The Mathematics of Derivatives Securities with Applications in MATLAB: (The Wiley Finance Series)

Quantitative Finance is expanding rapidly. One of the aspects of the recent financial crisis is that, given the complexity of financial products, the demand for people with high numeracy skills is likely to grow and this means more recognition will be given to Quantitative Finance in existing and new course structures worldwide. Evidence has suggested that many holders of complex financial securities before the financial crisis did not have in-house experts or rely on a third-party in order to assess the risk exposure of their investments. Therefore, this experience shows the need for better understanding of risk associate with complex financial securities in the future. The Mathematics of Derivative Securities with Applications in MATLAB provides readers with an introduction to probability theory, stochastic calculus and stochastic processes, followed by discussion on the application of that knowledge to solve complex financial problems such as pricing and hedging exotic options, pricing American derivatives, pricing and hedging under stochastic volatility and an introduction to interest rates modelling. The book begins with an overview of MATLAB and the various components that will be used alongside it throughout the textbook. Following this, the first part of the book is an in depth introduction to Probability theory, Stochastic Processes and Ito Calculus and Ito Integral. This is essential to fully understand some of the mathematical concepts used in the following part of the book. The second part focuses on financial engineering and guides the reader through the fundamental theorem of asset pricing using the Black and Scholes Economy and Formula, Options Pricing through European and American style options, summaries of Exotic Options, Stochastic Volatility Models and Interest rate Modelling. Topics covered in this part are explained using MATLAB codes showing how the theoretical models are used practically. Authored from an academic’s perspective, the book discusses complex analytical issues and intricate financial instruments in a way that it is accessible to postgraduate students with or without a previous background in probability theory and finance. It is written to be the ideal primary reference book or a perfect companion to other related works. The book uses clear and detailed mathematical explanation accompanied by examples involving real case scenarios throughout and provides MATLAB codes for a variety of topics.

Table of Contents:
Preface xi 1 An Introduction to Probability Theory 1 1.1 The Notion of a Set and a Sample Space 1 1.2 Sigma Algebras or Field 2 1.3 Probability Measure and Probability Space 2 1.4 Measurable Mapping 3 1.5 Cumulative Distribution Functions 4 1.6 Convergence in Distribution 5 1.7 Random Variables 5 1.8 Discrete Random Variables 6 1.9 Example of Discrete Random Variables: The Binomial Distribution 6 1.10 Hypergeometric Distribution 7 1.11 Poisson Distribution 8 1.12 Continuous Random Variables 9 1.13 Uniform Distribution 9 1.14 The Normal Distribution 9 1.15 Change of Variable 11 1.16 Exponential Distribution 12 1.17 Gamma Distribution 12 1.18 Measurable Function 13 1.19 Cumulative Distribution Function and Probability Density Function 13 1.20 Joint, Conditional and Marginal Distributions 17 1.21 Expected Values of Random Variables and Moments of a Distribution 19 2 Stochastic Processes 25 2.1 Stochastic Processes 25 2.2 Martingales Processes 26 2.3 Brownian Motions 29 2.4 Brownian Motion and the Reflection Principle 32 2.5 Geometric Brownian Motions 35 3 Ito Calculus and Ito Integral 37 3.1 Total Variation and Quadratic Variation of Differentiable Functions 37 3.2 Quadratic Variation of Brownian Motions 39 3.3 The Construction of the Ito Integral 40 3.4 Properties of the Ito Integral 41 3.5 The General Ito Stochastic Integral 42 3.6 Properties of the General Ito Integral 43 3.7 Construction of the Ito Integral with Respect to Semi-Martingale Integrators 44 3.8 Quadratic Variation of a General Bounded Martingale 46 4 The Black and Scholes Economy 55 4.1 Introduction 55 4.2 Trading Strategies and Martingale Processes 55 4.3 The Fundamental