The Mathematics of Derivatives Securities with Applications in MATLAB

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                                 Appendix 1: Bernoulli law of large numbers Appendix 2:  Conditional expectations Appendix 3: Hilbert spaces.                 Chapter 3             Stochastic Processes                                 Martingales processes Stopping times                                 The optional stopping theorem                                 Local martingales and semi-martingales Brownian motions Brownian motions and reflection principle                                 Martingales separation theorem of Brownian motions                                 Appendix 1: Working with Brownian motions.                                   Chapter 4             Ito Calculus and Ito Integral                                 Quadratic variation of Brownian motions                                 The construction of Ito integral with elementary process                                 The general Ito integral Construction of the Ito integral with respect to semi-martingales integrators                                 Quadratic variation and general bounded martingales                                 Ito lemma and Ito formula                                 Appendix 1: Ito Integral and Riemann-Stieljes integral Part 2 Chapter 5             The Black and Scholes Economy and Black and Scholes Formula                                 The fundamental theorem of asset pricing                                 Martingales measures                                         The Girsanov Theorem                                 The Randon-Nikodym                                 The Black and Scholes Model                                 The Black and Scholes formula                                 The Black and Scholes in practice                                 The Feyman-Kac formula                                                Appendix 1: The Kolmogorov Backword equation                                 Appendix 2: Change of numeraire Chapter 6             Monte Carlo Methods for Options Pricing                 Basic concepts and pricing European style options                 Variance reduction techniques                                 Pricing path dependent options Projections methods in finance                                 Estimations of Greeks by Monte Carlo methods.                                 Chapter 7             American Option Pricing                                 A review of the literature on pricing American put options                                 Optimal stopping times and American put options                                 A dynamic programming approach to price American options                                 The Losgstaff and Schwartz (2001) approach                                 The Glasserman and Yu (2004) approach                                 Estimation of the upper bound                                 Cerrato (2008) approach to compute upper bounds.                                 Chapter 8             Exotic Options                                 Digital and binary                                 Asian options                                 Forward start options                                 Barrier options                                 Hedging barrier options        Chapter 9             Stochastic Volatility Models                                 Square root diffusion models                                 The Heston Model                                 Processes with jumps                                 Monte Carlo methods to price derivatives under stochastic volatility                                 Euler methods and stochastic differential equations                                 Exact simulation of Greeks under stochastic volatility                                 Computing Greeks for exotics using simulations Chapter 10           Interest Rate Modeling                                 A general framework                                 Affine models                                 The Vasicek model                                 The Cox, Ingersoll & Ross Model                                 The Hull and White (HW) Model                                 Bond options