Stochastic Analysis for Finance with Simulations

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.  

Table of Contents

Part I Introduction to Financial Mathematics

1 Fundamental Concepts

2 Financial Derivatives

Part II Probability Theory

3 The Lebesgue Integral

4 Basic Probability Theory

5 Conditional Expectation

6 Stochastic Processes

Part Ill Brownian Motion

7 Brownian Motion

8 Girsanov’s Theorem

9 The Reflection Principle of Brownian Motion

Part IV lt6 Calculus

10 The lt6 Integral

11 The Ito Formula

12 Stochastic Differential Equations

13 The Feynman-Kac Theorem

Part V Option Pricing Methods

14 The Binomial Tree Method for Option Pricing

15 The Black-Scholes-Merton Differential Equation

16 The Martingale Method

Part VI Examples of Option Pricing

17 Pricing of Vanilla Options

18 Pricing of Exotic Options

19 American Options

Part VII Portfolio Management

20 The Capital Asset Pricing Model

21 Dynamic Programming

Part VIII Interest Rate Models

22 Bond Pricing

23 Interest Rate Models

24 Numeraires

Part IX Computational Methods

25 Numerical Estimation of Volatility

26 Time Series

27 Random Numbers

28 The Monte Carlo Method for Option Pricing

29 Numerical Solution of the Black-Scholes-Merton Equation

30 Numerical Solution of Stochastic Differential Equations

A Basic Analysis

B linear Algebra

C Ordinary Differential Equations

D Diffusion Equations

E Entropy

F Matlab Programming

Solutions for Selected Problems