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The Mathematics of Derivatives Securities with Applications in MATLAB

The Mathematics of Derivatives Securities with Applications in MATLAB

  • Author:
  • Publisher: John Wiley & Sons
  • ISBN: 9780470683699
  • Published In: February 2012
  • Format: Hardback , 248 pages
  • Jurisdiction: International ? Disclaimer:
    Countri(es) stated herein are used as reference only

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    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.

  • 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

  • Mario Cerrato is a Senior Lecturer (Associate Professor) in Financial Economics at the University of Glasgow Business School. He holds a PhD in Financial Econometrics and an MSc in Economics from London Metropolitan University, and a first degree in Economics from the University of Salerno. Mario’s research interests are in the area of financial derivatives, security design and financial market microstructures. He has published in leading finance journals such as Journey of Money Credit and Banking,

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