This book provides basic stochastic analysis techniques for mathematical finance. The book consists of seven chapters. Chapter 1 is an introduction. In Chapter 2, the author presents a brief review of probability theory. It is considered the notion of information structure and how it relates to a probability space and a random variable. Also the notion of conditional probability and conditional expectation and their calculation are discussed. In Chapter 3, discrete-time stochastic processes are presented. The author considers a random walk and uses it to illustrate important concepts associated with stochastic processes, such as change of probability measures, martingales, stopping times and applications of the optional sampling theorem. Discrete-time Markov chains are presented. Some applications of random walks, such as binomial models for a stock price, option pricing and interest rate models are discussed.
Chapter 4 deals with continuous-time stochastic processes. Brownian motion is introduced as a limit of a sequence of random walks. The reflection principle and barrier hitting time distributions are studied. An alternative approach using the Poisson process is considered. The notions of change of probability measures, martingales and stopping times, the optional sampling theorem are discussed for continuous-time stochastic processes. Chapter 5 deals with the Itô integral as it is the most useful for financial applications. Tools for Itô integrals such as Itô’s lemma and integration by parts are presented. Stochastic differential equations and Itô processes as solutions to stochastic differential equations are introduced. It is discussed how to solve stochastic differential equations and how to construct martingales associated with Itô processes. Some examples are given, including the famous Black-Scholes option pricing formula.
Chapter 6 is devoted to advanced topics in stochastic calculus. The Feynman-Kac formula, the Girsanov theorem and complex barrier hitting times as well as high-dimensional stochastic differential equations are presented. Chapter 7 deals with several applications in insurance valuation. The first application is the valuation of variable annuities. The approach developed for variable annuities is extended to valuate equity-indexed annuities. The third application involves the valuation of guaranteed annuity options. Finally a universal life is considered.