔ) − ρ(hΔ))⏉ → N(0, WΔ ),

.

n → ∞,

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where the asymptotic covariance matrix is given by .WΔ = (wpq;Δ )p,q=0,...,h ∈ ℝh,h with wpq;Δ = (vpq;Δ − ρ(pΔ)v0q;Δ − ρ(qΔ)vp0;Δ + ρ(pΔ)ρ(qΔ)v00;Δ )/γ 2 (0)  (η − 3)κ22 = (Gp;Δ (x, u) − G0;Δ (x, u)ρ(pΔ)) γ 2 (0) ℝ×[0,Δ]

.

· (Gq;Δ (x, u) − G0;Δ (x, u)ρ(qΔ))dxdu +

∞ 

[ρ((l + q)Δ)ρ((l + p)Δ) + ρ((l − q)Δ)ρ((l + p)Δ)

l=−∞

− 2ρ((l + q)Δ)ρ(lΔ)ρ(pΔ) − 2ρ(lΔ)ρ((l + p)Δ)ρ(qΔ) +2ρ(pΔ)ρ(qΔ)ρ 2 (lΔ) . If, in addition, the function .(x, u) I→ [0, Δ]), then we have

∞

j =−∞ |f (x, u

+ j Δ)| is in .L2 (ℝ ×

√ d n( ρn,Δ (Δ) − ρ(Δ), . . . , ρ n,Δ (hΔ) − ρ(hΔ))⏉ → N(0, WΔ ),

.

n → ∞,

with .WΔ defined as above. Remark 6 Note that the assumptions used in the above theorems typically rule out long-memory settings. We discuss the detailed implications in the context of (periodic) trawl processes in the Appendix, see section “Verifying the Assumptions of Theorem 2 for Selected Periodic Trawl Processes”.

5 Inference for Periodic Trawl Processes Using Methods of Moments Based on two working examples, we will now develop an estimation and inference methodology for the parameters determining the serial correlation of (periodic) trawl processes using a method-of-moments approach and present the corresponding asymptotic theory. Our methodology can in principle be applied to general trawl functions, but in order to simplify the exposition, we will concentrate on two particularly relevant specifications, namely the exponential trawl and the supGamma trawl function.

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5.1 Exponential Trawl Function Recall that, in the case of an exponential trawl function .g(x) = exp(−λx), λ > 0, x ≥ 0, the autocorrelation is, for all .t > 0, given by ρ(t) = Cor(Y0 , Yt ) = c(t) exp(−λt),

.

which is equivalent to

λ = − log

.

 ρ(t) /t. c(t)

Setting .t = Δ and assuming (for now) that .c(Δ) is known (for a (non-periodic) trawl process, we would have .c ≡ 1), we get the following estimator  ˆ = − log .λ

∗ (Δ) ρn;Δ



c(Δ)

/Δ,

where .ρ (·) denotes the empirical autocorrelation function. We now establish the corresponding limit theorem. From Theorem 2, assuming the corresponding assumptions hold, we deduce that √ d ∗ n(ρn;Δ (Δ) − ρ(Δ)) → N(0, w11;Δ ).

.

Next, we define the function .φ : (0, ∞) → ℝ, φ(x) := − log(x/c(Δ))/Δ, 1 and set which is continuously differentiable with .φ ' (x) = − xΔ ∗  .λ := φ(ρn;Δ (Δ)).

An application of the delta method, see [13, Proposition 6.4.3], leads to √ √ d ∗ (Δ)) − φ(ρ(Δ))) → N(0, Σλ;Δ ), n( λ − λ) = n(φ(ρn;Δ

.

where Σλ;Δ := φ ' (ρ(Δ))w11;Δ φ ' (ρ(Δ)) =

.

w11;Δ 2 ρ (Δ)Δ2

=

w11;Δ exp(2λΔ) . c2 (Δ)Δ2

Remark 7 In the case of a nonperiodic trawl, we would have .c(Δ) = 1, which simplifies the asymptotic result. For periodic trawls, in the case of unknown ˆ = .c(Δ), a plug-in estimator of the following form could be considered: .λ   − log

∗ (Δ) ρn;Δ  c(Δ)

/Δ.

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Now, suppose that .c(Δ) is not known, but the period .τ of the periodic function is known. I.e. we have that .c(x + τ ) = c(x), for all x. Furthermore, assume that there exists a . τΔ ∈ ℕ such that .τ =  τΔ Δ. To simplify the notation, we shall also define .Δ := (1 +  τΔ ). Then, we have c(Δ Δ) = c(Δ).

.

We can now construct an estimator for the memory parameter based on two empirical autocorrelations. More precisely, note that ρ(Δ) = e−λΔ c(Δ),

ρ(Δ Δ) = e−λΔ Δ c(Δ Δ) = e−λΔ Δ c(Δ),

.

which leaves us with two equations to estimate the two unknown parameters .λ and c(Δ). We can solve both equations for .c(Δ) and obtain

.

c(Δ) = ρ(Δ)eλΔ ,

.

c(Δ) = ρ(Δ Δ)eλΔ Δ .

Setting the two equations equal and solving for .λ leads to λ=

.

 ρ(Δ Δ) 1 log . Δ(1 − Δ ) ρ(Δ)

After .λ has been estimated, we can also estimate .c(lΔ) for .l = 1, . . . , τ˜Δ , using the relation c(lΔ) = ρ(lΔ)eλlΔ .

.

Altogether, we can consider the following estimator  . . . , c( ( λ, c(Δ), τ˜Δ Δ))⏉ 

.

 ∗  ρn;Δ (Δ Δ) 1 := log , ∗ (Δ) Δ(1 − Δ ) ρn;Δ  ∗    ρn;Δ (Δ Δ) l ∗ log ρn;Δ (lΔ) exp ∗ (Δ) (1 − Δ ) ρn;Δ

l=1,...,τ˜Δ

 .

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97

 . . . , c( We note that the vector .( λ, c(Δ), τ˜Δ Δ))⏉ can be represented as a function ∗ (lΔ)) of the empirical autocorrelation functions .(ρn;Δ l=1,...,Δ . As such, we can apply the delta method to derive a joint central limit theorem as follows. More precisely, we have ∗  . . . , c( ( λ, c(Δ), τ˜Δ Δ))⏉ = F ((ρn;Δ (lΔ))l=1,...,Δ )

.

∗ ∗ (lΔ))l=1,...,Δ ), . . . , FΔ ((ρn;Δ (lΔ))l=1,...,Δ ))⏉ , = (F1 ((ρn;Δ

where the functions .F1 , . . . , FΔ are given by

 ρΔ 1 log .F1 (ρ1 , . . . , ρΔ ) = F1 (ρ1 , ρΔ ) = , ρ1 Δ(1 − Δ ) F2 (ρ1 , . . . , ρΔ ) = F2 (ρ1 , ρΔ ) = ρ1 exp(ΔF1 (ρ1 , ρΔ )), F3 (ρ1 , . . . , ρΔ ) = F3 (ρ1 , ρ2 , ρΔ ) = ρ2 exp(2ΔF1 (ρ1 , ρΔ )), F4 (ρ1 , . . . , ρΔ ) = F4 (ρ1 , ρ3 , ρΔ ) = ρ3 exp(3ΔF1 (ρ1 , ρΔ )), .. . FΔ (ρ1 , . . . , ρΔ ) = FΔ (ρ1 , ρΔ −1 , ρΔ ) = ρΔ −1 exp((Δ − 1)ΔF1 (ρ1 , ρΔ )). Under the assumptions of Theorem 2, √  . . . , c( n(( λ, c(Δ), τ˜Δ Δ))⏉ − (λ, c(Δ), . . . , c(Δ Δ))

.

converges to a multivariate normal random vector with zero mean and variance given by .DWΔ D ' , where .WΔ is the .Δ × Δ -matrix whose elements are the ones defined in Theorem 2. The matrix D is a .Δ × Δ -matrix ⏉ .[(∂Fi /∂ρj )(ρ(Δ), . . . , ρ(Δ Δ)) ], where .

∂F1 1 ∂F1 (ρ1 , ρΔ ) = , (ρ1 , . . . , ρΔ ) = ∂ρ1 Δ(Δ − 1)ρ1 ∂ρ1 ∂F1 ∂F1 (ρ1 , . . . , ρΔ ) = 0, (ρ1 , . . . , ρΔ ) = · · · = ∂ρΔ −1 ∂ρ2

∂F1 1 ∂F1 (ρ1 , ρΔ ) = , (ρ1 , . . . , ρΔ ) = ∂ρΔ Δ(1 − Δ )ρΔ ∂ρΔ ∂F2 Δ ∂F2 exp(ΔF1 (ρ1 , ρΔ )), (ρ1 , ρΔ ) = (ρ1 , . . . , ρΔ ) = ∂ρ1 Δ − 1 ∂ρ1 ∂F2 ∂F2 (ρ1 , . . . , ρΔ ) = 0, (ρ1 , . . . , ρΔ ) = · · · = ∂ρΔ −1 ∂ρ2

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∂F2 ρ1 ∂F2 1 (ρ1 , . . . , ρΔ ) = (ρ1 , ρΔ ) = exp(ΔF1 (ρ1 , ρΔ )), ∂ρΔ ∂ρΔ 1 − Δ ρΔ ∂F3 ∂F3 2 ρ2 (ρ1 , . . . , ρΔ ) = (ρ1 , ρ2 , ρΔ ) = exp(2ΔF1 (ρ1 , ρΔ )), ∂ρ1 ∂ρ1 Δ − 1 ρ1 ∂F3 ∂F3 (ρ1 , . . . , ρΔ ) = (ρ1 , ρ2 , ρΔ ) = exp(2ΔF1 (ρ1 , ρΔ )), ∂ρ2 ∂ρ2 ∂F3 ∂F3 (ρ1 , . . . , ρΔ ) = · · · = (ρ1 , . . . , ρΔ ) = 0, ∂ρ3 ∂ρΔ −1 ∂F3 ρ2 ∂F3 2 (ρ1 , . . . , ρΔ ) = (ρ1 , ρ2 , ρΔ ) = exp(2ΔF1 (ρ1 , ρΔ )), ∂ρΔ ∂ρΔ 1 − Δ ρΔ .. . ∂FΔ ∂FΔ (ρ1 , . . . , ρΔ ) = (ρ1 , ρΔ −1 ρΔ ) ∂ρ1 ∂ρ1 ρ −1 = Δ exp((Δ − 1)ΔF1 (ρ1 , ρΔ )), ρ1 ∂FΔ ∂FΔ (ρ1 , . . . , ρΔ ) = · · · = (ρ1 , . . . , ρΔ ) = 0, ∂ρ2 ∂ρΔ −2 ∂FΔ ∂FΔ (ρ1 , . . . , ρΔ ) = (ρ1 , ρΔ −1 , ρΔ ) = exp((Δ − 1)ΔF1 (ρ1 , ρΔ )), ∂ρΔ −1 ∂ρΔ −1 ∂FΔ ∂FΔ (ρ1 , . . . , ρΔ ) = (ρ1 , ρΔ −1 , ρΔ ) ∂ρΔ ∂ρΔ ρ −1 = − Δ exp((Δ − 1)ΔF1 (ρ1 , ρΔ )). ρΔ

5.2 SupGamma Trawl Function Let us now focus on the case of a supGamma trawl function that allows for both short- and long-memory settings. We note that the inference for the long-memory case appears numerically unstable when approximating the corresponding integrals appearing in the autocorrelation function (in simulation experiments not reported here). Hence we suggest, as before, using the mean value theorem result for inference on the memory

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99

 −H parameter. Let .g(x) = 1 + αx , for .α > 0, H > 1. As before, there exists a .τ -periodic function c such that, for .t ≥ 0,

t .ρ(t) = Cor(Y0 , Yt ) = c(t) 1 + α

1−H .

For .H ∈ (1, 2], we are in the long-memory case and for .H > 2 in the short-memory case. We are interested in estimating the Hurst index H and therefore, in the following, assume that the parameter .α is known. Using the analogous procedure as in the short-memory case, we can write that, for all .t ≥ 0,

 ρ(t) ρ(t) . = exp((1 − H ) log(1 + t/α)) ⇔ log = (1 − H ) log(1 + t/α). c(t) c(t) This suggests the following estimator of the Hurst index:  .

 = 1 − log H

∗ (Δ) ρn;Δ

c(Δ)

 / log(1 + Δ/α),

assuming that .c(Δ) and .α are known.   x Now, define the function .φ : (0, ∞) → ℝ, φ(x) := 1 − log c(Δ) / log(1 + 1 1 Δ/α), which is continuously differentiable with .φ ' (x) = − log(1+Δ/α) x and set .

 := φ(ρ ∗ (Δ)). H n;Δ

An application of the delta method, see [13, Proposition 6.4.3], leads to √ √ d  − H ) = n(φ(ρ ∗ (Δ)) − φ(ρ(Δ))) → n(H N(0, ΣH ;Δ ), n;Δ

.

where w11;Δ (log(1 + Δ/α))2 ρ 2 (Δ)  2H −2 w11;Δ 1 + Δ w11;Δ α = 2−2H = (log(1 + Δ/α))2 c2 (Δ) .  (log(1 + Δ/α))2 c2 (Δ) 1 + Δ

ΣH ;Δ := φ ' (ρ(Δ))w11;Δ φ ' (ρ(Δ)) =

.

α

Now suppose that .τ is known and proceed as in the exponential case: Assume that there exists a . τΔ ∈ ℕ such that .τ =  τΔ Δ and set .Δ := (1 +  τΔ ). Then, we have c(Δ Δ) = c(Δ).

.

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We can now construct an estimator for the memory parameter based on two empirical autocorrelations. More precisely, note that ρ(Δ) = (1 + Δ/α)H −1 c(Δ),

.

ρ(Δ Δ) = (1 + Δ Δ/α)H −1 c(Δ Δ) = (1 + Δ Δ/α)H −1 c(Δ), which leaves us with two equations to estimate the two unknown parameters H and c(Δ), assuming that .α is known. We can solve both equations for .c(Δ) and obtain

.

c(Δ) = ρ(Δ)(1 + Δ/α)1−H ,

.

c(Δ) = ρ(Δ Δ)(1 + Δ Δ/α)1−H .

Setting the two equations equal and solving for H leads to

H = 1 + log

.



 α+Δ ρ(Δ Δ) / log . ρ(Δ) α + Δ Δ

After H has been estimated, we can also estimate .c(lΔ) for .l = 1, . . . , τ˜Δ , using the relation c(lΔ) = ρ(lΔ)(1 + lΔ/α)1−H .

.

Altogether, we can consider the following estimator  . . . , c( , c(Δ), (H τ˜Δ Δ))⏉  

.

:= 1 + log ⎛

∗ ( Δ) ρn;Δ Δ ∗ (Δ) ρn;Δ

⎝ρ ∗ (lΔ)(1 + lΔ/α) n;Δ



/ log

 α+Δ , α + Δ Δ

∗   ⎞ ρ (Δ Δ) α+Δ log n;Δ / log ∗ α+ Δ ρ (Δ) n;Δ

Δ

⎞

⎠

⎠. l=1,...,τ˜Δ

 . . . , c( , c(Δ), We note that the vector .(H τ˜Δ Δ))⏉ can be represented as a ∗ (lΔ)) function of the empirical autocorrelation functions .(ρn;Δ l=1,...,Δ . That is, we have ∗  . . . , c( , c(Δ), (H τ˜Δ Δ))⏉ = F ((ρn;Δ (lΔ))l=1,...,Δ )

.

∗ ∗ (lΔ))l=1,...,Δ ), . . . , FΔ ((ρn;Δ (lΔ))l=1,...,Δ ))⏉ , = (F1 ((ρn;Δ

Periodic Trawl Processes

101

where the functions .F1 , . . . , FΔ are given by

F1 (ρ1 , . . . , ρΔ ) = F1 (ρ1 , ρΔ ) = 1 + log

.

ρΔ ρ1



/ log

 α+Δ , α + Δ Δ

F2 (ρ1 , . . . , ρΔ ) = F2 (ρ1 , ρΔ ) = ρ1 (1 + Δ/α)1−F1 (ρ1 ,ρΔ ) F3 (ρ1 , . . . , ρΔ ) = F3 (ρ1 , ρ2 , ρΔ ) = ρ2 (1 + 2Δ/α)1−F1 (ρ1 ,ρΔ ) , F4 (ρ1 , . . . , ρΔ ) = F4 (ρ1 , ρ3 , ρΔ ) = ρ3 (1 + 3Δ/α)1−F1 (ρ1 ,ρΔ ) , .. . FΔ (ρ1 , . . . , ρΔ ) = FΔ (ρ1 , ρΔ −1 , ρΔ ) = ρΔ −1 (1 + (Δ − 1)Δ/α)1−F1 (ρ1 ,ρΔ ) . Under the assumptions of Theorem 2 √  . . . , c( , c(Δ), n((H τ˜Δ Δ))⏉ − (H, c(Δ), . . . , c(Δ Δ))

.

converges to a multivariate normal random vector with zero mean and variance given by .DWΔ D ' , where .WΔ is the .Δ × Δ -matrix whose elements are the ones defined in Theorem 2. The matrix D is a .Δ × Δ -matrix ⏉ .[(∂Fi /∂ρj )(ρ(Δ), . . . , ρ(Δ Δ)) ]. Remark 8 Note that the assumptions of Theorem 2 are not satisfied in the case of a long-memory supGamma trawl function, i.e. when .H ∈ (1, 2], see section “Verifying the Assumptions of Theorem 2 for Selected Periodic Trawl Processes” in the Appendix.

