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.
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.
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