# Mathematical Methods and Quantum Mathematics for Economics and Finance 9789811566103, 9789811566110

Given the rapid pace of development in economics and finance, a concise and up-to-date introduction to mathematical meth

407 124 6MB

English Pages 566 [439] Year 2020

Preface
Synopsis
Abstract
I. Introduction
II. Linear Algebra
III. Calculus
IV. Probability Theory
V. Quantum Mathematics
Contents
Part I Introduction
1 Series and Functions
1.1 Introduction
1.2 Basic Algebra
1.2.1 Binomial Expansion
1.3.1 Higher Order Polynomial
1.4 Finite Series
1.4.1 Example: NPV
1.4.2 Example: Coupon Bond
1.4.3 Example: Valuation of a Firm
1.4.4 Example: 1/3
1.5 Infinite Series
1.6 Cauchy Convergence
1.6.1 Example: Geometric Series
1.7 Expansion of sqrt1+x
1.8 Problems
1.9 Functions
1.10 Exponential Function
1.10.1 Parametric Representation
1.10.2 Continuous Compounding
1.10.3 Logarithm Function
1.11 Supply and Demand Functions
1.12 Option Theory Payoff
1.13 Interest Rates; Coupon Bonds
1.14 Problems
Part II Linear Algebra
2 Simultaneous Linear Equations
2.1 Introduction
2.2 Two Commodities
2.3 Vectors
2.4 Basis Vectors
2.4.1 Scalar Product
2.5 Linear Transformations; Matrices
2.6 EN: N-Dimensional Linear Vector Space
2.7 Linear Transformations of EN
2.8 Problems
3 Matrices
3.1 Introduction
3.2 Matrix Multiplication
3.3 Properties of NtimesN Matrices
3.4 System of Linear Equations
3.5 Determinant: 2times2 Case
3.6 Inverse of a 2times2 Matrix
3.7 Tensor (Outer) Product; Transpose
3.7.1 Transpose
3.8 Eigenvalues and Eigenvectors
3.8.1 Matrices: Spectral Decomposition
3.9 Problems
4 Square Matrices
4.1 Introduction
4.2 Determinant: 3times3 Case
4.3 Properties of Determinants
4.4 NtimesN Determinant
4.4.1 Inverse of a NtimesN Matrix
4.5 Leontief Input-Output Model
4.5.1 Hawkins-Simon Condition
4.5.2 Two Commodity Case
4.6 Symmetric Matrices
4.7 2times2 Symmetric Matrix
4.8 NtimesN Symmetric Matrix
4.9 Orthogonal Matrices
4.10 Symmetric Matrix: Diagonalization
4.10.1 Functions of a Symmetric Matrix
4.11 Hermitian Matrices
4.12 Diagonalizable Matrices
4.12.1 Non-symmetric Matrix
4.13 Change of Basis States
4.13.1 Symmetric Matrix: Change of Basis
4.13.2 Diagonalization and Rotation
4.13.3 Rotation and Inversion
4.13.4 Hermitian Matrix: Change of Basis
4.14 Problems
Part III Calculus
5 Integration
5.1 Introduction
5.2 Sums Leading to Integrals
5.3 Definite and Indefinite Integrals
5.4 Applications in Economics
5.5 Multiple Integrals
5.5.1 Change of Variables
5.6 Gaussian Integration
5.6.1 Gaussian Integration for Options
5.7 N-Dimensional Gaussian Integration
5.8 Problems
6 Differentiation
6.1 Introduction
6.2 Differentiation: Inverse of Integration
6.3 Rules of Differentiation
6.4 Integration by Parts
6.5 Taylor Expansion
6.6 Minimum and Maximum
6.6.1 Maximizing Profit
6.7 Integration; Change of Variable
6.8 Partial Derivatives
6.8.1 Cobb-Douglas Production Function
6.8.2 Chain Rule; Jacobian
6.8.3 Polar Coordinates; Gaussian integration
6.9 Hessian Matrix: Critical Points
6.10 Firm's Profit Maximization
6.11 Constrained Optimization: Lagrange Multiplier
6.11.1 Interpretation of λc
6.12 Line Integral; Exact and Inexact Differentials
6.13 Problems
7 Ordinary Differential Equations
7.1 Introduction
7.2 Separable Differential Equations
7.3 Linear Differential Equations
7.3.1 Dynamics of Price: Supply and Demand
7.4 Bernoulli Differential Equation
7.5 Swan-Solow Model
7.6 Homogeneous Differential Equation
7.7 Second Order Linear Differential Equations
7.7.1 Special Case
7.8 Riccati Differential Equation
7.9 Inhomogeneous Second Order Differential Equations
7.9.1 Special Case
7.10 System of Linear Differential Equations
7.11 Problems
Part IV Probability Theory
8 Random Variables
8.1 Introduction: Risk
8.1.1 Example
8.2 Probability Theory
8.3 Discrete Random Variables
8.3.1 Bernoulli Random Variable
8.3.2 Binomial Random Variable
8.3.3 Poisson Random Variable
8.4 Continuous Random Variables
8.4.1 Uniform Random Variable
8.4.2 Exponential Random Variable
8.4.3 Normal (Gaussian) Random Variable
8.5 Problems
