The Mathematics of Derivatives Securities with Applications in MATLAB

by Cerrato, Mario

• ISBN13: 9780470683699
• ISBN10: 0470683694
• Edition: 1st
• Format: Hardcover
• Copyright: 2012-03-19
• Publisher: Wiley

Summary

The book is divided into two parts - the first part introduces probability theory, stochastic calculus and stochastic processes before moving on to the second part which instructs readers on how to apply the content learnt in part one to solve complex financial problems such as pricing and hedging exotic options, pricing American derivatives, pricing and hedging under stochastic volatility, and interest rate modelling. Each chapter provides a thorough discussion of the topics covered with practical examples in MATLAB so that readers will build up to an analysis of modern cutting edge research in finance, combining probabilistic models and cutting edge finance illustrated by MATLAB applications.Most books currently available on the subject require the reader to have some knowledge of the subject area and rarely consider computational applications such as MATLAB. This book stands apart from the rest as it covers complex analytical issues and complex financial instruments in a way that is accessible to those without a background in probability theory and finance, as well as providing detailed mathematical explanations with MATLAB code for a variety of topics and real world case examples. Contents: Chapter 1: Introduction Overview of MatLab Using various MatLab's toolboxes Mathematics with MatLab Statistics with MatLab Programming in MatLab Part 1; Chapter 2: Probability Theory Set and sample space Sigma algebra, probability measure and probability space Discrete and continuous random variables Measurable mapping Joint, conditional and marginal distributions Expected values and moment of a distribution.

Author Biography

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, Journal of Banking and Finance, International Journal of Theoretical and Applied Finance, and many others. He is generally involved in research collaboration with leading financial firms in the City of London and Wall Street.