6 Inference for Periodic Trawl Processes Using a Generalised-Method-of-Moments Approach In the previous section, we showed how the memory parameter and periodic function of a periodic trawl process can be inferred if the period is known. More generally, if we would like to estimate all model parameters, including the parameters of the Lévy basis, we can proceed by using the generalised method of moments (GMM). Hence, we will now develop the asymptotic theory for estimating the parameters of a periodic trawl process using a generalised method of moments (GMM). This extends the work presented in [11] for (integer-valued) trawl processes to the case of periodic trawl processes.

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Recall that we denote the periodic trawl process by  Yt =

.

ℝ×ℝ

p(t − s)𝕀(0,g(t−s)) (x)𝕀[0,∞) (t − s)L(dx, ds),

and we denote the corresponding trawl process, where .p ≡ 1, by  Xt =

.

ℝ×ℝ

𝕀(0,g(t−s)) (x)𝕀[0,∞) (t − s)L(dx, ds).

(16)

6.1 Weak Dependence We will first show that periodic trawl processes are .θ -weakly dependent, see [16, Definition 3.2]. We note that a periodic trawl process is a special case of a causal mixed moving average process as defined in [16, Definition 3.3]. Hence, under the assumption that

2 . |ξ |>1 |ξ | 𝓁(dξ ) < ∞, we can deduce from [16, Corollary 3.4] that the periodic trawl process is .θ -weakly dependent, see [16, Definition 3.2] with coefficient, for .r ≥ 0,

 ' .θY (r) = Var(L ) p2 (−s)𝕀2(0,g(−s)) (x)𝕀2[0,∞) (−s)dxds (−∞,−r)×ℝ

   + 𝔼(L' )

(−∞,−r)×ℝ

 '



= Var(L )  = Var(L' )

−r −∞



∞ r

2 1/2  p(−s)𝕀(0,g(−s)) (x)𝕀[0,∞) (−s)dxds  '

p (−s)g(−s)ds + (𝔼(L )) 2

p2 (s)g(s)ds + (𝔼(L' ))2

 2

−∞



2 1/2

−r

p(−s)g(−s)ds 2 1/2

∞

p(s)g(s)ds

.

r

Since the periodic function is continuous and bounded, we note that the weakθY such that .θY (r) ≤  dependence coefficient can be bounded by a function . θY (r) for all .r ≥ 0, where 1/2   θY (r) = c1 Cov(X0 , Xr ) + c2 (𝔼(L' ))2 (Cov(X0 , Xr ))2 ,

.



where .𝔼(L' ) = γ + |ξ |>1 𝓁(dξ ), and .Var(L' ) = a + ℝ ξ 2 𝓁(dξ ) and .c1 , c2 > 0 are constants. Let us briefly consider the special case when .L' is of finite variation, i.e. when the characteristic triplet is given by .(γ , 0, 𝓁) with . ℝ |ξ |𝓁(ξ ) < ∞. This case is

Periodic Trawl Processes

103

of interest since it includes the well-known class of integer-valued trawl processes. Here, the weak dependence coefficient is, for .r ≥ 0, given by 

 θY (r) =

|p(−s)𝕀(0,g(−s)) (x)𝕀[0,∞) (−s)ξ |𝓁(dξ )dxds

.

(−∞,−r)×ℝ ℝ

 +

(−∞,−r)×ℝ

 =

ℝ

|p(−s)𝕀(0,g(−s)) (x)𝕀[0,∞) (−s)γ0 |dxds 

|ξ |𝓁(dξ )

−∞

 =

ℝ

−r

 |p(−s)|g(−s)ds + |γ0 | 

∞

|ξ |𝓁(dξ ) + |γ0 |

−r

−∞

|p(−s)|g(−s)ds

|p(s)|g(s)ds.

r

As before, we note that the weak-dependence coefficient can be bounded by a function . θY (r) for all .r ≥ 0, where θY such that .θY (r) ≤   θY (r) = cCov(X0 , Xr ),

.

for a positive constant .c > 0 and .γ0 = γ −



|ξ |≤1 ξ 𝓁(dξ ).

Remark 9 We note that the .θ-dependence coefficient of the periodic trawl process Y can be related to the .θ-dependence coefficient of X via θY (r) ≤  θY (r) = O(θX (r)).

.

6.2 GMM Estimation for Periodic Trawl Processes In this section, we will describe how the parameters of a periodic trawl process can be estimated via the generalised method of moments (GMM). As before, we consider the equidistantly sampled process .YΔ , Y2Δ , . . . , YnΔ , for .Δ = T /n > 0, T > 0, n ∈ ℕ. We will consider a GMM estimator, which is based on the sample mean, sample variance and sample autocovariances up to lag .m ≥ 2. Consider the vector Yt(m) = (YtΔ , Y(t+1)Δ , . . . , Y(t+m)Δ ),

.

for .t = 1, . . . , n − m. We denote by .Θ the parameter space of the periodic trawl process and set .μ := μ(θ ) = 𝔼(Y0 ) and .D(k) := D(k, θ ) := 𝔼(Y0 YkΔ ), for .k = 0, . . . , m. So, as soon as we specify a parametric model for Y , then .D(k) is a function of the model parameter(s) .θ.

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Next, we define the measurable function .h : ℝm+1 × Θ → ℝm+2 by ⎛

⎞ ⎛ ⎞ (m) hE (Yt , θ ) YtΔ − μ(θ ) ⎜ ⎟ (m) 2 − D(0, θ ) ⎟ YtΔ ⎜ h0 (Yt , θ ) ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ (m) ⎜ ⎟ (m) ⎜ ⎟ Y − D(1, θ ) Y (Y , θ ) h tΔ (t+1)Δ = .h(Yt , θ) = ⎜ 1 t ⎜ ⎟. ⎟ ⎜ ⎟ .. .. ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ . . (m) YtΔ Y(t+m)Δ − D(m, θ ) hm (Yt , θ ) Moreover, define the corresponding sample moments as ⎛ gn,m (θ ) =

.

⎜ n−m ⎜ 1  (m) h(Yt , θ ) = ⎜ ⎜ n−m ⎝ t=1

⎞

(m) 1 n−m n−m t=1 hE (Yt , θ ) ⎟ (m) n−m 1 t=1 h0 (Yt , θ ) ⎟ n−m ⎟

.. ⎟. ⎠ . (m) 1 n−m h (Y , θ ) t=1 m t n−m

Suppose that the true parameter (vector) is denoted by .θ0 , say, which can be estimated by minimising the objective function of the GMM, i.e. the GMM estimator is given by n,m  θ0,GMM = argmingn,m (θ )⏉ An,m gn,m (θ ),

.

(17)

for a positive-definite weight matrix .An,m . We shall now derive the weak consistency and asymptotic normality of the GMM estimator under suitable (standard) assumptions. First, we present the assumptions which guarantee the weak consistency of the estimator, cf. [39, Assumptions 1.1–1.3]. Assumption 1 (m)

(i) Suppose that the expectation .𝔼(h(Yt , θ )) exists and is finite for all .θ ∈ Θ and for all t. (m) (m) (ii) Set .h(m) t (θ ) = 𝔼(h(Yt , θ )). There exists a .θ0 ∈ Θ such that .ht (θ ) = 0 for all t if and only if .θ = θ0 . n−m (m) Assumption 2 Let .h(m) (θ ) = t=1 ht (θ ) and denote the j th component of (m) the .m + 2-dimensional vectors .h(m) (θ ) and .gn,m (θ ) by .hj (θ ) and .gn,m;j (θ ), respectively. Suppose that, for .j = 1, . . . , m + 2, as .n → ∞, .

ℙ

sup |h(m) j (θ ) − gn,m;j (θ )| → 0.

θ∈Θ

Assumption 3 There exists a sequence of non-random, positive definite matrices ℙ

An,m such that, as .n → ∞, .|An,m − An,m | → 0

.

Periodic Trawl Processes

105

From [39, Theorem 1.1], we deduce the following result. Theorem 3 Assume that Assumptions 1, 2, 3 hold. Then the GMM estimator n,m  defined in (17) is weakly consistent. θ0,GMM

.

Next, we formulate the assumptions needed for the central limit theorem. Assumption 4 .Θ is a compact parameter space which includes the true parameter θ0 .

.

Assumption 5 The weight matrix .An,m converges in probability to a positive definite matrix A. Assumption 6 The covariance matrix .Σa defined in (18) below is positive definite. Theorem 4 Consider a periodic trawl process Y with characteristic triplet .(ζ, a, 𝓁)

and suppose that . |ξ |>1 |ξ |4+δ 𝓁(dξ ) < ∞, for some .δ > 0 and suppose that the .θ −α weak dependence  coefficient  of the periodic trawl process satisfies .θY (r) ≤ O(r ), 1 for .α > 1 + 1δ 1 + 2+δ . Suppose that Assumptions 1, 4, 5, 6 hold. Then, as .n → ∞, √ n,m d n( θ0,GMM − θ0 ) → N(0, MΣa M ⏉ ),

.

where Σa =



.

(m)

l∈ℤ

M=

(m)

Cov(h(Y0 , θ0 ), h(Yl

−1 ⏉ (G⏉ 0 AG0 ) G0 A,

(18)

, θ0 )), 

(m)

∂h(Yt , θ ) where G0 = e ∂θ ⏉

 . θ=θ0

Proof (Proof of Theorem 4) The proof follows, with very minor modifications, the steps of the proof presented for trawl processes in [11], which is based on the arguments of the proofs of [39, Theorem 1.2], see also [16, Proof of Theorem 6.2] for the case of a supOU process. ⨆ ⨅ Remark 10 We note that the assumption on the .θ -weak dependence index rule out long-memory settings, see [11] for a more detailed discussion in the case of trawl processes.

7 Empirical Illustration on Electricity Day-Ahead Prices In this section, we will illustrate how the proposed methodology for estimating the kernel function of periodic trawl processes developed in Sect. 5 can be used in practice, see [54] for the corresponding data and R code.

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Fig. 3 Time series plot of the day-ahead electricity baseload prices for Germany and Luxembourg from 1st October 2018 to 1st January 2023 recorded in EUR/MWh Price

600

400

200

0 2019

2020

2021

2022

2023

Time

We consider day-ahead electricity baseload prices for Germany and Luxembourg from 1st October 2018 to 1st January 2023 recorded in EUR/MWh. The data have been downloaded from the website https://www.smard.de/en, which is maintained by the Bundesnetzagentur in Germany. The time series is depicted in Fig. 3. Given the turmoil experienced by the electricity market in recent years, a single stationary stochastic process model would not fit such data appropriately. Hence, we rather split the dataset into two parts, representing the relatively calm period from 01.10.2018–31.12.2020 (823 observations, referred to as TS1) and the volatile period from 01.01.2021–01.01.2023 (731 observations, referred to as TS2). We remark that more sophisticated methods, e.g. from change-point-detection, could be applied to split the dataset. Figure 4 depicts the time series, the empirical autocorrelation functions and the empirical densities of both time series. We note a rather slowly decaying serial correlation with a distinct weakly periodic pattern. The marginal distributions are rather different for the calm and the volatile regime: In the calm regime, we observe the well-known fact of a slightly skewed distribution, which has slightly heavier tails than the normal distribution and includes negative prices. For the volatile regime, the distribution appears to be skewed, multimodal, heavy-tailed and with significantly larger empirical mean and variance than in the calm regime. We will now focus on estimating the kernel function of a periodic trawl process. First, we consider the relatively calm period from 01.10.2018–31.12.2020 and estimate an exponential trawl function with period .τ = 7. We set .Δ = 1, = 0.055 and .λ representing one day. We obtain the following estimates:   c(1),  . . . , c(6))  = (1.000, 0.705, 0.496, 0.451, 0.445, 0.490, 0.642). The .(c(0), empirical and fitted autocorrelation functions are depicted in Fig. 5a. We note that the exponential decay appears to be too fast to capture the empirical autocorrelation function well. Hence we also fit a supGamma periodic trawl function. We first ran a GMM estimation on the empirical autocorrelation function to estimate .α and then set .α = 1.2. We then employ our method of  = 1.264, indicating a long-memory regime, moment estimator and obtain .H

Periodic Trawl Processes

107

600

Price

Price

50

0

400

200

0 2021−01 2021−07 2022−01 2022−07 2023−01

2019−01 2019−07 2020−01 2020−07 2021−01

Time

Time

1.00

Autocorrelation

Autocorrelation

1.00 0.75 0.50 0.25

0.75 0.50 0.25 0.00

0.00 0

25

50

75

100

0

25

50

Lag

75

100

Lag

0.006

Density

Density

0.03

0.02

0.004

0.002

0.01

0.00

0.000 −50

0

50

Price

0

200

400

600

Price

Fig. 4 Exploratory data analysis of the day-ahead electricity baseload prices for Germany and Luxembourg from 1st October 2018 to 1st January 2023 recorded in EUR/MWh, which has been split into two times series (TS1 and TS2). (a) TS1: Daily prices from 01.10.2018 to 31.12.2020. (b) TS2: Daily prices from 01.01.2021 to 01.01.2023. (c) Empirical ACF of TS1. (d) Empirical ACF of TS2. (e) Empirical density and scaled histogram of TS1. (f) Empirical density and scaled histogram of TS2

 c(1),  . . . , c(6))  = (1.000, 0.789, 0.582, 0.539, 0.534, 0.583, 0.752). We and .(c(0), note that the model fit looks very good, see Fig. 5b. Note that the sensitivity to the particular choice of .α appeared low, but this needs to be investigated in more detail in the future.

108

A. E. D. Veraart 1.00

Autocorrelation

Autocorrelation

1.00 0.75 0.50 0.25 0.00

0.50 0.25 0.00

0

25

50

75

100

0

25

50

Lag

Lag

(a)

(b)

1.00

75

100

75

100

1.00

Autocorrelation

Autocorrelation

0.75

0.75 0.50 0.25 0.00

0.75 0.50 0.25 0.00

0

25

50

75

100

0

25

50

Lag

Lag

(c)

(d)

Fig. 5 Empirical and fitted exponential-periodic and supGamma periodic trawl for the time series TS1 and TS2. (a) Exponential-periodic trawl for TS1. (b) SupGamma-periodic trawl for TS1. (c) Exponential-periodic trawl for TS2. (d) SupGamma-periodic trawl for TS2

We repeat the analysis on the more volatile time period from 01.01.2021– 01.01.2023. For the exponential-periodic trawl process we obtain the estimates  .λ =  c(1),  . . . , c(6))  = (1.000, 0.689, 0.473, 0.421, 0.406, 0.437, 0.558), 0.032 and .(c(0), which results in a decay which is too fast, see Fig. 5c. Again, the fit of the supGamma-periodic trawl model appears better, where we set .α = 5.297  = 1.298, which as (based on a GMM-estimation) and then find that .H  c(1),  . . . , c(6))  before falls into the long-memory setting, and .(c(0), = (1.000, 0.958, 0.901, 0.887, 0.888, 0.912, 0.957), see Fig. 5d. This small illustration demonstrates how easily the kernel function of a periodic trawl process can be fitted to periodic autocorrelation functions. In future work, it will be interesting to further extend and fine-tune the methodology, in particular also the GMM methodology derived in Sect. 6.2 to also estimate the parameters of the driving Lévy noise (under suitable identifiability conditions) in practical settings. Here, a multistep approach, where the parameters of the kernel functions are inferred first (as in this article), followed by the parameters of the Lévy basis, might be a useful direction to investigate further. We also remark that in cases when the period is not known, preliminary simulation experiments not reported here

Periodic Trawl Processes

109

indicate that the period can be estimated using a smoothed empirical periodogram, which is in line with existing literature. It will also be worthwhile to explore extensions beyond the stationary setting, by for instance considering a linear combination of an additive seasonal function and a periodic trawl process to allow for non-stationarities in the mean.

