This book discusses the interplay of stochastics (applied probability theory) and numerical analysis in the field of qua

*3,079*
*688*
*9MB*

*English*
*Pages 1310*
*Year 2019*

*Table of contents : ContentsPrefaceAcknowledgmentUsing this Book1 Basics about Stochastic Processes 1.1 Stochastic variables 1.1.1 Density function, expectation, variance 1.1.2 Characteristic function 1.1.3 Cumulants and moments 1.2 Stochastic processes, martingale property 1.2.1 Wiener process 1.2.2 Martingales 1.2.3 Iterated expectations (Tower property) 1.3 Stochastic integration, It ˆo integral 1.3.1 Elementary processes 1.3.2 Ito isometry 1.3.3 Martingale representation theorem 1.4 Exercise set2 Introduction to Financial Asset Dynamics 2.1 Geometric Brownian motion asset price process 2.1.1 Ito process 2.1.2 Ito’s lemma 2.1.3 Distributions of S(t) and logS(t) 2.2 First generalizations 2.2.1 Proportional dividend model 2.2.2 Volatility variation 2.2.3 Time-dependent volatility 2.3 Martingales and asset prices 2.3.1 P-measure prices 2.3.2 Q-measure prices 2.3.3 Parameter estimation under real-world measure P 2.4 Exercise set3 The Black-Scholes Option Pricing Equation 3.1 Option contract definitions 3.1.1 Option basics 3.1.2 Derivation of the partial differential equation 3.1.3 Martingale approach and option pricing 3.2 The Feynman-Kac theorem and the Black-Scholes model 3.2.1 Closed-form option prices 3.2.2 Green’s functions and characteristic functions 3.2.3 Volatility variations 3.3 Delta hedging under the Black-Scholes model 3.4 Exercise set4 Local Volatility Models 4.1 Black-Scholes implied volatility 4.1.1 The concept of implied volatility 4.1.2 Implied volatility; implications 4.1.3 Discussion on alternative asset price models 4.2 Option prices and densities 4.2.1 Market implied volatility smile and the payoff 4.2.2 Variance swaps 4.3 Non-parametric local volatility models 4.3.1 Implied volatility representation of local volatility 4.3.2 Arbitrage-free conditions for option prices 4.3.3 Advanced implied volatility interpolation 4.3.4 Simulation of local volatility model 4.4 Exercise set5 Jump Processes 5.1 Jump diffusion processes 5.1.1 Ito’s lemma and jumps 5.1.2 PIDE derivation for jump diffusion process 5.1.3 Special cases for the jump distribution 5.2 Feynman-Kac theorem for jump diffusion process 5.2.1 Analytic option prices 5.2.2 Characteristic function for Merton’s model 5.2.3 Dynamic hedging of jumps with the Black-Scholes model 5.3 Exponential Levy processes 5.3.1 Finite activity exponential L´evy processes 5.3.2 PIDE and the L´evy triplet 5.3.3 Equivalent martingale measure 5.4 Infinite activity exponential Levy processes 5.4.1 Variance Gamma process 5.4.2 CGMY process 5.4.3 Normal inverse Gaussian process 5.5 Discussion on jumps in asset dynamics 5.6 Exercise set6 The COS Method for European Option Valuation 6.1 Introduction into numerical option valuation 6.1.1 Integrals and Fourier cosine series 6.1.2 Density approximation via Fourier cosine expansion 6.2 Pricing European options by the COS method 6.2.1 Payoff coefficients 6.2.2 The option Greeks 6.2.3 Error analysis COS method 6.2.4 Choice of integration range 6.3 Numerical COS method results 6.3.1 Geometric Brownian Motion 6.3.2 CGMY and VG processes 6.3.3 Discussion about option pricing 6.4 Exercise set7 Multidimensionality, Change of Measure, Affine Processes 7.1 Preliminaries for multi-D SDE systems 7.1.1 The Cholesky decomposition 7.1.2 Multi-D asset price processes 7.1.3 Ito’s lemma for vector processes 7.1.4 Multi-dimensional Feynman-Kac theorem 7.2 Changing measures and the Girsanov theorem 7.2.1 The Radon-Nikodym derivative 7.2.2 Change of num´eraire examples 7.2.3 From P to Q in the Black-Scholes model 7.3 Affine processes 7.3.1 Affine diffusion processes 7.3.2 Affine jump diffusion processes 7.3.3 Affine jump diffusion process and PIDE 7.4 Exercise set8 Stochastic Volatility Models 8.1 Introduction into stochastic volatility models 8.1.1 The Sch¨ obel-Zhu stochastic volatility model 8.1.2 The CIR process for the variance 8.2 The Heston stochastic volatility model 8.2.1 The Heston option pricing partial differential equation 8.2.2 Parameter study for implied volatility skew and smile 8.2.3 Heston model calibration 8.3 The Heston SV discounted characteristic function 8.3.1 Stochastic volatility as an affine diffusion process 8.3.2 Derivation of Heston SV characteristic function 8.4 Numerical solution of Heston PDE 8.4.1 The COS method for the Heston model 8.4.2 The Heston model with piecewise constant