Table of Contents

Preface	p. xi
An Introduction to Probability Theory	p. 1
The Notion of a Set and a Sample Space	p. 1
Sigma Algebras or Field	p. 2
Probability Measure and Probability Space	p. 2
Measurable Mapping	p. 3
Cumulative Distribution Functions	p. 4
Convergence in Distribution	p. 5
Random Variables	p. 5
Discrete Random Variables	p. 6
Example of Discrete Random Variables: The Binomial Distribution	p. 6
Hypergeometric Distribution	p. 7
Poisson Distribution	p. 8
Continuous Random Variables	p. 9
Uniform Distribution	p. 9
The Normal Distribution	p. 9
Change of Variable	p. 11
Exponential Distribution	p. 12
Gamma Distribution	p. 12
Measurable Function	p. 13
Cumulative Distribution Function and Probability Density Function	p. 13
Joint, Conditional and Marginal Distributions	p. 17
Expected Values of Random Variables and Moments of a Distribution	p. 19
The Bernoulli Law of Large Numbers	p. 20
Conditional Expectations	p. 22
Stochastic Processes	p. 25
Stochastic Processes	p. 25
Martingales Processes	p. 26
Brownian Motions	p. 29
Brownian Motion and the Reflection Principle	p. 32
Geometric Brownian Motions	p. 35
An Application of Brownian Motions	p. 36
Ito Calculus and Ito Integral	p. 37
Total Variation and Quadratic Variation of Differentiable Functions	p. 37
Quadratic Variation of Brownian Motions	p. 39
The Construction of the Ito Integral	p. 40
Properties of the Ito Integral	p. 41
The General Ito Stochastic Integral	p. 42
Properties of the General Ito Integral	p. 43
Construction of the Ito Integral with Respect to Semi-Martingale Integrators	p. 44
Quadratic Variation of a General Bounded Martingale	p. 46
Quadratic Variation of a General Bounded Martingale	p. 48
Ito Lemma and Ito Formula	p. 48
The Riemann-Stieljes Integral	p. 51
The Black and Scholes Economy	p. 55
Introduction	p. 55
Trading Strategies and Martingale Processes	p. 55
The Fundamental Theorem of Asset Pricing	p. 56
Martingale Measures	p. 58
Girsanov Theorem	p. 59
Risk-Neutral Measures	p. 62
The Randon-Nikodym Condition	p. 64
Geometric Brownian Motion	p. 65
The Black and Scholes Model	p. 67
Introduction	p. 67
The Black and Scholes Model	p. 67
The Black and Scholes Formula	p. 68
Black and Scholes in Practice	p. 70
The Feynman-Kac Formula	p. 71
The Kolmogorov Backward Equation	p. 73
Change of Numeraire	p. 74
Black and Scholes and the Greeks	p. 76
Monte Carlo Methods	p. 79
Introduction	p. 79
The Data Generating Process (DGP) and the Model	p. 79
Pricing European Options	p. 80
Variance Reduction Techniques	p. 81
Antithetic Variate Methods	p. 81
Control Variate Methods	p. 82
Common Random Numbers	p. 84
Importance Sampling	p. 84
Monte Carlo European Options	p. 85
Variance Reduction Techniques - First Part	p. 86
Monte Carlo 1	p. 87
Monte Carlo 2	p. 88
Monte Carlo Methods and American Options	p. 91
Introduction	p. 91
Pricing American Options	p. 91
Dynamic Programming Approach and American Option Pricing	p. 92
The Longstaff and Schwartz Least Squares Method	p. 93
The Glasserman and Yu Regression Later Method	p. 95
Upper and Lower Bounds and American Options	p. 96
Multiassets Simulation	p. 97
Pricing a Basket Option Using the Regression Methods	p. 98
American Option Pricing: The Dual Approach	p. 101
Introduction	p. 101
A General Framework for American Option Pricing	p. 101
A Simple Approach to Designing Optimal Martingales	p. 104
Optimal Martingales and American Option Pricing	p. 104
A Simple Algorithm for American Option Pricing	p. 105
Empirical Results	p. 106
Computing Upper Bounds	p. 107
Empirical Results	p. 109
	p. 110
Estimation of Greeks using Monte Carlo Methods	p. 113
Finite Difference Approximations	p. 113
Pathwise Derivatives Estimation	p. 114
Likelihood Ratio Method	p. 116
Discussion	p. 118
Pathwise Greeks using Monte Carlo	p. 118
Exotic Options	p. 121
Introduction	p. 121
Digital Options	p. 121
Asian Options	p. 122
Forward Start Options	p. 123
Barrier Options	p. 123
Hedging Barrier Options	p. 125
Digital Options	p. 126
Pricing and Hedging Exotic Options	p. 129
Introduction	p. 129
Monte Carlo Simulations and Asian Options	p. 129
Simulation of Greeks for Exotic Options	p. 130
Monte Carlo Simulations and Forward Start Options	p. 131
Simulation of the Greeks for Exotic Options	p. 132
Monte Carlo Simulations and Barrier Options	p. 132
The Price and the Delta of Forward Start Options	p. 134
The Price of Barrier Options Using Importance Sampling	p. 134
Stochastic Volatility Models	p. 137
Introduction	p. 137
The Model	p. 137
Square Root Diffusion Process	p. 138
The Heston Stochastic Volatility Model (HSVM)	p. 139
Processes with Jumps	p. 143
Application of the Euler Method to Solve SDEs	p. 143
Exact Simulation Under SV	p. 144
Exact Simulation of Greeks Under SV	p. 146
Stochastic Volatility Using the Heston Model	p. 147
Implied Volatility Models	p. 151
Introduction	p. 151
Modelling Implied Volatility	p. 152
Examples	p. 153
Local Volatility Models	p. 157
An Overview	p. 157
The Model	p. 159
Numerical Methods	p. 161
	p. 164
An Introduction to Interest Rate Modelling	p. 167
A General Framework	p. 167
Affine Models (AMs)	p. 169
The Vasicek Model	p. 171
The Cox, Ingersoll and Ross (CIR) Model	p. 173
The Hull and White (HW) Model	p. 174
The Black Formula and Bond Options	p. 175
Interest Rate Modelling	p. 177
Some Preliminary Definitions	p. 177
Interest Rate Caplets and Floorlets	p. 178
Forward Rates and Numeraire	p. 180
Libor Futures Contracts	p. 181
Martingale Measure	p. 183
Binomial and Finite Difference Methods	p. 185
The Binomial Model	p. 185
Expected Value and Variance in the Black and Scholes and Binomial Models	p. 186
The Cox-Ross-Rubinstein Model	p. 187
Finite Difference Methods	p. 188
The Binomial Method	p. 189
An Introduction to MATLAB	p. 191
What is MATLAB?	p. 191
Starting MATLAB	p. 191
Main Operations in MATLAB	p. 192
Vectors and Matrices	p. 192
Basic Matrix Operations	p. 194
Linear Algebra	p. 195
Basics of Polynomial Evaluations	p. 196
Graphing in MATLAB	p. 196
Several Graphs on One Plot	p. 197
Programming in MATLAB: Basic Loops	p. 199
M-File Functions	p. 200
MATLAB Applications in Risk Management	p. 200
MATLAB Programming: Application in Financial Economics	p. 202
Mortgage Backed Securities	p. 205
Introduction	p. 205
The Mortgage Industry	p. 206
The Mortgage Backed Security (MBS) Model	p. 207
The Term Structure Model	p. 208
Preliminary Numerical Example	p. 210
Dynamic Option Adjusted Spread	p. 210
Numerical Example	p. 212
Practical Numerical Examples	p. 213
Empirical Results	p. 214
The Pre-Payment Model	p. 215
Value at Risk	p. 217
Introduction	p. 217
Value at Risk (VaR)	p. 217
The Main Parameters of a VaR	p. 218
VaR Methodology	p. 219
Historical Simulations	p. 219
Variance-Covariance Method	p. 220
Monte Carlo Method	p. 221
Empirical Applications	p. 222
Historical Simulations	p. 222
Variance-Covariance Method	p. 223
Fat Tails and VaR	p. 224
Generalized Extreme Value and the Pareto Distribution	p. 224
Bibliography	p. 227
References	p. 229
Index	p. 233
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