8 Conclusion and Outlook This article introduced an alternative definition of periodic trawl processes compared to the one given in [9], which appears to have slightly higher analytical tractability. We derived some of the key probabilistic properties of periodic trawl processes and have studied relevant examples including both short- and longmemory settings. We showed that a slice method can be used to simulate periodic trawl processes effectively. Under suitable technical conditions, which currently only cover short-memory scenarios, we have proved the asymptotic normality of the sample mean, sample autocovariances, sample autocorrelations and the GMMestimator. We have implemented the proposed methodology in R and showed that the serial correlation of periodic trawl processes describes the empirical one of electricity spot prices very well. These are promising results, which suggest that it will be worthwhile to further advance the theory and statistical methodology for periodic trawl processes. For instance, for the GMM approach to be applicable in practice, we need to specify suitable identifiability conditions for fully parametric settings. It will also be interesting to account for multiple periods, for instance, including weekly and yearly periodicities, and tailor methods from spectral analysis to such settings. Moreover, model selection tools are needed for choosing the appropriate Lévy basis. For the particular application considered here, we note that our framework allows for a smooth (Gaussian) noise terms as well as for jumps, where the latter can model the positive and even more dominant negative jumps, as in [55], but also the more volatile behaviour observed in the most recent electricity prices. That said, in future research, we should also investigate whether including a stochastic volatility term, either via temporal or spatial stochastic scaling of the periodic trawl process is needed to describe the data even better, which would bring us into the general framework of ambit fields and processes, see [10]. From a theoretical point of view, it will be interesting to develop an asymptotic theory for periodic trawl processes which can allow for long memory settings. We would then need to check how well such an asymptotic theory works in finite samples to decide whether confidence bounds for the obtained parameters can be constructed based on asymptotic guarantees or whether bootstrap approaches might be more useful in practice. Acknowledgments I would like to thank Paul Doukhan for suggesting a study of periodic trawl processes and for helpful discussions, as well as Michele Nguyen, Fred Espen Benth and an anonymous referee for commenting on an earlier version of this article.

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Appendix The appendix contains the proofs of all the technical results presented in the main paper, additional examples and a discussion of when the technical assumptions needed in our main theorems hold for periodic trawl processes.

Proof of the Second Order Properties First, we derive the joint characteristic/cumulant function. Proposition 8 Let .t1 < t2 and .θ1 , θ2 ∈ ℝ. Then Log(𝔼(exp(i(θ1 , θ2 )(Yt1 , Yt2 )⏉ ))) = Log(𝔼(exp(iθ1 Yt1 + iθ2 Yt2 )))  CL' (θ1 p(t1 − s)𝕀(0,g(t1 −s)) (x) + θ2 p(t2 − s)𝕀(0,g(t2 −s)) (x))dxds =

.

 +

(−∞,t1 ]×ℝ

(t1 ,t2 ]×ℝ

CL' (θ2 p(t2 − s)𝕀(0,g(t2 −s)) (x))dxds,

where .CL' denotes the cumulant function of the Lévy seed .L' associated with the Lévy basis L. Proof Let .t1 < t2 and .θ1 , θ2 ∈ ℝ. Then the joint characteristic function is given by 𝔼(exp(i(θ1 , θ2 )(Yt1 , Yt2 )⏉ )) = 𝔼(exp(iθ1 Yt1 + iθ2 Yt2 )) 

 p(t1 − s)𝕀(0,g(t1 −s)) (x)L(dx, ds) = 𝔼 exp iθ1

.

(−∞,t1 ]×ℝ

 + iθ2

(−∞,t1 ]×ℝ



+iθ2

(t1 ,t2 ]×ℝ

p(t2 − s)𝕀(0,g(t2 −s)) (x)L(dx, ds)  p(t2 − s)𝕀(0,g(t2 −s)) (x)L(dx, ds)



 = 𝔼 exp i

(−∞,t1 ]×ℝ

{θ1 p(t1 − s)𝕀(0,g(t1 −s)) (x)

+ θ2 p(t2 − s)𝕀(0,g(t2 −s)) (x)}L(dx, ds)   p(t2 − s)𝕀(0,g(t2 −s)) (x)L(dx, ds) +iθ2 (t1 ,t2 ]×ℝ

 = exp

(−∞,t1 ]×ℝ

C(θ1 p(t1 − s)𝕀(0,g(t1 −s)) (x)

Periodic Trawl Processes

111

+θ2 p(t2 − s)𝕀(0,g(t2 −s)) (x); L' )dxds

  · exp C(θ2 p(t2 − s)𝕀(0,g(t2 −s)) (x); L' )dxds . (t1 ,t2 ]×ℝ

I.e. .

log(𝔼(exp(iθ1 Yt1 + iθ2 Yt2 )))  = C(θ1 p(t1 − s)𝕀(0,g(t1 −s)) (x) + θ2 p(t2 − s)𝕀(0,g(t2 −s)) (x); L' )dxds  +

(−∞,t1 ]×ℝ

(t1 ,t2 ]×ℝ

C(θ2 p(t2 − s)𝕀(0,g(t2 −s)) (x); L' )dxds. ⨆ ⨅

We can now easily derive the second-order properties of the periodic trawl process: Proof (Proof of Proposition 5) For .t, t1 , t2 ∈ ℝ, t1 < t2 , we have 𝔼(Yt ) = 𝔼(L' )



t

p(t − s)g(t − s)ds = 𝔼(L' )

.

−∞  ∞

Var(Yt ) = Var(L' )



∞

p(u)g(u)du, 0

p2 (u)g(u)du,

0

  ∂2 Cov(Yt1 , Yt2 ) = − log(𝔼(exp(iθ1 Yt1 + iθ2 Yt2 ))) ∂θ1 ∂θ2 θ1 =θ2 =0  t1 = Var(L' ) p(t1 − s)p(t2 − s) min(g(t1 − s), g(t2 − s))ds,

t1 Cor(Yt1 , Yt2 ) =

−∞

−∞ p(t1

− s)p(t2 − s) min(g(t1 − s), g(t2 − s))ds

∞ . 2 0 p (u)g(u)du

Recall that we assume that g is monotonically decreasing, i.e. if .x ≤ y, then .g(x) ≥ g(y). Since .t1 < t2 , we have .t1 − s < t2 − s for .s < t1 and .

min(g(t1 − s), g(t2 − s)) = g(t2 − s),

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A. E. D. Veraart

hence the above expressions simplify to Cov(Yt1 , Yt2 ) = Var(L' )



t1

.

= Var(L' )

∞ Cor(Yt1 , Yt2 ) =

0

−∞  ∞

p(t1 − s)p(t2 − s)g(t2 − s)ds p(u)p(t2 − t1 + u)g(t2 − t1 + u)du,

0

p(u)p(t2 − t1 + u)g(t2 − t1 + u)du

∞ . 2 0 p (u)g(u)du ⨆ ⨅

Proof (Proof of Proposition 6) Recall that

∞ Cor(Y0 , Yt ) =

.

0

p(u)p(t + u)g(t + u)du

∞ . 2 0 p (u)g(u)du

We consider a constant .M > τ . Since p is periodic with period .τ , there exist .ξ1 , ξ2 ∈ [0, τ ] such that, 



M

M

p(u)p(t + u)g(t + u)du = p(ξ1 )p(ξ1 + t)

.

0



M

 (p(u))2 g(u)du = (p(ξ2 ))2

0

g(t + u)du,

0 M

g(t + u)du,

0

by the mean value theorem. We note that p is assumed to be continuous, and since it is also periodic, it is bounded. Also, the integrability conditions in (7) guarantee the existence of the integrals when taking the limit as .M → ∞. Taking the limit and setting .c(t) = p(ξ1 )p(ξ1 + t)/(p(ξ2 ))2 leads the result; since .Cor(Y0 , Y0 ) = 1, we deduce that .c(0) = 1. Also, we observe that c is proportional to the .τ -periodic ⨆ ⨅ function p and is hence .τ -periodic itself. Remark 11 As mentioned in Remark 2, Barndorff-Nielsen et al. [9] proposed adding a periodic function as a multiplicative factor to g rather than as kernel function as in (8), which results in a process .(Zt )t≥0 with  Zt =

.

ℝ×ℝ

𝕀(0,p(t−s)g(t−s)) (x)𝕀[0,∞) (t − s)L(dx, ds),

compared to our earlier definition of .(Yt )t≥0 with  Yt =

.

ℝ×ℝ

p(t − s)𝕀(0,g(t−s)) (x)𝕀[0,∞) (t − s)L(dx, ds).

(19)

Periodic Trawl Processes

113

The autocorrelation function of the process Z is of the form, for .t1 < t2 ,

t1 Cor(Zt1 , Zt2 ) =

.

−∞ min(p(t1

− s)g(t1 − s), p(t2 − s)g(t2 − s))ds

∞ , 0 p(u)g(u)du

which is potentially slightly more difficult to deal with than the autocorrelation function of our proposed periodic trawl process Y .

Proofs of the Asymptotic Theory The following proofs extend the ideas presented in the work by Cohen and Lindner [14]. Alternatively, we could have deduced the results from the more recent work by Curato and Stelzer [16]. Proof (Proof of Theorem 1) The proof is a straightforward extension of the arguments provided in the proof of Theorem 2.1 in [14]. For the convenience of the reader and to keep this article self-contained, we will present the steps to extend the proof by Cohen and Lindner [14] to our more general setting of mixed moving average processes driven by homogeneous Lévy bases. First of all, we continue the function .FΔ periodically on .ℝ by setting ∞ 

FΔ (x, u) =

|f (x, u + j Δ)|,

.

u ∈ ℝ,

j =−∞

where .FΔ (x, u) = FΔ (x, u + j Δ) for all .j ∈ ℤ, .u, x ∈ ℝ. We note that the autocovariance function of Y satisfies  .|γf (j Δ)| ≤ κ2 |f (x, −s)||f (x, j Δ − s)|dxds, ℝ×ℝ

for any .j ∈ ℤ and .

∞ ∞   1  |γf (j Δ)| ≤ |f (x, −s)||f (x, j Δ − s)|dxds κ2 ℝ×ℝ j =−∞

j =−∞

 ≤

ℝ×ℝ

 =  =

ℝ×ℝ

ℝ×ℝ

|f (x, −s)|

∞ 

|f (x, j Δ − s)|dxds

j =−∞

|f (x, −s)|FΔ (x, −s)dxds |f (x, s)|FΔ (x, s)dxds

(20)

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A. E. D. Veraart

=

∞   j =−∞ ℝ×[0,Δ]

 =  =

∞  ℝ×[0,Δ] j =−∞

ℝ×[0,Δ]

|f (x, j Δ + s)|FΔ (x, s)dxds

|f (x, j Δ + s)|FΔ (x, s)dxds

FΔ2 (x, s)dxds < ∞.

The ∞ above computations can be repeated without the modulus, which implies that j =−∞ γf (j Δ) = VΔ . To simplify the exposition, we shall now assume that .μ = 0. We proceed as in [14]. Define the function .fm;Δ (x, s) := f (x, s)𝕀(−mΔ,mΔ) (s), for .m ∈ ℕ, x, s ∈ ℝ, and set   m .Y := f (x, s)L(dx, ds) = f (x, s)L(dx, ds) m;Δ j ;Δ

.

ℝ×ℝ

 =

ℝ×(−mΔ,mΔ)

ℝ×((−m+j )Δ,(m+j )Δ)

f (x, j Δ − s)L(dx, ds).

Since L is independently scattered, we can deduce that .(Yj(m) ;Δ )j ∈ℤ is a .(2m − 1)dependent sequence, which is also strictly stationary. Hence, by Brockwell and Davis [13, Theorem 6.4.2], we know that n  √ (m) (m) d (m) Yj ;Δ → ZΔ , n Y n;Δ = n−1/2

.

as n → ∞,

j =1

d

(m) (m) where the random variable .ZΔ satisfies .ZΔ = N(0, V (m) ), where

(m)

VΔ

.

2m 

=

γfm (j Δ),

j =−2m

for (m)

(m)



γfm (j Δ) = Cov(Y0;Δ , Yj ;Δ ) = κ2

.

 =

ℝ×((−m+j )Δ,(m+j )Δ)

ℝ×ℝ

fm;Δ (x, −s)fm;Δ (x, j Δ − s)dxds

f (x, −s)f (x, j Δ − s)dxds.

Periodic Trawl Processes

115

We observe that .limm→∞ γfm (j Δ) = γf (j Δ) for all .j ∈ ℤ; also  |γfm (j Δ)| ≤ κ2

.

ℝ×ℝ

|f (x, −s)||f (x, j Δ − s)|dxds,

 and . ∞ j =−∞ ℝ×ℝ |f (x, −s)||f (x, j Δ − s)| < ∞ by the computations in (20). (m)

Hence, Lebesgue’s Dominated Convergence Theorem implies that .limm→∞ VΔ VΔ and we get

(m) d that .ZΔ →

=

d

ZΔ , where .Z = N(0, VΔ ). (m)

). We argue as follows. It remains to control the difference .n1/2 (Y n;Δ − Y n;Δ Using similar arguments as above, we note that .limm→∞ ∞ j =−∞ γf −fm;Δ (j Δ) = 0. Hence, we have that (m)

.

lim lim Var(n1/2 (Y n;Δ − Y n;Δ )) ⎛ ⎞ n   (f (x, j Δ − s) − fm;Δ (x, j Δ − s))L(dx, ds)⎠ = lim lim nVar ⎝n−1

m→∞ n→∞

m→∞ n→∞

∞ 

(✶)

= lim

m→∞

j =1 ℝ×ℝ

γf −fm;Δ (j Δ) = 0,

j =−∞

where the equality .(✶) follows from [13, Theorem 7.1.1]. Chebyshev’s inequality allows us to conclude that, for any .ϵ > 0, (m)

.

lim lim sup ℙ(n1/2 |Y n;Δ − Y n;Δ | > ϵ) = 0.

m→∞ n→∞

As stated in [14], the final step of the proof consists of an application of a Slutskytype theorem as presented in [13, Proposition 6.3.9]. ⨆ ⨅ Proof (Proof of Lemma 1) For .t1 , t2 , t3 , t4 ∈ ℝ, we have, for any .a1 , a2 , a3 , a4 ∈ ℝ, the following expression for the joint characteristic function ψ((a1 , a2 , a3 , a4 ); (Yt1 , Yt2 , Yt3 , Yt4 )) := 𝔼(exp(i(a1 Yt1 + a2 Yt2 + a3 Yt3 + a4 Yt4 )) ⎡ ⎛ ⎞ ⎞⎤ ⎛  4  ⎝ = 𝔼 ⎣exp ⎝i aj f (x, tj − s)⎠ L(ds, dx)⎠⎦

.

ℝ×ℝ

⎡ = exp ⎣

 ℝ×ℝ

j =1

⎛ ⎞ ⎤ 4  C⎝ aj f (x, tj − s); L' ⎠ dxds ⎦ , j =1

where .C(·; L' ) denotes the cumulant function of the Lévy seed .L' , which we will present next.

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Suppose .L' has characteristic triplet .(c, A, ν) w.r.t. the truncation function .τ (y) = 𝕀[−1,1] (y). I.e. we have the following representation for its characteristic function, for any .θ ∈ ℝ, 

 1 𝔼(exp(iθ L' )) = exp icθ − Aθ 2 + (eiyθ − 1 − iθyτ (y))ν(dy) . 2 ℝ

.

We recall that .𝔼(L' ) = we are assuming that .𝔼(L' ) =

c+ ℝ y(1−τ (y))ν(dy). Since

0, we get that .c = − ℝ y(1 − τ (y))ν(dy) = − ℝ y𝕀[−1,1]c (y)ν(dy) and, hence, 

 1 2 iyθ (e − 1 − iθy)ν(dy) . .𝔼(exp(iθ L )) = exp − Aθ + 2 ℝ '

I.e. the corresponding cumulant function is given by 1 C(θ ; L' ) = − Aθ 2 + 2



.

ℝ

(eiyθ − 1 − iθy)ν(dy).

Moreover,  C((a1 , a2 , a3 , a4 ); (Yt1 , Yt2 , Yt3 , Yt4 )) :=

.

1 =− A 2

⎛



⎝ ℝ×ℝ

 +

 ℝ×ℝ ℝ

⎛

4 

ℝ×ℝ

⎛ ⎞ 4  C⎝ aj f (x, tj − s); L' ⎠ dxds j =1

⎞2 aj f (x, tj − s)⎠ dxds

j =1

⎝eiy

4

j =1 aj f (x,tj −s)

−1−i

4 

⎞ aj f (x, tj − s)y ⎠ ν(dy)dxds

j =1

 4 1  =− A aj ak f (x, tj − s)f (x, tk − s)dxds 2 ℝ×ℝ j,k=1

 +

 ℝ×ℝ ℝ

⎛ ⎝eiy

1  2 aj =− A 2 4

j =1

4

j =1 aj f (x,tj −s)

−1−i

4  j =1

 ℝ×ℝ

f 2 (x, tj − s)dxds

⎞ aj f (x, tj − s)y ⎠ ν(dy)dxds

Periodic Trawl Processes

1 − A 2

 aj ak

ℝ×ℝ

j,k=1,j /=k

⎛



 +

4 

117

ℝ×ℝ ℝ

⎝eiy

f (x, tj − s)f (x, tk − s)dxds

4

j =1 aj f (x,tj −s)

−1−i

4 

⎞ aj f (x, tj − s)y ⎠ ν(dy)dxds.

j =1

Next, we compute the fourth moments, where we recall that   ∂4 ψ((a1 , a2 , a3 , a4 ); (Yt1 , Yt2 , Yt3 , Yt4 )) ∂a1 ∂a2 ∂a3 ∂a4 a1 =a2 =a3 =a4 =0

.

= 𝔼(Yt1 Yt2 Yt3 Yt4 ). We now abbreviate the functions to .ψ and C without stating their arguments and a subscript denotes the corresponding partial derivative, e.g. .Ca1 = ∂ ∂a1 C((a1 , a2 , a3 , a4 ); (Yt1 , Yt2 , Yt3 , Yt4 ) and similarly for higher order partial derivatives. Since .ψ = exp(C), we have ψa1 = ψCa1 ,

.

