Stochastic Models of Financial Mathematics

Description

This book presents a short introduction to continuous-time financial models. An overview of the basics of stochastic analysis precedes a focus on the Black–Scholes and interest rate models. Other topics covered include self-financing strategies, option pricing, exotic options and risk-neutral probabilities. Vasicek, Cox-Ingersoll-Ross, and Heath–Jarrow–Morton interest rate models are also explored. The author presents practitioners with a basic introduction, with more rigorous information provided for mathematicians. The reader is assumed to be familiar with the basics of probability theory. Some basic knowledge of stochastic integration and differential equations theory is preferable, although all preliminary information is given in the first part of the book. Some relatively simple theoretical exercises are also provided. Front Cover Stochastic Models of Financial Mathematics Copyright Contents Preface Notations Chapter 1. Overview of the Basics of Stochastic Analysis 1.1. Brownian motion 1.2. Stochastic integrals 1.3. Martingales, Itô processes and general Itô’s formula 1.4. Stochastic differential equations 1.5. Change of probability: the Girsanov theorem Chapter 2. The Black–Scholes Model 2.1. Introduction: what is an option? 2.2. Self-financing strategies 2.3. Option pricing problem: the Black–Scholes model 2.4. The Black–Scholes formula 2.5. Risk-neutral probabilities: alternative derivation of the Black–Scholes formula 2.6. American options in the Black–Scholes model 2.7. Exotic options Chapter 3. Models of Interest Rates 3.1. Modeling principles 3.2. The Vašíček model 3.3. The Cox–Ingersoll–Ross model 3.4. The Heath–Jarrow–Morton model Bibliography Index Back Cover