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