ψa1 ,a2 = ψ[Ca1 ,a2 + Ca1 Ca2 ], ψa1 ,a2 ,a3 = ψ[(Ca1 ,a2 + Ca1 Ca2 )Ca3 + Ca1 ,a2 ,a3 + Ca1 ,a3 Ca2 + Ca1 Ca2 ,a3 ] = ψ[Ca1 Ca2 Ca3 + Ca1 Ca2 ,a3 + Ca2 Ca1 ,a3 + Ca3 Ca1 ,a2 + +Ca1 ,a2 ,a3 ], ψa1 ,a2 ,a3 ,a4 = ψ[Ca1 Ca2 Ca3 Ca4 + Ca1 Ca2 ,a3 Ca4 + Ca2 Ca1 ,a3 Ca4 + Ca3 Ca1 ,a2 Ca4 + Ca1 ,a2 ,a3 Ca4 + Ca1 ,a4 Ca2 Ca3 + Ca1 Ca2 ,a4 Ca3 + Ca1 Ca2 Ca3 ,a4 + Ca1 ,a4 Ca2 ,a3 + Ca1 Ca2 ,a3 ,a4 + Ca2 ,a4 Ca1 ,a3 + Ca2 Ca1 ,a3 ,a4 + Ca3 ,a4 Ca1 ,a2 + Ca3 Ca1 ,a2 ,a4 + Ca1 ,a2 ,a3 ,a4 ]. Here we have Cai |a1 =a2 =a3 =a4 =0 = 0,

.

Cai ,aj |a1 =a2 =a3 =a4 =0

for i = 1, 2, 3, 4,   =− A+ y 2 ν(dy)

ℝ

ℝ×ℝ

f (x, ti − s)f (x, tj − s)dxds,

for i, j = 1, 2, 3, 4,  Ca1 ,a2 ,a3 ,a4 =

4 $ ℝ×ℝ j =1

 f (x, tj − s)dxds

ℝ

y 4 ν(dy).

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The above results imply that 𝔼(Yt1 Yt2 Yt3 Yt4 )

2  2 = A+ y ν(dy)

.

ℝ

 ℝ×ℝ



f (x, t1 − s)f (x, t2 − s)dxds

ℝ×ℝ

 +

ℝ×ℝ

 f (x, t1 − s)f (x, t3 − s)dxds

 +  +

ℝ×ℝ

ℝ

f (x, t3 − s)f (x, t4 − s)dxds

ℝ×ℝ

f (x, t2 − s)f (x, t4 − s)dxds 

 f (x, t1 − s)f (x, t4 − s)dxds 

y 4 ν(dy)

ℝ×ℝ

ℝ×ℝ

f (x, t2 − s)f (x, t3 − s)dxds

f (x, t1 − s)f (x, t2 − s)f (x, t3 − s)f (x, t4 − s)dxds.



We note that .κ4 := ℝ y 4 ν(dy) = (η − 3)κ22 and .κ2 = A + ℝ y 2 ν(dy). We can further simplify the above formula as follows: 𝔼(Yt1 Yt2 Yt3 Yt4 )

2  2 = A+ y ν(dy)

.

ℝ

 ℝ×ℝ



f (x, t1 − t2 + s)f (x, s)dxds

 +

ℝ×ℝ

 +

ℝ×ℝ

ℝ

f (x, t3 − t4 + s)f (x, s)dxds

 f (x, t1 − t3 + s)f (x, s)dxds

 +

ℝ×ℝ

ℝ×ℝ

f (x, t2 − t4 + s)f (x, s)dxds 

 f (x, t1 − t4 + s)f (x, s)dxds

y 4 ν(dy)

 ℝ×ℝ

ℝ×ℝ

f (x, t2 − t3 + s)f (x, s)dxds

f (x, t1 − t3 + s)f (x, t2 − t3 )f (x, s)f (x, t4 − t3 + s)dxds

= γ (t1 − t2 )γ (t3 − t4 ) + γ (t1 − t3 )γ (t2 − t4 ) + γ (t1 − t4 )γ (t2 − t3 )  f (x, t1 − t3 + s)f (x, t2 − t3 )f (x, s)f (x, t4 − t3 + s)dxds. + κ4 ℝ×ℝ

⨆ ⨅ Proof (Proof of Proposition 7) We first expand the covariance of the sample autocovariances as follows ∗ ∗ ∗ ∗ ∗ ∗ Cov( γn;Δ (Δp),  γn,Δ (Δq)) = 𝔼( γn;Δ (Δp) γn;Δ (Δq)) − 𝔼( γn;Δ (Δp))𝔼( γn;Δ (Δq)),

.

Periodic Trawl Processes

119

where ∗ 𝔼( γn;Δ (Δp)) =

.

∗ (Δq)) = 𝔼( γn;Δ

1 1 𝔼(Yj Δ Y(j +p)Δ ) = γ (pΔ) = γ (pΔ), n n n

n

j =1

j =1

1 1 𝔼(YkΔ Y(k+q)Δ ) = γ (qΔ) = γ (qΔ). n n n

n

k=1

k=1

Also, ∗ ∗ 𝔼( γn;Δ (Δp) γn;Δ (Δq)) =

.

n n 1  𝔼(Yj Δ Y(j +p)Δ YkΔ Y(k+q)Δ ), n2 j =1 k=1

where 𝔼(Yj Δ Y(j +p)Δ YkΔ Y(k+q)Δ )

.

= γ (pΔ)γ (qΔ) + γ ((k − j )Δ)γ ((j + p − k − q)Δ) + γ ((j − k − q)Δ)γ ((j + p − k)Δ)  + κ4 f (x, (j − k)Δ + s)f (x, (j − k + p)Δ + s)f (x, s) ℝ×ℝ

f (x, qΔ + s)dxds = γ (pΔ)γ (qΔ) + γ ((j − k)Δ)γ ((j − k + p − q)Δ) + γ ((j − k − q)Δ)γ ((j − k + p)Δ)  + κ4 f (x, (j − k)Δ + s)f (x, (j − k + p)Δ + s)f (x, s) ℝ×ℝ

f (x, qΔ + s)dxds l=j −k

=

γ (pΔ)γ (qΔ) + γ (lΔ)γ ((l + p − q)Δ) + γ ((l − q)Δ)γ ((l + p)Δ)  f (x, lΔ + s)f (x, (l + p)Δ + s)f (x, s)f (x, qΔ + s)dxds. + κ4 ℝ×ℝ

Now we subtract .γ (pΔ)γ (qΔ), we set .l = j − k, interchange the order of summation and use the stationarity to obtain ∗ ∗ Cov( γn;Δ (Δp),  γn;Δ (Δq))

.

=

n n 1  𝔼(Yj Δ Y(j +p)Δ YkΔ Y(k+q)Δ ) − γ (pΔ)γ (qΔ) n2 j =1 k=1

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A. E. D. Veraart

=

n n 1  (γ ((j − k)Δ)γ ((j − k + p − q)Δ) n2 j =1 k=1

+ γ ((j − k − q)Δ)γ ((j − k + p)Δ)  f (x, (j − k)Δ + s)f (x, (j − k + p)Δ + s) + κ4 ℝ×ℝ

f (x, s)f (x, qΔ + s)dxds =



n−|l| 1   (γ (lΔ)γ ((l + p − q)Δ) + γ ((l − q)Δ)γ ((l + p)Δ) n2 |l| 0, H > 2, i.e. we require a shortmemory setting, .x ≥ 0, we have  γ (h) =

∞

.

|h|

g(x)dx =

 α |h| 1−H . 1+ H −1 α

Then VΔ =

∞ 

.

γ (j Δ) =

j =−∞

=

α H −1

 ∞  |j |Δ 1−H α 1+ H −1 α j =−∞

 ∞ 1−H Δ 1−H   α + |j | α Δ j =−∞

     α H −1 Δ 1−H  α 1−H α 2 ζ (H − 1, α/Δ) − 1 = H −1 α Δ Δ     α H −1 α 2 ζ (H − 1, α/Δ) − 1 , = H −1 Δ

where .ζ denotes the Hurwitz Zeta function defined by .ζ (s, a) = .Re(s) > 1.

∞

1 k=0 (k+a)s ,

for

128

A. E. D. Veraart

Verifying the Assumptions of Theorem 2 for Selected Periodic Trawl Processes For the applications discussed in Sect. 5, we need to verify the condition (11) from Proposition 7 and Assumption (15) from Theorem 2 assuming that the corresponding moment assumptions for the Lévy seed hold. For both conditions, it is sufficient to check that a (non-periodic) trawl process satisfies the stated conditions since the periodic function is bounded. Hence, in the following, we shall set .p ≡ 1.

Verifying Condition (11) from Proposition 7 We need to check that ⎛ .

∞ 

⎝ℝ × [0, Δ] → ℝ, (x, u) I→

⎞ f 2 (x, u + j Δ)⎠ ∈ L2 (ℝ × [0, Δ]).

j =−∞

This condition holds for trawl processes if, for .(x, u) ∈ ℝ × [0, Δ], ∞  j =−∞

=

∞ 

f 2 (x, u + j Δ) =

.

f (x, u + j Δ)

j =−∞ ∞ 

𝕀(0,g(u+j Δ)) (x)𝕀[0,∞) (u + j Δ) = FΔ (x, u) ∈ L2 (ℝ × [0, Δ]).

j =−∞

This is equivalent to checking that  ℝ×[0,Δ]



= .

|FΔ (x, u)|2 dxdu ∞ 

∞ 

ℝ×[0,Δ] j =−∞ k=−∞

𝕀(0,g(u+j Δ)) (x)𝕀[0,∞) (u + j Δ)𝕀(0,g(u+kΔ)) (x) (22)

𝕀[0,∞) (u + kΔ)dxdu ∝

∞ 

γ (j Δ) < ∞.

j =−∞

This condition is satisfied both for an exponential trawl function and also for a supGamma trawl function with short memory. In the latter case, we have that

Periodic Trawl Processes

129

γ (x) ∝ (1 + |x|/α)1−H for .α > 0, H > 2. Then, the finiteness of (22) follows using the .ζ -function representation.

.

Verifying Assumption (15) from Theorem 2 We need to verify ∞   .

j =−∞

ℝ×ℝ

2 |f (x, u)||f (x, u + j Δ)|dxdu < ∞.

Using very similar computations as before, we find that the above condition is equivalent to ∞   .

j =−∞

=

ℝ×ℝ

∞   ℝ×ℝ

j =−∞

∝

2 |f (x, u)||f (x, u + j Δ)|dxdu

∞ 

2 𝕀(0,g(u) (x)𝕀[0,∞) (u)𝕀(0,g(u+j Δ)) (x)𝕀[0,∞) (u + j Δ)dxdu

γ 2 (j Δ) < ∞,

j =−∞

which is satisfied by the exponential trawl function and the supGamma trawl functions with .H > 3/2, which includes some long-memory settings.

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49. Shephard, N., Yang, J.J.: Continuous time analysis of fleeting discrete price moves. J. Am. Stat. Assoc. 112(519), 1090–1106 (2017). https://doi.org/10.1080/01621459.2016.1192544 50. Surgailis, D., Rosinski, J., Mandrekar, V., Cambanis, S.: Stable mixed moving averages. Probab. Theory Relat. Fields 97, 543–558 (1993). https://doi.org/10.1007/BF01192963 51. Talarczyk, A., Treszczotko, L.: Limit theorems for integrated trawl processes with symmetric Lévy bases. Electron. J. Probab. 25, 1–24 (2020). https://doi.org/10.1214/20-EJP509 52. Veraart, A.E.D.: Modeling, simulation and inference for multivariate time series of counts using trawl processes. J. Multivar. Anal. 169, 110–129 (2019). https://doi.org/10.1016/j.jmva. 2018.08.012 53. Veraart, A.E.D.: ambit: Simulation and Estimation of Ambit Processes. R package version 0.1.2 (2022). https://cran.r-project.org/web/packages/ambit/index.html 54. Veraart, A.E.D.: PeriodicTrawl-Energy. R code, release v1.0.0 (2023). https://doi.org/10.5281/ zenodo.7706091 55. Veraart, A.E.D., Veraart, L.A.M.: Modelling electricity day-ahead prices by multivariate Lévy semistationary processes. In: Benth, F.E., Kholodnyi, V.A., Laurence, P. (eds.) Quantitative Energy Finance: Modeling, Pricing, and Hedging in Energy and Commodity Markets, pp. 157– 188. Springer, New York (2014). https://doi.org/10.1007/978-1-4614-7248-3_6 56. Wolpert, R.L., Brown, L.D.: Stationary infinitely-divisible Markov processes with nonnegative integer values. Working paper, April 2011 (2011). https://faculty.wharton.upenn.edu/ wp-content/uploads/2011/09/2011d-Stationary-Infinitely-Divisible-Markov-Processes-withNon-negative-Integer-Values.pdf 57. Wolpert, R.L., Taqqu, M.S.: Fractional Ornstein–Uhlenbeck Lévy processes and the Telecom process: upstairs and downstairs. Signal Process. 85, 1523–1545 (2005). https://doi.org/10. 1016/j.sigpro.2004.09.016 58. Woodward, W.A., Cheng, Q.C., Gray, H.L.: A k-factor GARMA long-memory model. J. Time Ser. Anal. 19(4), 485–504 (1998). https://doi.org/10.1111/j.1467-9892.1998.00105.x 59. Yajima, Y.: Semiparametric estimation of the frequency of unbounded spectral densities. J. Stat. Stud. 26, 143–155 (2007). http://www.jstor.org/stable/27639901

Part II

Energy Transition

Fuelling the Energy Transition: The Effect of German Wind and PV Electricity Infeed on TTF Gas Prices Christoph Halser and Florentina Paraschiv

Abstract Previous research shows that renewable energies have a direct negative marginal effect on electricity prices. Gas plants play an essential role in the electricity generation in several fuel-based energy systems through balancing out intermittent renewable energies, which is why it is labeled “green” in the EU Taxonomy. We show the substitution effect between renewable energies, wind and PV, and gas, in the context of a threshold model. Applied to daily Dutch natural gas prices (TTF) between 2016 and 2020, we determine the effect of demand/supply price drivers and lay special emphasis on the asymmetric effects of the day-ahead forecasts of wind and PV infeed. Results show a negative marginal effect of the dayahead wind and PV infeed forecasts on day-ahead natural gas prices. Employing threshold models we find that in regimes with low wind infeed, marginal increases in the wind and PV infeed forecasts decrease gas prices faster than in regimes with high infeed. Our findings further reveal that the day-ahead TTF price is positively associated with heating demand, supplier concentration, coal, and CO.2 prices. We discuss these findings in the context of the debate on the usage of gas for the European energy transition.

1 Introduction Fears of supply shortages and import cut-offs from Russia following the Russian invasion of Ukraine in 2022 led to unprecedented levels in European natural gas prices. At the Dutch Title Transfer Facility (TTF), Europe’s most liquid and relevant gas hub [32], day-ahead prices peaked above 300 EUR/MWh between August 25– 29 2022, representing a more than ten-fold increase from the average daily closing C. Halser () Norwegian University of Science and Technology, Trondheim, Norway e-mail: [email protected] F. Paraschiv Zeppelin University, Friedrichshafen, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. E. Benth, A. E. D. Veraart (eds.), Quantitative Energy Finance, https://doi.org/10.1007/978-3-031-50597-3_4

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price of 19.23 EUR/MWh between 2010 and 2020. In Europe, where in 2021 LNG shipments amounted to only 20% of total natural gas imports [21], the price spike can be attributed to an acute shortage of gas, due to a tight LNG market and a lack of pipeline import alternatives. While characterized by supply disruptions from Russia for instance through the Nord Stream I pipeline, the European gas crisis of 2022 was initiated by inadequate storage fillings by Gazprom during the summer of 2021. In the context of an accelerated phase-out from coal and nuclear, the energy crisis has elevated the awareness towards the acute reliance on natural gas as a bridging fuel in the European energy transition. We therefore set out to derive a renewed understanding of fundamental natural gas price drivers and the role of gas for balancing out the intermittency of renewable generation.

1.1 European Natural Gas Consumption The significance of natural gas as an energy source for Europe is exemplified by its 32.7% share in gross energy consumption in 2020 [23]. Households and industry, with 29.8 and 28.0% respectively, contributed to most of the consumption in 2020 [25]. While households consume gas primarily for heating and use in boilers, industrial consumption occurs in a variety of industrial processes, largely for process heat. Domestic natural gas production in Europe, however, declined by 62.4% between 2010 and 2020, amounting to only 45 billion cubic meters (bcm) in 2020 [22], equating roughly 10% of total EU consumption in 2021 [21]. Yet, over the last 20 years, the importance of natural gas for the European energy system has gradually increased in the energy transition picture: In the early 2000s, countries with high-carbon coal-based electricity and heat generation sought to diversify their portfolios with a cleaner gas alternative. Consequently, the share of natural gas in the EU’s gross electricity production rose from 12.5% in 2000 to 20.5% in 2008, while the shares of coal and oil-fired generation declined from 30.1 and 6.5% respectively in 2000 to 25.3 and 3.4% in 2008. The use of oil-indexed gas, however, declined again to 12.5% by 2014, driven by the rise of oil prices to above 100 USD/bbl, which made the use of gas financially unattractive. In the 2010s, however, the European energy transition was characterized by the increasing deployment of renewable generation, in particular wind and photovoltaic (PV) generation, which continued to replace coal. The share of wind and PV generation in the EU’s power supply rose from 3.8 and 0.2% respectively in 2008 to 14.3 and 5% in 2020, jointly exceeding coal for the first time in 2019 [25].