parameters 8.4.3 The Bates model 8.5 Exercise set9 Monte Carlo Simulation 9.1 Monte Carlo basics 9.1.1 Monte Carlo integration 9.1.2 Path simulation of stochastic differential equations 9.2 Stochastic Euler and Milstein schemes 9.2.1 Euler scheme 9.2.2 Milstein scheme: detailed derivation 9.3 Simulation of the CIR process 9.3.1 Challenges with standard discretization schemes 9.3.2 Taylor-based simulation of the CIR process 9.3.3 Exact simulation of the CIR model 9.3.4 The Quadratic Exponential scheme 9.4 Monte Carlo scheme for the Heston model 9.4.1 Example of conditional sampling and integrated variance 9.4.2 The integrated CIR process and conditional sampling 9.4.3 Almost exact simulation of the Heston model 9.4.4 Improvements of Monte Carlo simulation 9.5 Computation of Monte Carlo Greeks 9.5.1 Finite differences 9.5.2 Pathwise sensitivities 9.5.3 Likelihood ratio method 9.6 Exercise set10 Forward Start Options; Stochastic Local Volatility Model 10.1 Forward start options 10.1.1 Introduction into forward start options 10.1.2 Pricing under the Black-Scholes model 10.1.3 Pricing under the Heston model 10.1.4 Local versus stochastic volatility model 10.2 Introduction into stochastic-local volatility model 10.2.1 Specifying the local volatility 10.2.2 Monte Carlo approximation of SLV expectation 10.2.3 Monte Carlo AES scheme for SLV model 10.3 Exercise set11 Short-Rate Models 11.1 Introduction to interest rates 11.1.1 Bond securities, notional 11.1.2 Fixed-rate bond 11.2 Interest rates in the Heath-Jarrow-Morton framework 11.2.1 The HJM framework 11.2.2 Short-rate dynamics under the HJM framework 11.2.3 The Hull-White dynamics in the HJM framework 11.3 The Hull-White model 11.3.1 The solution of the Hull-White SDE 11.3.2 The HW model characteristic function 11.3.3 The CIR model under the HJM framework 11.4 The HJM model under the T-forward measure 11.4.1 The Hull-White dynamics under the T-forward measure 11.4.2 Options on zero-coupon bonds under Hull-White model 11.5 Exercise set12 Interest Rate Derivatives and Valuation Adjustments 12.1 Basic interest rate derivatives and the Libor rate 12.1.1 Libor rate 12.1.2 Forward rate agreement 12.1.3 Floating rate note 12.1.4 Swaps 12.1.5 How to construct a yield curve 12.2 More interest rate derivatives 12.2.1 Caps and floors 12.2.2 European swaptions 12.3 Credit Valuation Adjustment and Risk Management 12.3.1 Unilateral Credit Value Adjustment 12.3.2 Approximations in the calculation of CVA 12.3.3 Bilateral Credit Value Adjustment (BCVA) 12.3.4 Exposure reduction by netting 12.4 Exercise set13 Hybrid Asset Models, Credit Valuation Adjustment 13.1 Introduction to affine hybrid asset models 13.1.1 Black-Scholes Hull-White (BSHW) model 13.1.2 BSHW model and change of measure 13.1.3 Schobel-Zhu Hull-White (SZHW) model 13.1.4 Hybrid derivative product 13.2 Hybrid Heston model 13.2.1 Details of Heston Hull-White hybrid model 13.2.2 Approximation for Heston hybrid models 13.2.3 Monte Carlo simulation of hybrid Heston SDEs 13.2.4 Numerical experiment, HHW versus SZHW model 13.3 CVA exposure profiles and hybrid models 13.3.1 CVA and exposure 13.3.2 European and Bermudan options example 13.4 Exercise set14 Advanced Interest Rate Models and Generalizations 14.1 Libor market model 14.1.1 General Libor market model specifications 14.1.2 Libor market model under the HJM framework 14.2 Lognormal Libor market model 14.2.1 Change of measure in the LMM 14.2.2 The LMM under the terminal measure 14.2.3 The LMM under the spot measure 14.2.4 Convexity correction 14.3 Parametric local volatility models 14.3.1 Background, motivation 14.3.2 Constant Elasticity of Variance model (CEV) 14.3.3 Displaced diffusion model 14.3.4 Stochastic volatility LMM 14.4 Risk management: The impact of a financial crisis 14.4.1 Valuation in a negative interest rates environment 14.4.2 Multiple curves and the Libor rate 14.4.3 Valuation in a multiple curves setting 14.5 Exercise set15 Cross-Currency Models 15.1 Introduction into the FX world and trading 15.1.1 FX markets 15.1.2 Forward FX contract 15.1.3 Pricing of FX options, the Black-Scholes case 15.2 Multi-currency FX model with short-rate interest rates 15.2.1 The model with correlated, Gaussian interest rates 15.2.2 Pricing of FX options 15.2.3 Numerical experiment for the FX-HHW model 15.2.4 CVA for FX swaps 15.3 Multi-currency FX model with interest rate smile 15.3.1 Linearization and forward characteristic function 15.3.2 Numerical experiments with the FX-HLMM model 15.4 Exercise setReferencesIndex*