Mario Cerrato

The Mathematics of Derivatives Securities with Applications in MATLAB

• Broschiertes Buch

Produktbeschreibung

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.

Produktdetails

• Wiley Finance Series
• Verlag: Wiley & Sons
• 1. Auflage
• Seitenzahl: 248
• Erscheinungstermin: 19. März 2012
• Englisch
• Abmessung: 236mm x 161mm x 27mm
• Gewicht: 505g
• ISBN-13: 9780470683699
• ISBN-10: 0470683694
• Artikelnr.: 31192871

Autorenporträt

Mario Cerrato, Glasgow, Scotland is a Lecturer in Economics at the University of Glasgow, Department of Economics. He previously held posts at London Metropolitan University, Banca del Salento and Expedia Capital Management Ltd. He has been actively involved in various consultancies in the area of financial engineering over the last five years. He has published numerous articles in the area of financial econometrics and financial derivatives in international journals like the International Journal of Finance & Economics, International Journal of Theoretical and Applied Finance, Computational Statistics and Data Analysis.

Inhaltsangabe

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