1.2 Gas Key Player in the Energy Transition The increasing reliance on intermittent renewable generation, however, intensified the need for a flexible and low-carbon balancing tool, which provides electricity in

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times of low wind and PV output. Critics of the German Energiewende, such as [50], have voiced concern over the necessity of requiring a fossil fuel-based dual buffering infrastructure, also because of a lack of seasonal European power storage. Consequently, after the oil price collapse in 2014, the share of gas-fired electricity generation in the EU rose from 12.5 to 20.1% by 2020 [25]. Regulatory support for this strategy was later drafted in form of the EU taxonomy, which grants investments into natural gas infrastructure access to sustainable finance, if hydrogen-ready or otherwise aligning to the long-term decarbonization targets [20]. The need for a bridging technology has been particularly vital for Germany, which, unlike many other European countries, seeks to phase out both coal and nuclear technology. Made possible by its fast switching times, natural gas became the chosen technology and a flexible tool for balancing out the intermittency of renewables. In times of low wind and PV infeed, which is otherwise fed into the grid with priority (see [35]), grid operators increase the production from natural gas, leading to a substitution effect between renewables and gas, while coal plants can be turned off only at high administrative costs [42]. We find visual evidence for the substitution effect in Fig. 1, depicting the electricity generation portfolio in Germany between January 21–30, 2019. Each color thereby represents one generation source. Intuitively in January, the time frame is characterized by a small infeed from PV (yellow), which amounted to only 1.9% of the total infeed during this time frame. On- and offshore wind infeed, here jointly depicted, however, amounted to 20.5% and show an intermittent pattern. Although wind infeed was moderate between 21 and January 24, we find that the infeed from wind drastically increased between January 25 and 27, before again slightly decreasing between January 28 and 30. While renewable infeed is fed into the grid with priority, we find large differences in the output response of other generation sources. Nuclear (red), Hydro run-of-river (dark blue), and Biomass

90,000 80,000 70,000

PH

60,000 50,000 40,000 30,000 20,000 10,000 0 21

22

23

24

25

26

27

28

29

January Nuclear Hard coal Hydro pump

Hydro river Oil Other

Biomass Natural gas Waste

Lignite Geothermal Wind (on & offshore)

Coal-derived gas Hydro reservoir Solar PV

Fig. 1 Public net electricity generation in Germany, January 21–30, 2019. Source: [26]

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(dark green) contribute to the base load and operators can decrease the output of these generation sources only slowly and only to a small degree. Therefore, we find very little decrease in the output of these sources. Lignite (light brown), hard coal (dark brown), and fossil gas (orange) contribute the majority of the load in the depicted time frame with 23.8, 20.5, and 15.8% share in the total infeed. We find that lignite shows the most sluggish reaction to the increase in wind infeed. Hard coal and gas-fired generation, however, show a dynamic response to the increasing infeed from wind between January 25–27. As the electricity from wind is fed into the grid, operators are able to reduce the generation from sources with higher marginal cost and consequently less natural gas is burned in gas-fired generation (see for instance [42]). We therefore hypothesize that the decrease in gas used in electricity production reflects on the demand via the market price for gas. We thereby further observe that gas infeed dynamically responds to the changing wind infeed between January 21 and 24, while the infeed from lignite remains rather stable. The substitution dynamic between renewable generation and gas can also be visually identified with regards to PV infeed. While similarly suggesting a substitution between wind and gas between October 10–11 and between October 15–16, Fig. 2 visualizes the substitution dynamic between gas and PV generation. We observe that the infeed from PV only occurs during the day time, peaking around noon, while intuitively no infeed occurs during the night. This leads to a sharply rising and falling load pattern, whereby the PV infeed on October 11–12 was larger than on October 13 and 14. Figure 2 suggests that in response to the large PV infeed on October 11 and 12, the infeed from fossil gas is dynamically reduced around noon time, as the infeed from PV is fed into the grid with priority. On October 13 and 14, however, gas fired-generation remains constant even during peak hours. Therefore, as also PV infeed determines the amount of gas burnt in electricity generation, we

Fig. 2 Public net electricity generation in Germany, October 10–19, 2022. Source: [26]

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expect the demand fluctuations to reflect on the day-ahead price for natural gas, such as the TTF.

1.3 Gas Pricing Factors Historically, pipeline import to Europe has been indexed against the long-term price of crude oil, originating from the occurrence of gas as a byproduct in the discovery of oil reservoirs and the possibility of fuel switching. Asche et al. [4] show that the British National Balancing Point (NBP) shared a long-run equilibrium with crude oil prices, while these prices decouple during the winter, when heating demand intensifies. Nick and Thoenes [39] for German gas hub prices between 2008 and 2012 find an intact oil link as well, which is short-term impacted by temperature, storage, and supply disruptions. Chiappini et al. [11] outline the inability of oilindexed prices to reflect the fundamentals of natural gas in the wake of oil prices above 100 USD between 2011 and 2014. These prices led European utilities to demand more transparent hub pricing. The increase in more flexible liquefied natural gas (LNG) trade further drove the transition towards gas market-based pricing in Europe, which saw gas-on-gas pricing increase to 77% in total European gas imports in 2021 [34]. Consequently, [11] identify that the oil-gas link weakened after 2010, while [55] show eventual decoupling in a sample until June 2016. Hulshof et al. [33] in a fundamental model investigate the drivers of daily TTF day-ahead prices between 2011 and 2014. While oil prices remain a significant factor for the shortterm price formation process, gas prices are found to be further driven by heating demand, storage levels, and the infeed from German wind. Contrary to what the substitution effect between gas and renewables would suggest, [33] find that the wind infeed forecast increases day-ahead natural gas prices. The authors attribute this finding to the fact that in a regime of large wind infeed in Germany, the Dutch transmission system operator (TSO) reduces the cross-border flow in response to loop flows. However, the authors’ sample covers a period of declining natural gas use for electricity, driven by oil indexation and high oil prices. Table 1 shows the electricity production in Germany between 2005 and 2020. We find that between 2011 and 2014 the share of gas in electricity generation decreased from 14.1 to 9.8% [8]. Wind and PV generation at the same time, only rose from 8.2 and 3.2% respectively in 2011, to 9.4 and 5.6% in 2014. As the marginal cost of electricity production from natural gas dropped after 2014 as a consequence to a shift to hub pricing [51], however, the share of natural gas in electricity production in Germany increased to 16.7% in 2020. Driven by the phase-out of coal and nuclear, the share of wind and PV generation rose to 23.3 and 8.6% respectively in 2020 [8]. The increase in wind and PV generation has raised the substitution potential between intermittent renewable and gas-fired generation. Additionally, phase shifters have been installed on the Dutch-German border to block loop flows [40]. Therefore, in this study, we focus on the marginal effect of renewables infeed on gas prices, in light of the substitution effect with gas in electricity production. Our

Electricity generation Nuclear Lignite Hard Coal Natual Gas Oil Renewable energies Wind (on- offshore) Hydropower Biomass Photovoltaic Waste-to-energy Others

2005 26.5 25.0 21.8 11.7 1.9 10.3 4.5 3.2 1.9 0.2 0.5 2.8

2010 22.4 23.3 18.7 14.2 1.4 16.8 6.2 3.4 4.6 1.9 0.8 3.2

2011 17.8 24.7 18.5 14.1 1.2 20.4 8.2 2.9 5.3 3.2 0.8 3.2

2012 16.0 25.8 18.7 12.2 1.2 23.0 8.3 3.5 6.2 4.2 0.8 3.1

2013 15.4 25.5 20.2 10.6 1.1 24.0 8.4 3.6 6.4 4.8 0.9 3.2

2014 15.7 25.1 19.1 9.8 0.9 26.0 9.4 3.2 6.8 5.6 1.0 3.4

Table 1 Gross electricity generation in Germany without imports/exports, percentage [8] 2015 14.3 24.1 18.4 9.6 1.0 29.2 12.6 3.0 7.0 5.8 0.9 3.4

2016 13.2 23.3 17.5 12.5 0.9 29.3 12.5 3.2 7.0 5.7 0.9 3.4

2017 11.8 23.0 14.4 13.3 0.9 33.3 16.4 3.1 7.0 5.9 0.9 3.4

2018 12.0 23.0 13.0 12.9 0.8 35.1 17.4 2.8 7.0 6.9 1.0 3.3

2019 12.5 18.9 9.6 15.0 0.8 40.0 20.9 3.4 7.4 7.4 1.0 3.3

2020 11.4 16.2 7.6 16.7 0.8 44.0 23.3 3.3 7.8 8.6 1.0 3.3

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model builds on the fundamental model by Hulshof et al. [33], which we extend by variables for PV infeed forecast and forecasting errors. We further employ threshold regression models following [7], to detect asymmetric adjustment of gas prices to the forecast and realization of renewable infeed, but also to forecasting errors. Our findings reveal a significant effect of coal and CO.2 prices, as well as of heating demand, supplier concentration, and renewable infeed forecast on the gas formation process. In light of the substitution effect, we find a significant negative impact of the wind and PV infeed forecast on natural gas prices in the day-ahead market. Using threshold models, we report that at low regimes of realized and forecasted wind at the time the gas price is formed, we observe a higher speed of adjustment to wind and PV forecasts. Low levels of renewables boost the need to burn extra gas for electricity production, which sets upwards pressure on the day-ahead gas price. Conversely, high regimes of infeed from wind-generated electricity push gas-fired generation out of the production mix and the effect of further infeed is small. To our knowledge, no existing literature quantifies the relationship between gas-fired and renewable technology in a comprehensive analysis and the focus on the asymmetric adjustment is novel. The introduction of this paper is followed by a literature review in Sect. 2. The data is outlined in Sect. 3, after which Sect. 4 describes the methodology of the study. Results and discussion are presented in Sect. 5, followed by a conclusion in Sect. 6.

2 Literature Review The pricing for natural gas in continental Europe underwent significant changes over the course of the last two decades. While market pricing of natural gas occurred first in the 1990s in the deregulated and largely self-supplied UK gas market and its virtual trading hub the NBP, imports to import-dependent continental Europe were historically priced indexed against the price of oil under long-term contracts. These contracts, which often lasted over decades, granted the purchaser only limited flexibility through renegotiation, whereby the German border price served as the most prominent price benchmark [48]. Following the example of the UK, but also of the North-American gas market after the discovery and exploration of shale gas, pricing gas imports by prices at trading hubs (gas-on-gas competition) became increasingly prominent. This was specifically driven by the inability of long-term oil-indexed prices to reflect the fundamentals of gas demand, leading deregulated European utilities to respond to the demand for more transparent hub prices [51]. Consequently, the share of hub-pricing in Northwest European imports increased from 29% in 2005 to 93% by 2015 [34]. Today, the Dutch TTF has become the European benchmark price for natural gas in Europe, having overtaken the NBP in trading volume, impacting other European natural gas prices, and appearing to overtake crude oil as a benchmark for energy commodities between 2008 and 2016 [13]. The role of the Dutch TTF further manifests in its churn rate, i.e. the amount of

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trading per physical gas, which amounted to 21.4 in 2021 for its Northwest European area of influence [32]. TTF thereby reflects a mix of gas delivered by pipeline to continental Europe from the North Sea (largely from Norway, but also the UK and Netherlands) and LNG imports. Because of their historical decoupling from oil after the shale gas revolution, North-American Henry Hub prices serve as a historical reference point for understanding natural gas price drivers also for Europe. Brown and Yucel [9] for instance show that the long-run relationship between Henry Hub gas and crude oil, prior to the shale gas boom, was short-term impacted by weather, seasonality, storage, and supply disruptions. Also [31] show a long run relationship between oil and natural gas prices, similarly affected by storage, weather, seasonal factors, and supply shocks, however, further impacted by the exchange rate. Ramberg and Parsons [45] thereby outline that the oil-gas relationship leaves large amounts of unexplained volatility at short horizons. After the loosening of the relationship oil-gas in NorthAmerica [55], empirical studies show evidence for a fundamentally-driven gas market. Thus, [53] outline that explanatory factors evolve over time and show yearly seasonal patterns, with heating demand being primarily responsible for driving gas demand during the winter. Instead of oil, the impact of supply and demand variables, such as storage and weather increases. Wang et al. [53] note the increasing impact of financial variables on the gas price formation process, including stock market volatility and speculative behavior. Rubaszek and Uddin [46], similarly, note the importance of underground storage on the development of US natural gas prices, while [54] identify time-varying supply and demand shocks as the main drivers and speculative behavior as a secondary driver between 1993 and 2015. Hailemariam and Smyth [27] however, note that supply shocks only played a role in shaping natural gas price movement and volatility prior to the shale boom, while the effect wears off over the course of the supply increase. Similar to the North-American market, [4] confirm an intact long-term equilibrium relation between oil and the British NBP before 2010, while [5] shows time-varying mean reversion of gas to oil prices for the UK gas market. Between 1997 and 2014 the authors find oil-gas decoupling during the winter, when gasspecific heating and electricity demand are high, while the relationship with oil strengthens over the summer. Zhang et al. [55] in a VAR model find evidence to support the decoupling hypothesis from oil also for the German hub NetConnect (NCG),1 following the shale driven LNG oversupply and adoption of hub indexation. Similarly, [18] with a sample until 2010 indicate that the relationship between oil and gas prices weakened and that natural gas showed early signs of decoupling. Consistently, [28] find only temporary equilibria between TTF and monthly oil prices between 2015 and 2021, with oil market financialization being a significant factor for both prices. Chiappini et al. [11] note that for the majority of European hubs, a structural change in the relationship between oil and natural gas prices occurred in 2010. The authors find increasing integration among European gas hub

1 NCG

merged with Gaspool into the nationwide trading hub THE in 2021 [15].

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prices and between European hub and Henry hub prices, but that the long-term relationship between oil and gas prices in Europe weakened. Nick and Thoenes [39] show that, albeit an intact long-run link to oil and coal prices existed, NCG hub prices are short-term impacted by temperature, storage, and supply shortfalls, such as during the Russian-Ukrainian gas dispute in 2009. Altinay and Yalta [3] thereby reveal time-varying elasticities also of residential natural gas demand with sensitivity to the economic situation but also to fluctuations in weather. While most of the available literature analyzes the long-term price movement for natural gas, [52] developed a stochastic model for daily gas spot prices, which incorporates both yearly and weekly seasonality to account for heating and industrial demand respectively. The authors include normalized heating degree days in their ARMA model and find that sustained periods of temperatures below 15.◦ C significantly impact gas prices positively [52]. Müller et al. [38] extend this work by incorporating oil prices as a proxy for economic influences and as an approximation for oil-indexation in gas imports and achieve a significantly improved model fit. More recently, [33] investigate fundamental drivers for TTF day-ahead gas prices between 2011 and 2014. Apart from the continuous relevance of Brent crude oil prices, the authors find that TTF gas prices are driven by heating demand and the deviation from the 3-year average filling level of underground storage. Furthermore, the findings by Hulshof et al. [33] suggest a positive effect of the German wind electricity infeed forecast, hypothesized to originate from loop flows. The authors argue that, in response to unscheduled flows from German wind generation in peak hours, the Dutch grid operator reduces the cross-border connection capacity, leading to the need of higher gas-fired generation in the Netherlands. Hedging-related literature acknowledges the value of fundamental variables in refining hedging strategies. Ergen and Rizvanoghlu [19] highlight the impact of temperature deviations and storage levels on volatility, while [12] emphasize the importance of underground gas storage for spot and futures returns. Liang et al. [36] underscores the significance of extreme weather data for natural gas market volatility. Meanwhile, [16] suggest refining hedge ratio estimates by incorporating anticipated spot price changes, and [37] point to the influence of seasonal patterns. Shrestha et al. [49] and Pouliasis et al. [44] emphasize the role of traditional models in estimating hedge ratios and the robustness of a combined forecast approach, respectively. The substitution effect between natural gas and renewable generation for electricity production outlined in Sect. 1 has been discussed by Paraschiv et al. [42], who disentangle the effect of renewable energies on electricity prices for each price quantile separately. The authors identify negative correlation between wind and gas prices and find a direct effect of wind and PV infeed on the level of electricity prices and an indirect one, via the gas prices. The authors link these findings to the substitution effect, in which gas facilities, operating with higher marginal cost, are pushed out of action under high wind production. Kiesel and Paraschiv [35] apply a threshold model to identify the impact of updated forecasting errors of wind and PV on the intraday electricity prices. The authors note that with an expected average input from renewable energies of 20% in Germany, renewables are fed into

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the grid with priority, decreasing the residual demand. In case of excess demand, however, positive forecasting errors suppress the price, while negative forecasting errors increase the intraday price [35]. Building on these findings, we depart from the fundamental model of [33] and investigate asymmetries in the effect of wind and PV infeed forecasts on day-ahead TTF natural gas prices, applying OLS and threshold regression models. The following section outlines the data set and methodology used in the analysis.

