Stochastic Analysis for Finance with Simulations

Description

This book is an introduction to stochastic analysis and quantitative finance; it includes both theoretical and computational methods. Topics covered are stochastic calculus, option pricing, optimal portfolio investment, and interest rate models. Also included are simulations of stochastic phenomena, numerical solutions of the Black–Scholes–Merton equation, Monte Carlo methods, and time series. Basic measure theory is used as a tool to describe probabilistic phenomena.  The level of familiarity with computer programming is kept to a minimum. To make the book accessible to a wider audience, some background mathematical facts are included in the first part of the book and also in the appendices. This work attempts to bridge the gap between mathematics and finance by using diagrams, graphs and simulations in addition to rigorous theoretical exposition. Simulations are not only used as the computational method in quantitative finance, but they can also facilitate an intuitive and deeper understanding of theoretical concepts.   Stochastic Analysis for Finance with Simulations is designed for readers who want to have a deeper understanding of the delicate theory of quantitative finance by doing computer simulations in addition to theoretical study. It will particularly appeal to advanced undergraduate and graduate students in mathematics and business, but not excluding practitioners in finance industry.   Front Matter....Pages i-xxxii Front Matter....Pages 1-1 Fundamental Concepts....Pages 3-14 Financial Derivatives....Pages 15-22 Front Matter....Pages 23-23 The Lebesgue Integral....Pages 25-40 Basic Probability Theory....Pages 41-74 Conditional Expectation....Pages 75-89 Stochastic Processes....Pages 91-107 Front Matter....Pages 109-109 Brownian Motion....Pages 111-135 Girsanov’s Theorem....Pages 137-145 The Reflection Principle of Brownian Motion....Pages 147-156 Front Matter....Pages 157-157 The Itô Integral....Pages 159-175 The Itô Formula....Pages 177-202 Stochastic Differential Equations....Pages 203-223 The Feynman–Kac Theorem....Pages 225-235 Front Matter....Pages 237-237 The Binomial Tree Method for Option Pricing....Pages 239-253 The Black–Scholes–Merton Differential Equation....Pages 255-280 The Martingale Method....Pages 281-294 Front Matter....Pages 295-295 Pricing of Vanilla Options....Pages 297-320 Pricing of Exotic Options....Pages 321-335 American Options....Pages 337-350 Front Matter....Pages 351-351 The Capital Asset Pricing Model....Pages 353-378 Front Matter....Pages 351-351 Dynamic Programming....Pages 379-393 Front Matter....Pages 395-395 Bond Pricing....Pages 397-419 Interest Rate Models....Pages 421-441 Numeraires....Pages 443-454 Front Matter....Pages 455-455 Numerical Estimation of Volatility....Pages 457-467 Time Series....Pages 469-485 Random Numbers....Pages 487-499 Numerical Solution of the Black–Scholes–Merton Equation....Pages 501-517 Numerical Solution of Stochastic Differential Equations....Pages 519-534 Back Matter....Pages 535-544 ....Pages 545-657