9 Option Pricing and Binomial Model
9.1 Introduction
9.2 Pricing Options: Dynamic Replication
9.2.1 Single Step Binomial Model
9.3 Dynamic Portfolio
9.3.1 Single Step Option Price
9.4 Martingale: Risk-Neutral Valuation
9.5 N-Steps Binomial Option Price
9.6 N=2 Option Price: Binomial Tree
9.7 Binomial Option Price: Put-Call Parity
9.8 Summary
9.9 Problems
10 Probability Distribution Functions
10.1 Introduction
10.1.1 Cumulative Distribution Function
10.2 Axioms of Probability Theory
10.3 Joint Probability Density
10.4 Independent Random Variables
10.5 Central Limit Theorem: Law of Large Numbers
10.5.1 Binomial Random Variable
10.5.2 Limit of Binomial Distribution
10.6 Correlated Random Variables
10.6.1 Bernoulli Random Variables
10.6.2 Gaussian Random Variables
10.7 Marginal Probability Density
10.8 Conditional Expectation Value
10.8.1 Bernoulli Random Variables
10.8.2 Binomial Random Variables
10.8.3 Poisson Random Variables
10.8.4 Gaussian Random Variables
10.9 Problems
11 Stochastic Processes and Black–Scholes Equation
11.1 Introduction
11.2 Stochastic Differential Equation
11.3 Gaussian White Noise and Delta Function
11.3.1 Integrals of White Noise
11.4 Ito Calculus
11.5 Lognormal Stock Price
11.5.1 Geometric Mean of Stock Price
11.6 Linear Langevin Equation
11.6.1 Security's Random Paths
11.7 Black–Scholes Equation; Hedged Portfolio
11.8 Assumptions in the Black–Scholes
11.9 Martingale: Black–Scholes Model
11.10 Black–Scholes Option Price
11.10.1 Put-Call Parity
11.11 Black–Scholes Limit of the Binomial Model
11.12 Problems
12 Stochastic Processes and Merton Equation
12.1 Introduction
12.2 Firm's Stochastic Differential Equation
12.3 Contingent Claims on Firm
12.4 No Arbitrage Portfolio
12.5 Merton Equation
12.6 Risky Corporate Coupon Bond
12.7 Zero Coupon Corporate Bond
12.8 Zero Coupon Bond Yield Spread
12.9 Default Probability and Leverage
12.10 Recovery Rate of Defaulted Bonds
12.11 Merton's Risky Coupon Bond
Part V Quantum Mathematics
13 Functional Analysis
13.1 Introduction
13.2 Dirac Bracket: Vector Notation
13.3 Continuous Basis States
13.4 Dirac Delta Function
13.5 Basis States for Function Space
13.6 Operators on Function Space
13.7 Gaussian Kernel
13.8 Fourier Transform
13.8.1 Taylor Expansion
13.8.2 Green's Function
13.8.3 Black–Scholes Option Price
13.9 Functional Differentiation
13.10 Functional Integration
13.11 Gaussian White Noise
13.12 Simple Harmonic Oscillator
13.13 Acceleration Action
14 Hamiltonians
14.1 Introduction
14.2 Black–Scholes and Merton Hamiltonian
14.3 Option Pricing Kernel
14.4 Black–Scholes Pricing Kernel
14.5 Merton Oscillator Hamiltonian
14.6 Hamiltonian: Martingale Condition
14.7 Hamiltonian and Potentials
14.8 Double Knock Out Barrier Option
15 Path Integrals
15.1 Introduction
15.2 Feynman Path Integral
15.3 Path Dependent Options
15.4 Merton Lagrangian
15.5 Black-Scholes Discrete Path Integral
15.6 Path Integral: Time-Dependent Volatility
15.7 Black-Scholes Continuous Path Integral
15.8 Stationary Action: Euler-Lagrange Equation
15.9 Black-Scholes Euler-Lagrange Equation
15.10 Stationary Action: Time Dependent Volatility
15.11 Harmonic Oscillator Pricing Kernel
15.12 Merton Oscillator Pricing Kernel
15.13 Martingale: Merton Model
16 Quantum Fields for Bonds and Interest Rates
16.1 Introduction
16.2 Coupon Bonds
16.3 Forward Interest Rates
16.4 Action and Lagrangian
16.5 Correlation Function
16.6 Time Dependent State Space mathcalVt
16.7 Time Dependent Hamiltonian
16.8 Path Integral: Martingale
16.9 Numeraire
16.10 Zero Coupon Bond Call Option
16.11 Libor: Simple Interest Rate
16.12 Libor Market Model
16.13 Libor Forward Interest Rates
16.14 Libor Lagrangian
16.15 Libor Hamiltonian: Martingale
16.16 Black's Caplet Price
Appendix Mathematics of Numbers
A.1 Introduction
A.2 Integers
A.2.1 Prime Numbers
A.2.2 Irrational Numbers
A.2.3 Transcendental Numbers
A.3 Real Numbers
A.3.1 Decimal Expansion
A.3.2 Complex Numbers
A.4 Cantor's Diagonal Construction
A.5 Higher Order Infinities
A.6 Meta-mathematics
A.7 Gödel's Incompleteness
Appendix References
Index