Option Pricing

Numerical Methods for Option Pricing

Mark Richardson

March 2009

Contents

1 Introduction

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2 A brief introduction to derivatives and options

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3 The

3.1

3.2

3.3

3.4

Black-Scholes Model

Asset prices and Ito’s Lemma . . . .

Derivation of the Black-Scholes PDE

Boundary Conditions . . . . . . . . .

Closed Form Solutions . . . . . . . .

4 Numerical Methods

4.1 Finite Difference Methods

4.2 Risk-Neutral Valuation . .

4.3 Binomial Tree Methods .

4.4 Monte-Carlo Simulation .

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5 Pricing American Options

5.1 Boundary Conditions and Early Exercise Criteria .

5.2 Pricing American Options using a Binomial Tree .

5.3 Pricing American Options using Finite Differences

5.4 Comparing the Performance of the Two Methods .

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6 Conclusions

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7 Appendix

7.1 MATLAB

7.2 MATLAB

7.3 MATLAB

7.4 MATLAB

7.5 MATLAB

7.6 MATLAB

7.7 MATLAB

7.8 MATLAB

7.9 MATLAB

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