3 Data The data set of this study consists of 1261 TTF trading days between January 5, 2016 and December 31, 2020. We aim at a fundamental analysis that sheds light on more general dynamics of gas prices with no particular focus on event studies, which is why we first identify and exclude price outliers. In late February and early March 2018, the anticyclone Hartmut led to the British Isle Cold Wave in Northwest Europe, causing 95 human deaths in Europe, of which 17 in the UK [47]. The rare weather event drove an unprecedented rise in TTF gas prices, which rose from 22.05 EUR/MWh on February 22, 2018 to 76 EUR/MWh on March 01, 2018. We handle these extreme price changes as outliers and hereby exclude the six trading days between February 23 to March 2, 2018 from our sample. We further exclude the period succeeding December 2020, since price effects stemming from the prelude of the Russian invasion of Ukraine are out of the scope of this study. In particular, over the summer of 2021, Gazprom failed to adequately fill its storage, thereby spurring fears of supply shortages. These led to an unparalleled rise in natural gas prices above 100 EUR/MWh at the end of 2021. Table 2 presents the variables used in the analysis. TTF day-ahead, Brent spot, Rotterdam coal futures, as well as the European Energy Exchange (EEX) auction prices in the EU emission trading system (ETS) are derived from Refinitiv EIKON and Datastream databases. These variables are denoted in Euro per Megawatt hour (MWh), Euro per barrel (bbl), and Euro per metric tonne, respectively. German wind and PV infeed data stems from Netztransparenz, which aggregates infeed data marketed by the four German power transmission network operators 50Hertz, Amprion, TenneT, and TransnetBW.2 We perform log transformation for all variables that cannot assume zero values, such as storage, temperature, and forecasting errors. We extend the work of Hulshof et al. [33], which analyze fundamental dayahead TTF prices for the period between 2011 and 2014, by replicating the analysis on newer sample and by expanding the list of explanatory variables with PV infeed forecasts. We furthermore hypothesize that an increase in the infeed forecast

2 Note that the infeed marketed by the TSOs and published by Netztransparenz excludes direct marketing. This leads to a discrepancy with the infeed data available from ENTSO-E.

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Table 2 Variable definitions Variable .T T Ft .Coalt

.Brentt .CO2t

.I N Dm

.ST ORt

.H DDt

.H H It

.W ind.f oret

.P V .f oret

.W ind.errort

.P V .errort

.W ind.realt

.P V .realt

Description TTF day-ahead gas price Coal price, Rotterdam coal futures Brent crude oil spot price EEX price for EU carbon allowance (ETS) Gas consumption-weighted industrial production index (base January 2015) Differential of Northwest European underground storage from the average filling level of the previous 3 years Consumption-weighted NorthWest European heating degree days (base temperature 18.◦ C) Herfindahl-Hirschman index: Aggregate squared market share of daily gas flow into NorthWest Europe and production in UK and the Netherlands Electricity infeed forecast from German wind on day t for for the next calendar day Electricity infeed forecast from German PV on day t for the next calendar day Forecasting error wind: difference between .W ind.realt and forecast for trading day t, made on previous calendar day Forecasting error PV: difference between .P V .realt , and forecast for trading day t, made on previous calendar day Realized electricity infeed from German wind on trading day t Realized electricity infeed from German PV on trading day t

Source Refinitiv Eikon Refinitiv Eikon

Frequency & Unit Daily (EUR/MWh) Daily (EUR/tonne)

Refinitiv Eikon Datastream

Daily (EUR/bbl) Daily (EUR/tonne)

Eurostat

Monthly (Index)

Aggregated Gas Storage Inventory (AGSI) by Gas Daily (Percentage) infrastructure Europe (GIE) European Climate Daily (.◦ C) Assessment & Dataset, RMI (Belgium), Eurostat ENTSOG transparency Daily (Index)

Netztransparenz (50 Hertz, Amprion, TenneT, TransnetBW) Netztransparenz

Daily (MWh)

Netztransparenz

Daily (MWh)

Netztransparenz

Daily (MWh)

Netztransparenz

Daily (MWh)

Netztransparenz

Daily (MWh)

Daily (MWh)

We do not implement dummies for the discovery of gas resources. We argue that, while also insignificant in the results of [33], the long-term supply potential has no impact on the short-term price formation process in the day-ahead market, which is determined by available supply, which we capture with our variable for supplier concentration

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for the next calendar day from any intermittent generation source decreases the demand for gas-fired generation because of the substitution effect outlined in Sect. 1, subsequently leading to a decrease in the gas price. We further argue that the size of the substitution effect is determined by the total renewables infeed, since renewables are fed with priority into the electricity grid. Thereby, given that gas prices are available in daily resolution, unlike [33], we account not only for the infeed during peak hours, but for aggregate daily renewables infeed volumes derived from 15 minute data. We further hypothesize that the day-ahead gas price formation process reads in contemporaneous information of forecasting errors of wind and PV. Thereby, we analyze forecasting errors of wind and PV measured on the day t when the TTF dayahead price is formed. These variables capture the difference between the realized infeed on the trading day (.W ind.realt and .P V .realt ) and their respective forecasts from the previous day .t − 1. Positive forecasting errors for wind and PV infeed at day t affect the day-ahead gas price formed in the same day t negatively, as positive values indicate a surplus of gas planned at day .t − 1 due to inexact weather forecasts, materialized in larger-than-expected infeed from renewables. These gas amounts can be held in short-term storage at gas power plants and are available for use in gas-fired electricity generation on the next day, thereby decreasing the demand for new gas in the day-ahead market. Conversely, in the case of negative forecasting errors, we expect additional demand from electricity producers, as gas from short-term storage needs to be refilled. Netztransparenz publishes the infeed forecast for the next calendar day towards the end of the trading day of TTF futures at 17:30, prior to market closing, while actual infeed numbers become available continuously on a 15 minutes frequency throughout the day. Figure 3 shows the wind infeed variables used in the analysis.3 Although both the infeed forecast and the realized infeed show seasonality with higher values during the winter, we also observe large intra-seasonal variation in infeed size. Even though higher peaks are observable in quarters IV and I, the infeed size appears to fall and rise significantly. In addition to the infeed forecast for wind electricity, we also incorporate the infeed forecast for PV electricity in our model (.P V .f oret ), as well as a variable for the forecasting error (.P V .errort ), which is the difference between the realized PV infeed on the trading day (.P V .realt ) and its forecast value from the previous calendar day. Figure 4 shows the PV infeed variables used in the analysis and indicates a seasonal pattern in the PV infeed as well: Because PV generation increases with higher sun intensity, larger values for the realized and forecasted infeed are found over the summer months. In contrast to wind infeed, however, PV

3 .W ind.real (blue) corresponds to the realized infeed on the day t the day-ahead TTF price is t formed and .W ind.f oret (red) is the infeed forecast made on day t for the next calendar day. .P V .errort (green) is the difference between the realized infeed .W ind.realt and its forecast value, made on the previous calendar day.

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Fig. 3 Infeed variables of German wind electricity. January 05, 2016–December 31, 2020

Fig. 4 Infeed variables of German PV electricity. January 05, 2016–December 31, 2020

infeed shows less intra-seasonal variation, and appears to be more consistently large during the summer and consistently low during the winter. In addition to the 5.8% share of district heating in total European gas consumption in 2020, household consumption and the share of commercial and public services amounted to 24.0 and 10.6% of total consumption respectively [25]. Because these consumer groups use gas primarily for heating, we account for the price effect of heating demand, following the example of [33]. We apply Northwest European heating degree days (HDD), which for each TTF trading day indicate how many degrees Celsius of heating are needed to achieve a base temperature of 18 .◦ C. Higher values for HDD are therefore expected to be associated with higher gas prices. Because heating demand is higher during the winter months, we find cyclicality in the data, presented in Fig. 5. The variable is calculated as the consumption-weighted average of heating degree days in the seven Northwest European cities Innsbruck (Austria), Uccle (Belgium), Nancy (France), Hammer Odde Fyr (Denmark), Frankfurt (Germany), Bologna (Italy), and Schiphol (the

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Fig. 5 Northwest European heating degree days (HDD). Consumption-weighted averages for Innsbruck, Nancy, Uccle, Hammer Odde Fyr, Frankfurt (Main), Bologna, and Schiphol. Full sample, January 05, 2016–December 31, 2020

Netherlands). Temperature data for Uccle (Belgium) is derived from the Royal Meteorological Institute of Belgium, while data for all other cities is downloaded from the European Climate Assessment & Dataset. We use monthly natural gas consumption statistics from eurostat to calculate consumption weights per country, and form our HDD variable calculated as H DDt =

.

7  (ηi,m H DDi,t ),

(1)

i=1

where .ηi,m is the share of gas consumed in country i in total gas consumed by all seven countries in month m and .H DDi,t are heating degree days in country i on day t of the year. Because of the predominant use of gas for heating, most natural gas is consumed over the winter months. The high demand during the winter, however, exceeds the pipeline import capacity. Therefore, gas-consuming countries operate underground storage facilities, often in depleted fields or salt caverns. Assuming a continuous import flow throughout the year, storage is filled during lower demand over the summer months, while gas is withdrawn during the winter, thereby smoothing the cyclicality in demand. With 21.23 bcm in Germany for instance, total storage capacity equates about one quarter of annual consumption [1, 2]. As market participants, however, anticipate typical changes in storage levels, we incorporate not the absolute filling level, but the difference of the filling level at time t to the calendar’s day average of the previous 3-years. The theoretical effect of relative storage levels on gas prices has been well established for instance by Cartea and Williams [10] or Brown and Yucel [9]. We expect filling levels below the three year average to signal a relative supply shortage and therefore to create upward price pressure. Conversely, above-average filling levels are expected to decrease

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Fig. 6 Daily storage differential to the average filling level of the underground storage of the previous 3 years in Denmark, the Netherlands, Belgium, France, Germany, Italy, and Austria. Full sample, January 05, 2016–December 31, 2020

the demand for natural gas. Note, however, that natural gas storage accounts for seasonal shifts in heating demand only and not for inter-day fluctuations in the demand for gas-fired electricity generation. The storage differential variable is depicted in Fig. 6. With domestic supply in 2021 equating only 11.1% of annual consumption in the EU4 [24], Europe’s gas supply is highly dependent on imports. Additionally, most imports into Europe originate from only a handful of supplier countries and often lay in the hands of state-controlled national monopolies. Because a small set of exporters may exert oligopolistic market behaviour, the competitive structure of the European gas market is often described as a Cournot setting. Therefore, to account for the effect of supplier market power on gas prices, in line with [33] we apply a variable for market concentration. A common framework for quantifying suppliers’ market share is the Herfindahl-Hirschmann index (HHI). The HHI in our study is constructed by linking daily European natural gas imports to the countries of their origin, in addition to the production in the UK and in the Netherlands. In contrast to [33], we thereby do not account for pipeline gas and LNG imports to Spain, due to the limited pipeline capacity from Spain to France. A detailed list of relevant border crossing points can be found in Table 15. We derive our data from the transparency platform ENTSOG (European Network of Transmission System Operators for Gas). Note however, that the origin of LNG imports arriving at regasification facilities cannot be traced to exporting countries. LNG imports are therefore treated as an independent category. A visual representation of the development of market concentration can be found in Fig. 7. We observe a slight decrease in market concentration towards the winter months, while market concentration is slightly

4 Note

excluding UK.

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Fig. 7 Daily Herfindahl-Hirschman index for natural gas imports into Northwest Europe. Full sample, January 05, 2016–December 31, 2020

Fig. 8 Consumption-weighted industrial output. January 05, 2016–December 31, 2020

higher during the summer. The data indicates that the mild seasonal pattern is driven by an increase of LNG imports during the winter months. In the EU, the share of industry consumption in 2020 amounted to 22.6% of gross natural gas consumption [25]. In line with [33], we capture industrial demand with an index for monthly industrial output in the manufacturing sector. We apply weights for the share of consumption weight of Belgium, Denmark, Germany, France, Italy, the Netherlands, and Austria in the sum of their consumption. Consumption-weighted manufacturing output, depicted in Fig. 8, is calculated as I N Dm =

.

7  (ηi,m I N Di,m ), i=1

(2)

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where .ηi,m is the share of gas consumed in country i in total gas consumption in month m and .I N Di,t is the industrial output index in the manufacturing sector in country i for month m. Although the potential for short-term substitution is limited, natural gas stands in competition with other fossil energy sources, for instance for heating. Figure 9 depicts the price development of the TTF day-ahead, Rotterdam Coal, Brent crude, and ETS CO.2 prices. For TTF prices we observe a slight upward trend between 2016 and 2018, before in the context of economic slowdown, prices swiftly decline between the end of 2018 and early 2020. Coal prices appear to broadly mimic the TTF price development, suggesting a certain degree of substitutability between these energy sources in heat and power generation. Much like gas, coal-fired generation tends to have higher marginal fuel costs, compared to relatively low capital costs. Paraschiv et al. [42] describe the competitive relationship between coal and gas prices. In their quantile regression model, the authors show that marginal technologies switch according to gas and coal price spreads as well as shifts in the supply function induced by wind and solar generation. Therefore, we also include a coal price variable in the model. The ordering of coal and gas use, however, is also subject to carbon emission prices, with coal requiring roughly twice as many certificates as gas, per output unit [17]. Thus, because CO.2 prices affect the ordering of these fuels, we apply the price for 1 metric tonne of carbon dioxide equivalent from EEX. Lastly, although most natural gas imports to Europe today are indexed against gas hub prices instead of oil, in 2020 5% of imports to Northwest Europe remained oil-indexed, compared to 72% in 2005 [34]. Because oil-indexed contracts often employ a minimum take quantity, importers have an arbitrage opportunity, when oil-indexed prices are lower than hub prices. In this case, these importers do not exceed the minimum take and buy additional quantities at trading hub, consequently leading to upward price pressure at gas hubs. Additionally, albeit oil being a transport fuel, gas and oil remain partial competitors in consumption for instance in heating, as with the case of refined heating oil. Thus, to capture the effect of oil price changes on natural gas price changes, we incorporate European Brent prices, depicted in Fig. 9. Table 3 shows the descriptive statistics of the variables used in the analysis. We observe that all variables strongly reject normality and further observe a large standard deviation in the infeed forecasts for wind, supporting the large variation in infeed observable in Fig. 3. The volatility in the infeed forecast from wind further motivates the use of this variable as a threshold to detect asymmetric gas price adjustment speed. Table 14 shows the correlation of all variables used in the analysis in levels. We observe that TTF prices are highly correlated with coal prices, but show a strong negative correlation with the 3-year storage differential. Regarding the explanatory variables, we find that the heating degree days variable (.ln(H DDt )) exhibits high negative correlation with the PV infeed forecast .ln(P V .f oret ) and moderate negative correlation with .ln(H H It ), the variable for supplier market concentration. This is intuitive, as high infeed from PV occurs primarily in summer, while heating demand drives prices only in winter. Furthermore, supplier diversification

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Fig. 9 Price graphs for European TTF and Coal prices (row 1), Brent and .CO2 prices (row 2). Full sample, January 05, 2016–December 31, 2020 Table 3 Descriptive statistics: January 05, 2016–December 31, 2020 .T T Ft .Coalt .Brentt .CO2t .I N Dm .ST ORt .H DDt .H H It .W ind.f oret .W ind.errort .P V .f oret .P V .errort

Obs. 1262 1262 1262 1262 1262 1262 1262 1262 1262 1262 1262 1262

Mean Min. Max. Std. Dev. Skew. Kurt. 15.32 3.10 29.48 5.29 0.11 2.69 69.50 38.45 102.60 18.60 0.14 1.57 48.37 5.18 75.06 11.96 −0.46 3.08 15.33 3.91 32.86 9.07 0.08 1.35 102.83 71.46 116.76 9.31 −1.05 4.13 0.04 −0.17 0.29 0.12 0.38 2.23 6.72 0.00 22.06 5.75 0.36 1.86 2855.60 2475.50 3493.26 186.80 0.65 3.20 10079.11 397.76 57574.10 8633.09 1.73 6.55 738.35 −5276.12 9887.40 1543.17 1.31 8.00 78633.94 3552.04 185683.0 49807.66 0.21 1.77 −642.43 −28740.92 32760.69 6452.22 −0.05 4.87

J.-B. prob. 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

is slightly higher during winter (see Sect. 3). To avoid multicollinearity, driven by the opposing seasonality between .ln(H DDt ) and the other two variables, we use first differences for heating degree days in our analysis. Table 12 indicates that we avoid multicollinearity, as the highest variance inflation factor (VIF) in our

Fuelling the Energy Transition Table 4 Unit root and stationarity test results

153 Test statistic Series .ln(T T Ft ) .ln(Coalt ) .ln(Brentt ) .ln(CO2t ) .ln(I N Dm ) .ST ORt .H DDt .ln(H H It ) .ln(W ind.f oret ) .W ind.errort .ln(P V .f oret ) .P V .errort

ADF −1.744 −1.615 −2.780 −2.773 −5.228∗∗∗ −2.269 −3.548∗∗ −4.670∗∗∗ −18.381∗∗∗ −16.022∗∗∗ −2.928 −28.659∗∗∗

PP −1.900 −1.652 −2.606 −2.766 −5.570∗∗∗ −1.251 −3.958∗∗ −5.778∗∗∗ −19.212∗∗∗ −28.440∗∗∗ −9.186∗∗∗ −28.662∗∗∗

H0 ADF and PP: Unit root All tests including intercept and time trend Critical values at 1%: ADF .−3.971, PP .−3.971 Critical values at 5%: ADF .−3.416, PP .−3.416 Critical values at 10%: ADF .−3.130, PP .−3.130 Lag selection ADF by SIC *.p < 0.10, **.p < 0.05, ***.p < 0.01

estimation equates 3.810456 (.D3t ). We further test for serial correlation with a Breusch-Godfrey test, presented in Table 13. Table 4 presents the results of the Augmented Dickey-Fuller (ADF) and PhillipsPerron (PP) tests for unit roots. The results indicate insufficient evidence for rejecting a unit root in all log-transformed price series, TTF, Coal, Brent crude oil, and CO.2 prices, but also in the storage time series. We conclude that these series exhibit a unit root and use first differences. We further find unambiguous results for monthly industrial output from manufacturing .ln(I NDm ), the supplier concentration index .ln(H H It ), wind infeed forecast (.W ind.f oret ), as well as the forecasting errors from both wind and PV (.W ind.errort and .P V .errort ). For all these series we find evidence for unit root rejects in both the ADF and PP test at 1% significance level and consequently proceed using levels. While we similarly find also evidence for unit rejections at 5% for heating degree days (.H DDt ), we nevertheless use this variable in first differences to avoid possible multicollinearity with other seasonal variables, such as the PV infeed forecast. Lastly, for the PV infeed forecast (.P V .f oret ), we obtain mixed results but find evidence for the rejection of unit roots using the non-parametric PP test. We therefore use these series in levels.

