Honours Project

Final Draft Derivation and Application of the Black-Scholes Equation for Option Pricing

Author: Yeheng XU

Supervisor: Dr. David Amundsen

April 30, 2012

Abstract In this project, I will first study the concept of a stochastic process, and discuss some properties of Brownian Motion. Then I generalize Brownian Motion to what it called an Itˆ process. The above concepts will be used to derive the Black-Scholes Option Price o formula. Then an analytical solution for the equation will be provided by using mathematical tools such as Fourier Transformation and properties of the heat equation. Finally, I will implement a finite difference numerical scheme in MATLAB to simulate the original Black-Scholes equation for both European call and put options and compare to analytic solutions.

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Contents

1 Introduction and Background 1.1 1.2 1.3 1.4 What is financial mathematics? . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction of option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some economic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 3 4 5

2 Brownian Motion

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3 Itˆ’s Lemma o

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4 Black-Schloes Partial Differential Equation

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5 Analytical Solution of the Black-Scholes Equation 5.1 5.2 The I.B.V.P. for the Black-Scholes Equation . . . . . . . . . . . . . . . . . . . . Changing Variables for the Black-Scholes Equation . . . . . . . . . . . . . . . . 5.2.1 5.2.2 5.3 First set of changing variables . . . . . . . . . . . . . . . . . . . . . . . . Second set of changing variables . . . . . . . . . . . . . . . . . . . . . . .

18 18 19 19 22 24

Solving the Black-Scholes Equation . . . . . . . . . . . . . . . . . . . . . . . . .

6 Numerical Solution of the Black-Scholes Equation 6.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30 30

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6.2

Simulation for the Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 6.2.2 6.2.3 European Call Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . European Put Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison Exact Solution with Numerical Solution . . . . . . . . . . .

31 32 34 35

7 Conclusion

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Appendix A MATLAB Code for European Call Option

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Appendix B MATLAB Code for European Put Option

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1

Introduction and Background

In this section, I will introduce some concepts and background in financial mathematics that I will use in the discussion of my honours project.

1.1

What is financial mathematics?

Financial mathematics is a very popular word in the world nowadays, but what is financial mathematics? What do financial mathematics do actually? The first question is easy to answer; financial mathematics is a collection of mathematical techniques that have applications in finance. In other words, it is an applied math to solve real world financial market problems. The second question is hard to answer because there are too many applications in financial mathematics. The project I am focusing on is only a very small branch of the whole. In most of the cases, financial mathematics can solve problems by building the models of the problems and then, for example, to maximizing the profit or minimizing the risk. There are two main approaches for financial mathematics. One is stochastic processes and the other is differential equations. It is easily to understand why in most cases I need those two tools. Since financial mathematics is based on real life, it is important to introduce probability theory and stochastic process to simulate the real situation. By using differential equations in real world problems time can be considered as continuous in infinite steps, many problems can be solved analytically or...