Theorem of Asset Pricing 56 4.4 Martingale Measures 58 4.5 Girsanov Theorem 59 4.6 Risk-Neutral Measures 62 5 The Black and Scholes Model 67 5.1 Introduction 67 5.2 The Black and Scholes Model 67 5.3 The Black and Scholes Formula 68 5.4 Black and Scholes in Practice 70 5.5 The Feynman–Kac Formula 71 6 Monte Carlo Methods 79 6.1 Introduction 79 6.2 The Data Generating Process (DGP) and the Model 79 6.3 Pricing European Options 80 6.4 Variance Reduction Techniques 81 7 Monte Carlo Methods and American Options 91 7.1 Introduction 91 7.2 Pricing American Options 91 7.3 Dynamic Programming Approach and American Option Pricing 92 7.4 The Longstaff and Schwartz Least Squares Method 93 7.5 The Glasserman and Yu Regression Later Method 95 7.6 Upper and Lower Bounds and American Options 96 8 American Option Pricing: The Dual Approach 101 8.1 Introduction 101 8.2 A General Framework for American Option Pricing 101 8.3 A Simple Approach to Designing Optimal Martingales 104 8.4 Optimal Martingales and American Option Pricing 104 8.5 A Simple Algorithm for American Option Pricing 105 8.6 Empirical Results 106 8.7 Computing Upper Bounds 107 8.8 Empirical Results 109 9 Estimation of Greeks using Monte Carlo Methods 113 9.1 Finite Difference Approximations 113 9.2 Pathwise Derivatives Estimation 114 9.3 Likelihood Ratio Method 116 9.4 Discussion 118 10 Exotic Options 121 10.1 Introduction 121 10.2 Digital Options 121 10.3 Asian Options 122 10.4 Forward Start Options 123 10.5 Barrier Options 123 10.5.1 Hedging Barrier Options 125 11 Pricing and Hedging Exotic Options 129 11.1 Introduction 129 11.2 Monte Carlo Simulations and Asian Options 129 11.3 Simulation of Greeks for Exotic Options 130 11.4 Monte Carlo Simulations and Forward Start Options 131 11.5 Simulation of the Greeks for Exotic Options 132 11.6 Monte Carlo Simulations and Barrier Options 132 12 Stochastic Volatility Models 137 12.1 Introduction 137 12.2 The Model 137 12.3 Square Root Diffusion Process 138 12.4 The Heston Stochastic Volatility Model (HSVM) 139 12.5 Processes with Jumps 143 12.6 Application of the Euler Method to Solve SDEs 143 12.7 Exact Simulation Under SV 144 12.8 Exact Simulation of Greeks Under SV 146 13 Implied Volatility Models 151 13.1 Introduction 151 13.2 Modelling Implied Volatility 152 13.3 Examples 153 14 Local Volatility Models 157 14.1 An Overview 157 14.2 The Model 159 14.3 Numerical Methods 161 15 An Introduction to Interest Rate Modelling 167 15.1 A General Framework 167 15.2 Affine Models (AMs) 169 15.3 The Vasicek Model 171 15.4 The Cox, Ingersoll and Ross (CIR) Model 173 15.5 The Hull and White (HW) Model 174 15.6 The Black Formula and Bond Options 175 16 Interest Rate Modelling 177 16.1 Some Preliminary Definitions 177 16.2 Interest Rate Caplets and Floorlets 178 16.3 Forward Rates and Numeraire 180 16.4 Libor Futures Contracts 181 16.5 Martingale Measure 183 17 Binomial and Finite Difference Methods 185 17.1 The Binomial Model 185 17.2 Expected Value and Variance in the Black and Scholes and Binomial Models 186 17.3 The Cox–Ross–Rubinstein Model 187 17.4 Finite Difference Methods 188 Appendix 1 An Introduction to MATLAB 191 A1.1 What is MATLAB? 191 A1.2 Starting MATLAB 191 A1.3 Main Operations in MATLAB 192 A1.4 Vectors and Matrices 192 A1.5 Basic Matrix Operations 194 A1.6 Linear Algebra 195 A1.7 Basics of Polynomial Evaluations 196 A1.8 Graphing in MATLAB 196 A1.9 Several Graphs on One Plot 197 A1.10 Programming in MATLAB: Basic Loops 199 A1.11 M-File Functions 200 A1.12 MATLAB Applications in Risk Management 200 A1.13 MATLAB Programming: Application in Financial Economics 202 Appendix 2 Mortgage Backed Securities 205 A2.1 Introduction 205 A2.2 The Mortgage Industry 206 A2.3 The Mortgage Backed Security (MBS) Model 207 A2.4 The Term Structure Model 208 A2.5 Preliminary Numerical Example 210 A2.6 Dynamic Option Adjusted Spread 210 A2.7 Numerical Example 212 A2.8 Practical Numerical Examples 213 A2.9 Empirical Results 214 A2.10 The Pre-Payment Model 215 Appendix 3 Value at Risk 217 A3.1 Introduction 217 A3.2 Value at Risk (VaR) 217 A3.3 The Main Parameters of a VaR 218 A3.4 VaR Methodology 219 A3.5 Empirical Applications 222 A3.6 Fat Tails and VaR 224 Bibliography 227 References 229 Index 233