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4 Methodology In the analysis, we first implement the fundamental model put forward by Hulshof et al. [33], in which gas prices are determined in an equilibrium set by fundamental supply and demand variables, applying ordinary least squares (OLS). We then make variable extensions and implement threshold regression models to capture possible asymmetry in the adjustment process of natural gas to renewable electricity infeed.

4.1 Baseline Model OLS We apply log-transformation to all series, but those exhibiting zero values. Furthermore, we apply first differences to those variables for which the results of Table 4 provide insufficient evidence for rejecting the presence of a unit root. To contextualize the results of our threshold model, we employ a baseline model as defined below: Δ ln PtT T F = α + β1 (Δ ln PtCoal ) + β2 (Δ ln PtBrent ) + β3 (Δ ln PtCO2 ) +β4 (ln I N Dm ) + β5 (ΔST ORt ) + β6 (ΔH DDt ) + β7 ln(H H It ) +β8 (ln W ind.f oret ) + β9 (W ind.errort ) + β10 (ln P V .f oret ) .

+β11 (P V .errort ) +

5 

d ηn (Dn,t ) + ϵt ,

(3)

n=2

where .PtT T F is the daily day-ahead TTF gas prices, .PtCoal is the daily price for coal, .PtBrent the daily price for Brent crude, and .ln PtCO2 is the daily prices for carbon allowance. .I N Dm is the monthly index for consumption-weighted NorthEuropean industrial output, .ST ORt is the daily EU gas storage differential to the previous 3 years, .H DDt are daily heating degree days in Europe, and .H H It is the daily supplier concentration for natural gas in Northwest Europe. .W ind.f oret is the day-ahead forecast for German wind electricity infeed, while .W ind.errort is the daily forecasting error of wind infeed, calculated as the difference between the actual wind infeed on the trading day t (.W ind.realt ) and its forecast. Similarly, .P V .f oret is the day-ahead forecast for German PV electricity infeed made at time t, while .P V .errort is the residual PV supply that exceeded the forecast on day t, when the TTF price is formed. We capture day-of-the-week effects with dummy variables .Dnd and .ϵt is the error term with mean 0 and variance .σ 2 . The subscript t indicates the trading day on which the day-ahead TTF price is formed and m refers to the corresponding month. n indicates the day of the week, whereas .Δ indicates the use of first differences. To account for heteroskedasticity, we use Huber-WhiteHinkley heteroskedasticity consistent standard errors in all estimations (HC1).

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4.2 Threshold Regression Model We propose the use of a threshold regression model to investigate the sensitivity of TTF daily gas price fluctuations to the level of the renewables infeed. A general threshold model specification following [6] gives:

.

yt = Zt δ1 + ϵt

if − ∞ < qt < γt

yt = Zt δ2 + ϵt

if γt ≤ qt < ∞,

(4)

where .yt is the dependent variable, .Zt is a k x 1 vector of regime-sensitive independent variables, .qt is the threshold variable which splits the sample into two regimes, .γt is the threshold value, and .ϵt is an IID error term with mean 0 and variance .σ 2 . Threshold regression, in essence, is equivalent to a break-point regression (see [30, 43]), in which the data has been recorded with respect to the threshold variable. Therefore, no fundamental difference exists between the estimation of threshold and breakpoint models [7]. Gas-fired power plants are the primary tool for balancing out the intermittency of renewables in Germany [20]. Given the large fluctuations in wind infeed (Table 3) and limited on-site storage capacity of power-plant operators, we test whether at different regions of the chosen threshold variable, the day-ahead infeed forecast of either German wind or PV electricity affects the day-ahead gas price asymmetrically. In our analysis, we apply three threshold variables from wind and PV infeed in each estimated model. We assume that these variables drive the regimes of TTF price adjustments to other variables. In the case of wind, these are the infeed forecast (.W ind.f oret ), the infeed realization (.W ind.realt ), and the forecasting error (.W ind.errort ). For each threshold variable, we first estimate one model in which only the wind or PV infeed forecast is allowed to be regime sensitive, keeping all other variables non-regime dependent. We then test for asymmetric adjustment of the gas price to all fundamental variables, which follows the general threshold model specification of Eq. (4). Through sequential testing we derive the optimum number of thresholds m, which significantly minimizes the sum of squared residuals (SSR) across all possible sets of thresholds. Assuming a two regime model, with regime sensitivity of only the wind infeed forecast .W ind.realt to an observable threshold variable .W ind.realt and m observations with strictly increasing threshold values .(γ1 < γ2 < ... < γm ), the threshold regression model reads:

.

Δ ln PtT T F = Xt β + W ind.f oret δ1 + ϵt

if − ∞ < W ind.realt < γt

Δ ln PtT T F = Xt β + W ind.f oret δ2 + ϵt

if γt ≤ W ind.realt < ∞, (5)

where .PtT T F is the day-ahead TTF natural gas price, .Xt is a k x 1 vector of the region invariant parameters, which includes all explanatory variables from Eq. (3),

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besides the wind infeed forecast. .W ind.f oret is the exogenous variable for the wind infeed forecast for the next day, with region-specific coefficients .δ1 and .δ2 , and 2 .ϵt is an IID error term with mean 0 and variance .σ . Using an indicator function I that takes the value 1 if the expression is true and 0 otherwise, Eq. (5) can be rewritten as Δ ln PtT T F = Xt β + W ind.f oret δ1 I (−∞ < W ind.realt < γt ) +W ind.f oret δ2 I (γt ≤ W ind.realt < ∞) + ϵt . (6)

.

Lastly, for an unknown number of thresholds m we have Δ ln PtT T F = Xt β + .

m 

I (W ind.realt , γt )W ind.f oret δj + ϵt ,

j =0

(7)

where .(γ1 < γ2 < ... < γm ) are m ordered thresholds with .(γ0 = −∞ and γm+1 = ∞). Note that the procedure might yield more than two regimes as the most satisfying solution with regards to a statistically improved fit over the baseline OLS model. If no statistically significant threshold is found, we manually define the threshold selection with regards to highest significance of the regime sensitive infeed forecast variable. In those estimations, in which all fundamental variables are allowed to be regime sensitive, we restrict the total number of regimes to maximum of two. For the estimation of the threshold regression model we use Eviews and cross-reference our results in Stata. We apply the estimation procedure of [6] instead of the bootstrapping algorithm put forward by Hansen [29], which accounts for structural breaks in the lagged endogenous regressors of threshold autoregressive models (TAR). In our setting, we employ the Bai-Perron test of .L + 1 vs L sequentially determined thresholds, in a standard setting with trimming percentage 15.

.

5 Empirical Results and Discussion In the following, we present the results of two linear regression models (Table 5, models (1) and (2)) and a series of 15 threshold regression models (models (3)– (17)), in which we first allow the day-ahead TTF price to adjust asymmetrically to the wind and PV infeed forecast and in a next step to all fundamental variables. In all models, we correct for heteroskedasticity by using White heteroskedasticityconsistent standard errors and covariances (HC1). Because we cannot reject a unit root in the TTF price series, we estimate all models in first differences. Table 20 offers an overview of all implemented models.

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Table 5 Results, OLS estimation Sample Model .Δln(T T Ft ) .Δln(Coalt ) .Δln(Brentt ) .Δln(CO2t ) .ln(I N Dm ) .ΔST ORt .ΔH DDt .ln(H H It ) .ln(W ind.f oret ) .ln(P V .f oret ) .W ind.errort .P V .errort .D2t .D3t .D4t .D5t ._cons N .F -stat AI C 2 .Adj.R

05/01/2016–12/31/2020 (1) Coefficient Std. error 0.307290*** 0.119059 0.028800 0.017717 0.278050*** 0.036261 0.003247 0.016170 0.406134 0.396632 0.001979* 0.001022 0.025503* 0.014572 .−0.001723 0.001444 – – – – – – 0.000109 0.004642 .−0.004892 0.004526 .−0.005438 0.004592 .−0.002528 0.004651 .−0.200438 0.149752 1261 9.099484 .−3.543976 0.071614

(2) Coefficient 0.313660*** 0.028297 0.277923*** 0.002138 0.533008 0.001647* 0.042019*** .−0.002817* .−0.003648*** .−1.06E-07 .−3.83E-07** 0.000506 .−0.004338 .−0.004931 .−0.002452 .−0.277404* 1261 7.949362 .−3.546803 0.076409

Std. error 0.120206 0.017812 0.036320 0.016037 0.415648 0.000957 0.015883 0.001589 0.001340 6.69E-07 1.83E-07 0.004627 0.004534 0.004584 0.004614 0.155725

Estimation with White heteroskedasticity-consistent standard errors * .p < 0.1, ** .p < 0.05, *** .p < 0.01

5.1 OLS Model (1) replicates the analysis performed by Hulshof et al. [33] for the period from January 5, 2016 until December 31, 2020. The findings indicate a highly significant positive contribution of coal and CO.2 prices on the TTF price. We find that at 1% significance level, a 1% increase in coal prices and a 1% increase in CO.2 prices increase the day-ahead TTF gas price by 0.307 and 0.278%, respectively. This is intuitive, as coal and gas are historical substitutes in heating, while higher CO.2 prices shift the merit order curve in favor of gas as a marginal technology over coal. Because of the lower emissions of gas, higher CO.2 prices, which rose from around 5 EUR/tonne between 2016 and 2017 to 30 EUR/tonne by 2020 (Fig. 9), penalize the use of higher emission technologies, consequently increasing the demand for natural gas. In electricity generation, as the price for carbon allowances (certificates) increases, gas-fired generation moves down in the merit order. Contrary to the findings of [33], however, we find Brent prices to be insignificant for explaining the day-ahead TTF price between 2016 and 2020. This finding can be attributed

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to the drastic decline in oil-indexation, which in Northwest Europe declined from 72% in 2005 to 43% in 2010 and to less than 10% since 2015 [34].The decline of oil indexation lowers the potential for importers to arbitrage between oil-indexed prices and hub prices (see Sect. 3). Furthermore, for the duration of our study, the findings of model (1) indicate that the 3-year gas storage differential is insignificant in impacting the short-term price formation process, whereas a 1% increase in supplier market concentration (.ln(H H It )) increases the gas price by 0.0255% at 10% significance level. In accordance with [33], we find monthly industrial manufacturing output (.ln(I NDm )) to be insignificant, while a unit increase in heating degree days (.ΔH DDt ), on the other hand, increases the gas price by 0.1979% at 10% significance level. This finding is intuitive, as in 2020, household consumption accounted for 29.8% of total gas consumption in the EU [25]. Contrary to [33], the findings of model (1) suggest that the infeed forecast from wind electricity is negative, however, statistically insignificant. Departing from [33], we introduce three additional variables, one represented by the infeed from PV electricity and one each for the forecasting error from wind and PV infeed (Table 5, model (2)). We find that the introduction of additional variables improves the explanatory value of some of the variables already implemented in model (1). While significant only at 10% in model (1), the results of our proposed model (2) indicate that at 1% significance level, a 1% increase in the supplier concentration (.ln(H H It )) is expected to raise the day-ahead TTF price by 0.042% through the exercise of market power. This is a significant change to the findings of [33] for the period between 2011 and 2014 and is driven by the continuous decline of domestic European production and simultaneous rise in import dependence. Furthermore, the results highlight the substitution effect between gas and intermittent wind and PV generation, as infeed from these generation sources decreases the day-ahead gas price. We find that a 1% decrease in day-ahead wind infeed forecast is associated with a 0.00282% increase in the gas price, while a 1% decrease in the day-ahead PV infeed forecast is expected to increase the gas price by 0.00365%. These findings support our hypothesis that gas-fired power generation counterbalances the intermittency of renewable energies. Furthermore, the results of model (2) suggest that forecasting errors from PV also bring a significant contribution to the gas price formation process, while forecasting errors from wind do not. We find a statistically significant, yet marginally small negative effect of PV infeed forecasting errors on the day-ahead TTF gas price at 1% significance level. In particular, this finding suggests that when realized infeed from PV electricity is larger than forecasted (.P V .errort > 0), power plant operators have excess supply of gas, which can be used in gas-fired generation for the dayahead, thereby reducing the demand for additional quantities and hence suppressing the day-ahead price. Conversely, in the case of negative PV forecasting errors (.P V .errort < 0), more gas than forecasted has been used for electricity generation and power plant operators need to replenish their short-term storage with additional purchases, leading to an appreciation of the gas price. However, the results of model (2) (Table 5) indicate that only forecasting errors from PV infeed are significant for

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the day-ahead gas price, while wind forecasting errors are insignificant. In relative terms, forecasting errors from wind are positive with an average of 5.52%, while the mean forecasting errors from PV are only .−2.21%. Furthermore, based on the observation that PV infeed (Fig. 4) shows stronger seasonality than the infeed from wind (Fig. 3), one possible explanation of our results is that solar forecasts represent a more reliable information for the day-ahead gas price formation. Our finding of a negative effect from wind and PV electricity infeed forecasts on gas prices contradict the finding of [33]. For the period between 2011 and 2014, the authors found a significant positive contribution of wind electricity infeed forecasts on gas prices, resulting from loop flows, which have since been restricted through the installation of phase shifters on the Dutch-German border [40]. Furthermore, the cross-border capacity between Germany and the Netherlands increased with the installation of another 750 MW interconnector in 2018, totaling 4.5 GW in 2020 [14], which decreased the potential for grid bottlenecks. At the same time, Table 1 indicates that the gross electricity generation from wind and PV in Germany rose from 9.4 and 5.6%, respectively, in 2014, to 23.3 and 8.6%, respectively, in 2020 [8]. The rise in intermittent generation has increased the substitution need from flexible gas-fired generation, which rose from 9.8% in 2014 to 16.7% in 2020 [8]. Therefore, our finding of significant negative effects of wind and PV infeed forecasts on the day-ahead gas price can be linked to the systematic use of gas-fired generation as part of the energy transition in Europe, and especially in Germany. Our results are in line with the study of [42] who conclude that renewables have not only a directbut also an indirect effect on electricity prices, through gas prices, as a consequence of their substitution in production.

5.2 Threshold Regression Models In the following, we present the results of 15 different specifications of the threshold model defined in Sect. 4.1, which identify the asymmetric adjustments of the gas price to wind and PV infeed forecasts, infeed realizations, and forecasting errors. We allow for multiple thresholds. For wind and PV as threshold variables, we thereby first estimate three models in which only the forecast variable itself is allowed to be regime sensitive, keeping all other variables non-regime dependent. In a next step, all fundamental variables are allowed to shift between regimes, as defined by the PV- and wind threshold variables.

