**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**