Option Pricing: Lévy Process Pubblico

Abstract

We present an introduction and implementation of exponential Lévy model for option pricing. First of all, we provide a short introduction on two of most popular option pricing models, Binomal Price Tree and the celebrated Black-Scholes Model. Through the introduction, we also illustrate several mathematical and numerical concepts related to solve our Lévy model. Then, we demonstrate the idea to use Lévy process to approximate nancial market movements and to estimate option prices. Finally, we develop an explicit-implicit nite dierence scheme for solving the exponential Levy process, which has a parabolic partial integro-dierential equation (PIDE) with jump-diusion process. We implement this scheme based on European put option, and we calculate the call option price based on Put-Call Parity. We also compare the numerical results by our Levy model and the Black-Scholes model.

Table of Contents

1 Introduction 1
1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Derivatives and Option Pricing 4
2.1 European options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 The binomial price tree model . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 The Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6
2.3.1 Brownian motion: Wiener process . . . . . . . . . . . . . . . . . . . . .7
2.3.2 Itô Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.3 The Black-Scholes Equation . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Levy Processes 13
3.1 Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.1 Simulating Lévy processes . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.2 Characteristic function of a Lévy process . . . . . . . . . . . . . . . 18

3.2 No-arbitrage condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Explicit-implicit nite dierence scheme . . . . . . . . . . . . . . . . . . . .21
3.3.1 Truncate integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.2 Divide Lu into explicit and implicit parts . . . . . . . . . . . . . . . . .24

4 Numerical Approximation and Its Implementation 25
4.1 Explicit-implicit nite dierence scheme . . . . . . . . . . . . . . . . . . . . 25
4.2 A sketch of algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 MATLAB code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26
4.4 Explanation of MATLAB code . . . . . . . . . . . . . . . . . . . . . . . . . .33

5 Numerical Results 34
5.1 Dierent parameters and implied volatility errors . . . . . . . . . . . . . 34
5.2 Results with comparison to the Black-Scholes Model . . . . . . . . . .37

6 Conclusion 39

7 Appendix 40
7.1 Existence/uniqueness and errors: exp. Lévy model . . . . . . . . . . . 40

7.1.1 Lipschitz Condition . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 40
7.1.2 Viscosity solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
7.1.3 Bound on approximation error . . . . . . . . . . . . . . . . . . . . . . . . 42
7.2 Bound on localization error . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8 Reference 45

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	BBA
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Parola chiave	implied volatility error Black-Scholes model option pricing parabolic integro-differential equations finite difference methods Levy model
Committee Chair / Thesis Advisor	Veneziani, Alessandro, Emory University
Committee Members	Valerio III, Nicholas, Emory University Nagy, James, Emory University

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