5.2.1

Wind Infeed Variables as Thresholds

Tables 6 and 7 show the results for three threshold regression models in which we allow the gas price to adjust asymmetrically to the wind infeed forecast .W ind.f oret , when setting as threshold variables .W ind.f oret (model (3)), .W ind.realt (model (4)), and .W ind.errort (model (5)). We find evidence for a

160 Table 6 Automatic threshold selection: 05/01/2016–12/31/2020. Regime sensitive variable: wind infeed forecast. Threshold variable: wind infeed forecast

C. Halser and F. Paraschiv (3)†† Coefficient Std. error .W ind.f oret < 4318.545 (355 obs) 0.003867 .ln(W ind.f oret ) .−0.021301*** .4318.545 ≤ W ind.f oret < 16917.44 (689 obs) 0.003351 .ln(W ind.f oret ) .−0.018510*** .16917.44 ≤ W ind.f oret (217 obs) .ln(W ind.f oret ) .−0.016141*** 0.003003 Non-Threshold Variables 0.315261*** .Δln(Coalt ) 0.114871 0.017930 0.021405 .Δln(Brentt ) .Δln(CO2t ) 0.035333 0.282591*** 0.015848 .−6.31E-05 .ln(I N Dm ) .ΔST ORt 0.403779 0.614365 .ΔH DDt 0.000943 0.001991** 0.015830 .ln(H H It ) 0.040473** .ln(P V .f oret ) 0.001332 .−0.003273** .W ind.errort 6.76E-07 .−4.51E-07 1.80E-07 .P V .errort .−3.55E-07** .D2t 0.004527 0.000664 .D3t 0.004415 .−0.004485 0.004471 .−0.005486 .D4t .D5t 0.004484 .−0.002335 ._cons 0.157276 .−0.117280 1261 N .F -stat 9.356218 .−3.572580 AI C 2 .Adj.R 0.101320 Model

.Δln(T T Ft )

Estimation with White heteroskedasticityconsistent standard errors Non-Threshold Variables are not regime sensitive * .p < 0.1, ** .p < 0.05, *** .p < 0.01 The superscript †† indicates that the thresholds of model (3) have been selected at 5% significance level over the baseline OLS model

significant threshold and an asymmetric adjustment speed of the gas price to the wind infeed forecast in models (3) and (4), using the wind infeed forecast and the realized infeed from wind as threshold variables, respectively. The results of model (3) suggest that the negative effect of wind infeed forecast on the gas price decreases in the regime with largest wind forecasts. While for infeed forecasts smaller than 4318.545 MWh, a 1% increase in the infeed lowers the gas price by 0.0213%, an equivalent 1% increase lowers the gas price by 0.01851% for forecasts between 4318.545 MWh and 16917.44 MWh, and by 0.01614% only for infeed forecasts equal to or larger than 16917.44 MWh. While the effect of the infeed forecast is at

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Table 7 Automatic threshold selection, model (4); two fixed regimes, model (5): 05/01/2016– 12/31/2020. Regime sensitive variable: wind infeed forecast. Threshold variables: wind infeed realization, model (4); wind forecasting error, model (5) Model .Δln(T T Ft ) .ln(W ind.f oret )

.ln(W ind.f oret )

.Δln(Coalt ) .Δln(Brentt ) .Δln(CO2t ) .ln(I N Dm ) .ΔST ORt .ΔH DDt .ln(H H It ) .ln(P V .f oret ) .W ind.errort .P V .errort .D2t .D3t .D4t .D5t ._cons

N .F -stat

AI C .Adj.R

2

(4)†† Coefficient Std. error .W ind.realt < 11740.1 (852 obs) .−0.004851*** 0.001777 .11740.1 ≤ W ind.realt (409 obs) .−0.003942** 0.001658 Non-threshold variables 0.315360*** 0.119195 0.028688 0.017518 0.280954*** 0.036155 0.001271 0.016000 0.452231 0.412363 0.001741* 0.000958 0.044183*** 0.015837 .−0.003109** 0.001343 .−7.60E-07 6.89E-07 .−3.81E-07** 1.82E-07 0.000102 0.004599 .−0.004752 0.004499 .−0.005095 0.004576 .−0.002556 0.004605 .−0.280568* 0.155311 1261 7.964683 .−3.551280 0.081254

(5) Coefficient Std. error .W ind.errort < 68.42 (411 obs) .−0.002966* 0.001595 .68.42 ≥ W ind.errort (850 obs) .−0.002373 0.001606 Non-threshold variables 0.312020*** 0.120074 0.029794* 0.017654 0.276956*** 0.036261 0.001998 0.015956 0.531647 0.416498 0.001598* 0.000962 0.040479** 0.015962 .−0.003696*** 0.001344 .−1.17E-06 8.54E-07 .−3.93E-07** 1.82E-07 0.000496 0.004633 .−0.004469 0.004553 .−0.005095 0.004610 .−0.002566 0.004623 .−0.265370* 0.155763 1261 7.629854 .−3.547366 0.077651

Estimation with White heteroskedasticity-consistent standard errors Non-Threshold Variables are not regime sensitive * .p < 0.1, ** .p < 0.05, *** .p < 0.01 The superscript †† indicates that the thresholds of model (4) have been selected at 5% significance level over the baseline OLS model

1% significance level in all three regimes, model (3) also delivers overall improved fit over the baseline OLS model (2) at 5% significance levels . We next choose the realized wind infeed on the trading day as threshold variable (model (4)) and obtain a better fit than the baseline OLS model (2), as shown by the AIC and adjusted R-squared values. We find two relevant regimes: When the realized infeed on the day the TTF price is formed is smaller than 11740.1 MWh, a 1% increase in the infeed forecast decreases the gas price by 0.00485%, while the decrease is at 0.00394% for a realized infeed larger and equal to 11740.1 MWh. Conversely, when the realized infeed on the trading day is small, a further decrease in the infeed forecast raises the gas prices to a greater extent and at higher

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significance (1%), compared to trading days with a large infeed (5% significance level). This finding appears to capture the purchasing activity of gas power plant operators to prevent supply bottlenecks. Lastly, we estimate model (5) in which we allow for asymmetric adjustment of the gas price to the wind forecasting error (.W ind.errort ), keeping all other variables non-regime sensitive. Following BaiPerron (see [6]), however, we fail to find statistically significant regimes set by the wind forecasting error. One possible explanation is the overlap between the infeed error and realization variables, as the former is derived by the latter. Tables 8 and 9 present the results of three threshold models in which we allow the gas price to asymmetrically adjust to the PV forecast set by the wind infeed forecast for the day-ahead (model (6)), the realized wind infeed on the day the gas price is formed (model (7)), and by the wind infeed forecasting error (model (8)). At 5% significance level, we find that also the effect of the PV infeed forecast on the day-ahead gas price varies depending on the regime of the wind infeed forecast and realization. The results of model (6) suggest that the infeed from PV electricity only decreases the day-ahead TTF price when the infeed forecast for wind electricity is low. Likewise the results of model (7) suggest that regimes of low realized wind electricity infeed on the trading day are associated with a larger and more significant gas price reduction from PV infeed forecasts. These findings show that in high regimes of wind infeed, renewables generation exceeds the substitutable infeed from gas. Furthermore, the results of model (8) suggest that the infeed from PV electricity decreases the gas price more significantly in a regime of small and negative forecasting errors, while the thresholds of this model are found to be insignificant. Lastly, Tables 10 and 11 present the results for threshold models in which we allow all fundamental variables to be regime sensitive. The threshold variables are set as the wind infeed forecast (model (9)), the realized infeed on the trading day (model (10)), and the forecasting error (model (11)). The thresholds in models (9) and (10) are statistically not significant, yet indicating that coal prices (.Coalt ) and heating demand (.H DDt ) have a significant positive effect on gas prices in regimes of either a low wind infeed forecast (model (9)), or a low realized infeed on the trading day (model (10)). These findings are intuitive, as low infeed from wind and thus high gas usage for electricity raises the sensitivity of substitute technologies and other demand factors. Model (11) (Table 11), on the other hand, shows better fit than the baseline model (2) at 1% significance level. We find that in a regime of small and negative forecasting errors, industrial demand (.I N Dm ) increases the gas price at 5% significance. This finding hints towards the inflexibility of industrial consumers to substitute gas in case of a short-term supply shortage. Furthermore, for regimes of negative forecasting errors (model (11), regime 1), we find that an increase in forecasting errors from wind and PV infeed is expected to decrease the gas price at 10% and 5% significance level respectively. Intuitively, when the realized infeed from either generation source is smaller than expected, more gas has to be burned in substitution. Likewise in a regime of negative wind forecasting errors, the PV infeed forecast is expected to lower the gas price at 1% significance level, explained

Fuelling the Energy Transition Table 8 Automatic threshold selection: 05/01/2016–12/31/2020. Regime sensitive variable: PV infeed forecast. Threshold variable: wind infeed forecast

163 (6)†† Coefficient Std. error .W ind.f oret < 4318.54 (355 obs) .ln(P V .f oret ) .−0.005402*** 0.001380 .4318.54 ≤ W ind.f oret < 16917.44 (689 obs) .ln(P V .f oret ) .−0.003444*** 0.001324 .16917.44 ≤ W ind.f oret (217 obs) .ln(P V .f oret ) .−0.001077 0.001437 Non-threshold variables .Δln(Coalt ) 0.317554*** 0.114759 .Δln(Brentt ) 0.021047 0.017883 .Δln(CO2t ) 0.280306*** 0.035416 .ln(I N Dm ) .−0.001905 0.015906 .ΔST ORt 0.636367 0.405336 .ΔH DDt 0.001886** 0.000939 .ln(H H It ) 0.043269*** 0.015859 .ln(W ind.f oret ) .−0.018905*** 0.003306 .W ind.errort .−4.41E-07 6.75E-07 .P V .errort .−3.46E-07* 1.80E-07 .D2t 0.000674 0.004530 .D3t .−0.004487 0.004423 .D4t .−0.005469 0.004478 .D5t .−0.002332 0.004482 ._cons .−0.125617 0.156568 N 1261 .F -stat 9.355575 AI C .−3.572572 2 .Adj.R 0.101313 Model

.Δln(T T Ft )

Estimation with White heteroskedasticity-consistent standard errors Non-Threshold Variables are not regime sensitive *.p < 0.1, **.p < 0.05, ***.p < 0.01 The superscript †† indicates that the thresholds of model (6) have been selected at 5% significance level over the baseline OLS model

by the substitution effect outlined in Sect. 1.2. Conversely, in a regime of positive forecasting errors (model (11), regime 2), only the wind infeed forecast is significant at 10% for the day-ahead gas price, while the infeed forecast from PV, wind-, and PV forecasting errors are not statistically significant. This finding suggests asymmetry in the effect of forecasting errors from wind, which only affect the gas price when they are negative and are irrelevant for the short-term price formation process when positive. Furthermore, the findings of model (11) suggest that an increase in the supplier concentration raises the day-ahead TTF price at 1% significance in a regime

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Table 9 Automatic threshold selection, model (7); two fixed regimes, model (8): 05/01/2016– 12/31/2020. Regime sensitive variable: PV infeed forecast. Threshold variables: wind infeed realization, model (7); wind forecasting error model (8) Model .Δln(T T Ft ) .ln(P V .f oret )

.ln(P V .f oret )

.Δln(Coalt ) .Δln(Brentt ) .Δln(CO2t ) .ln(I N Dm ) .ΔST ORt .ΔH DDt .ln(H H It ) .ln(W ind.f oret ) .W ind.errort .P V .errort .D2t .D3t .D4t .D5t ._cons

N .F -stat

AI C .Adj.R

2

(7)†† Coefficient Std. error .W ind.realt < 11740.1 (852 obs) .−0.003440** 0.001335 .11740.1 ≤ W ind.realt (409 obs) .−0.002664* 0.001365 Non-threshold variables 0.315092*** 0.119273 0.028803* 0.017492 0.280325*** 0.036181 0.000782 0.015987 0.460542 0.412859 0.001699* 0.000956 0.045058*** 0.015863 .−0.004607*** 0.001719 .−7.33E-07 6.87E-07 .−3.76E-07** 1.82E-07 0.000130 0.004603 .−0.004767 0.004501 .−0.005148 0.004573 -0.002563 0.004605 .−0.283657* 0.155327 1261 7.965606 .−3.551291 0.081264

(8) Coefficient Std. error .W ind.errort < 68.42 (411 obs) .−0.004063*** 0.001356 .68.42 ≤ W ind.errort (850 obs) .−0.003513*** 0.001345 Non-threshold variables 0.312125*** 0.119960 0.030453* 0.017645 0.276878*** 0.036254 0.002083 0.015932 0.530526 0.416486 0.001594* 0.000963 0.039808** 0.015958 .−0.002588 0.001597 .−1.26E-06 8.31E-07 .−3.94E-07** 1.82E-07 0.000528 0.004630 .−0.004475 0.004555 .−0.005131 0.004614 .−0.002585 0.004627 .−0.260173* 0.155434 1261 7.710563 .−3.548311 0.078522

Estimation with White heteroskedasticity-consistent standard errors Non-Threshold Variables are not regime sensitive * .p < 0.1, ** .p < 0.05, *** .p < 0.01 The superscript †† indicates that the thresholds of model (7) have been selected at 5% significance level over the baseline OLS model

of negative forecasting errors, when more gas than expected is burned for electricity, while heating demand is only significant in case of positive forecasting errors. Negative forecasting errors raise the significance of coal as a substitute technology and our findings further indicate that the coefficients of coal prices are significant at 1% confidence levels only in low wind forecast (model (9)) and realization (model (10)) regimes. This finding is intuitive because coal is used as a substitution technology for intermittent wind and PV electricity generation (see Figs. 1 and 2) in addition to gas. Lastly, PV forecasting errors are only significant

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Table 10 Two fixed regimes: 05/01/2016–12/31/2020. Regime sensitive variables: All fundamental variables. Threshold variables: wind infeed forecast, model (9); wind infeed realization, model (10) Model .Δln(T T Ft )

.Δln(Coalt ) .Δln(Brentt ) .Δln(CO2t ) .ln(I N Dm ) .ΔST ORt .ΔH DDt .ln(H H It ) .ln(W ind.f oret ) .ln(P V .f oret ) .W ind.errort .P V .errort ._cons

.Δln(Coalt ) .Δln(Brentt ) .Δln(CO2t ) .ln(I N Dm ) .ΔST ORt .ΔH DDt .ln(H H It ) .ln(W ind.f oret ) .ln(P V .f oret ) .W ind.errort .P V .errort ._cons

.D2t .D3t .D4t .D5t

N .F -stat

AI C .Adj.R

2

(9) Coefficient Std. error .W ind.f oret < 4318.545 (355 obs) 0.615101*** 0.171021 0.042034 0.073374 0.231847*** 0.076353 .−0.024802 0.024825 2.244052* 1.199698 0.007478** 0.003708 0.039301 0.028377 .−0.023938*** 0.006942 .−0.003434 0.003323 .−1.27E-06 2.82E-06 .−7.22E-07** 3.62E-07 0.029187 0.262532 .4318.545 ≤ W ind.f oret (906 obs) 0.178168 0.139304 0.028338 0.017541 0.294660*** 0.042047 0.012633 0.020719 0.018807 0.353503 0.000650 0.000768 0.037225** 0.018851 .−0.004025 0.002557 .−0.002898* 0.001573 .−1.70E-07 6.99E-07 .−2.25E-07 2.05E-07 .−0.283137 0.200927 Non-threshold variables 0.000309 0.004515 .−0.004740 0.004326 .−0.005714 0.004373 .−0.001910 0.004524 1261 6.264390 .−3.564856 0.101373

(10) Coefficient Std. error .W ind.realt < 6074.727 (499 obs) 0.537788** 0.244522 -0.064642 0.063478 0.310910*** 0.059474 .−0.014282 0.024276 1.279749* 0.731660 0.002461* 0.001436 0.026576 0.025914 .−0.009613*** 0.002841 .−0.003632 0.002467 1.57E-07 2.48E-06 .−7.57E-07*** 2.69E-07 .−0.024829 0.253308 .6074.727 ≤ W ind.realt (762 obs) 0.193707 0.129912 0.045454** 0.018898 0.273769*** 0.044445 0.009745 0.019929 0.109286 0.469340 0.001594 0.001179 0.055620*** 0.019605 .−0.000239 0.002166 .−0.002084 0.001575 .−8.54E-07 6.93E-07 .−1.81E-07 2.52E-07 .−0.458457** 0.188477 Non-threshold variables 0.001442 0.004653 .−0.004637 0.004593 .−0.005261 0.004624 .−0.002577 0.004677 1261 5.372871 .−3.547540 0.085676

Estimation with White heteroskedasticity-consistent standard errors Non-Threshold Variables are not regime sensitive * .p < 0.1, ** .p < 0.05, *** .p < 0.01

166 Table 11 Automatic threshold selection: 05/01/2016–12/31/2020. Regime sensitive variables: All fundamental variables. Threshold variable: wind forecasting error

C. Halser and F. Paraschiv Model .Δln(T T Ft )

.Δln(Coalt ) .Δln(Brentt ) .Δln(CO2t ) .ln(I N Dm ) .ΔST ORt .ΔH DDt .ln(H H It ) .ln(W ind.f oret ) .ln(P V .f oret ) .W ind.errort .P V .errort ._cons

.Δln(Coalt ) .Δln(Brentt ) .Δln(CO2t ) .ln(I N Dm ) .ΔST ORt .ΔH DDt .ln(H H It ) .ln(W ind.f oret ) .ln(P V .f oret ) .W ind.errort .P V .errort ._cons

.D2t .D3t .D4t .D5t

N .F -stat

AI C .Adj.R

2

(11)††† Coefficient Std. error .W ind.errort