Academic journal article International Journal of Business

How to Price Efficiently European Options in Some Geometric Levy Processes Models?

Academic journal article International Journal of Business

How to Price Efficiently European Options in Some Geometric Levy Processes Models?

Article excerpt

ABSTRACT

This paper presents the implementation to the class of jump diffusion models of the approach used by Boyarchenko and Levendorskii (2002) in the case of exponential Levy models. We show that this approach is more computationally efficient than the semi-closed form solutions formerly obtained by Kou (2002). A brand new model is then presented. It extends and generalizes Kou model.

JEL classification: C63; G13

Keywords: Jump diffusion Models, Fourier Transform, Multiple Exponential Jumps, Kou Processes.

I. INTRODUCTION

It is now well recognized that the Gaussian hypothesis for financial assets returns is a convenient assumption but which is clearly rejected when returns are computed with high or medium frequencies. To depart from the traditional hypothesis a general modelling has been put forward for the recent years. It can be expressed in the following way: asset prices are exponentials of Levy processes, otherwise stated the returns are Levy processes or asset prices are geometric Levy processes. The class of Levy processes is very large and includes arithmetic Brownian motion. Amongst the many candidates to describe financial series and frequently used are: Generalized Hyperbolic, Normal Inverse Gaussian, Meixner, Variance Gamma, CGMY processes and of course jump diffusions. This latter group has been extensively analyzed and constitutes probably a simple and flexible choice. Furthermore any of the previous cited processes can be approximated as jump diffusions.

With geometric Levy processes the solutions to pricing and hedging problems rarely are given in closed forms. The aim of this research is to see how efficient the Fourier approach is.

This paper is organized as follows: Section II gives the definitions of the considered processes in this article, Section III briefly develops the pricing of options in this context, Section IV presents our numerical analysis and a conclusion ends the paper.

II. THE CONSIDERED JUMP DIFFUSIONS

In this section firstly we present a simple setting for jump diffusions, then we define Kou processes and in a third part we suggest a new model which extends Kou model to multiple jumps.

A. Assets Price Dynamics

The three usual ways to define a jump diffusion process are by specifying its dynamics or by expressing its exponential argument, or by giving its characteristic function. Let us denote by [S.sub.t] the financial asset price at time t. The first definition leads us to say that our jump diffusion process obeys the following stochastic differential equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The last term models the jumps. In fact a jump is modelled by a random variable Y which transforms the price [S.sub.t] to Y[S.sub.t]. The difference Y -1 is then the relative change in price when a Poisson jump occurs. The mathematical expectation of this relative change is given by [kappa] = E (Y - 1). The intensity or mean arrival rate of the jumps per unit time is [lambda]. W is a standard Brownian motion with [W.sub.0] = 0. The constants [mu] and [sigma] > 0 are respectively the drift and diffusive coefficient of the continuous part of the process. N is a Poisson process with intensity [lambda].

Using Ito's lemma, we have the expression of the process in an exponential form:

[S.sub.t] = [S.sub.0] exp ([X.sup.t])

where [S.sub.0] is the asset price at time zero and X can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where a = [mu] - [[sigma].sup.2]/2. The random variables [J.sub.i] are independent and identically distributed and model jump sizes. They are such that [J.sub.i] = In ([Y.sub.i]). We denote by [[phi].sub.j] (u) = E [[e.sup.iuJ]] their characteristic function. The processes W, N and the random variables [J.sub.i] are supposed to be independent. With this modelling, the characteristic function of the random variable [X. …

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