Stochastic Processes with Applications to Finance: (Chapman & Hall/CRC Financial Mathematics Series)

In recent years, modeling financial uncertainty using stochastic processes has become increasingly important, but it is commonly perceived as requiring a deep mathematical background. Stochastic Processes with Applications to Finance shows that this is not necessarily so. It presents the theory of discrete stochastic processes and their applications in finance in an accessible treatment that strikes a balance between the abstract and the practical. Using an approach that views sophisticated stochastic calculus as based on a simple class of discrete processes-"random walks"-the author first provides an elementary introduction to the relevant areas of real analysis and probability. He then uses random walks to explain the change of measure formula, the reflection principle, and the Kolmogorov backward equation. The Black-Scholes formula is derived as a limit of binomial model, and applications to the pricing of derivative securities are presented. Another primary focus of the book is the pricing of corporate bonds and credit derivatives, which the author explains in terms of discrete default models. By presenting important results in discrete processes and showing how to transfer those results to their continuous counterparts, Stochastic Processes with Applications to Finance imparts an intuitive and practical understanding of the subject. This unique treatment is ideal both as a text for a graduate-level class and as a reference for researchers and practitioners in financial engineering, operations research, and mathematical and statistical finance.

Table of Contents:
ELEMENTARY CALCULUS: TOWARDS ITO'S FORMULA Exponential and Logarithmic Functions Differentiation Taylor's Expansion Ito's Formula Integration Exercises ELEMENTS IN PROBABILITY The Sample Space and Probability Discrete Random Variables Continuous Random Variables Multivariate Random Variables Expectation Conditional Expectation Moment Generating Functions Exercises USEFUL DISTRIBUTIONS IN FINANCE Binomial Distributions Other Discrete Distribution Normal and Log-Normal Distributions Other Continuous Distributions Multivariate Normal Distributions Exercises DERIVATIVE SECURITIES The Money-Market Account Various Interest Rates Forward and Futures Contracts Options Interest-Rate Derivatives Exercises A DISCRETE-TIME MODEL FOR SECURITIES MARKET Price Processes The Portfolio Value and Stochastic Integral No-Arbitrage and Replication Portfolios Martingales and the Asset Pricing Theorem American Options Change of Measure Exercises RANDOM WALKS The Mathematical Definition Transition Probabilities The Reflection Principle Change of Measure Revisited A Binomial Securities Market Model Exercises THE BINOMIAL MODEL The Single-Period Model The Multi-Period Model The Binomial Model for American Options The Trinomial Model The Binomial Model for Interest-Rate Claims Exercises A DISCRETE-TIME MODEL FOR DEFAULTABLE SECURITIES The Hazard Rate A Discrete Hazard Model Pricing of Defaultable Securities Correlated Defaults Exercises MARKOV CHAINS Markov and Strong Markov Properties Transition Probabilities Absorbing Markov Chains Applications to Finance Exercises THE MONTE CARLO SIMULATION Mathematical Backgrounds The Idea of Monte Carlo Generation of Random Numbers Some Examples for Financial Engineering Variance Reduction Methods Exercises FROM DISCRETE TO CONTINUOUS: TOWARDS THE BLACK-SCHOLES Brownian Motions The Central Limit Theorem Revisited The Black-Scholes Formula More on Brownian Motions Poisson Processes Exercises BASIC STOCHASTIC PROCESSES IN CONTINUOUS TIME Diffusion Processes Sample Paths of Brownian Motions Martingales Stochastic Integrals Stochastic Differential Equations Ito's Formula Revisited Exercises A CONTINUOUS-TIME MODEL FOR SECURITIES MARKET Self-Financing Portfolio and No-Arbitrage Price Process Models The Black-Scholes Model The Risk-Neutral Method The Forward-Neutral Method The Interest-Rate Term Structure Pricing of Interest-Rate Derivatives Pricing of Corporate Debts Exercises REFERENCES