文档介绍：Journal of Banking & Finance 26 (2002) 1297–1316
ate/econbase
Pure jump Leevy processes for asset
price modelling
Heelyette Geman *
ESSEC, Finance Department and University Paris Dauphine, Place de Lattre de Tassigny,
75775 Paris Cedex 16, France
Abstract
The goal of the paper is to show that some types of Leevy processes such as the hyperbolic
motion and the CGMY are particularly suitable for asset price modelling and option pricing.
We wish to review some fundamental mathematic properties of Leevy distributions, such as the
one of infinite divisibility, and how they translate observed features of asset price
explain how these processes are related to Brownian motion, the central process in finance,
through stochastic time changes which can in turn be interpreted as a measure of the economic
, we focus on two particular classes of pure jump L eevy processes, the general-
ized hyperbolic model and the CGMY models, and report on the goodness of fit obtained both
on stock prices and option prices.
Ó 2002 Published by Elsevier Science .
JEL classification: G12; G13
Keywords: Stochastic time changes; Economic time; Leevy density; Quadratic variation; Option pricing
1. Introduction
Normality of asset returns has played a central role in financial theory for the last
few decades, starting with the Markowitz frontier and the Capital Asset Pricing
Model and more recently, as a convenient setting for Value at putations.
The normality of distributions has been augmented with the assumption of conti-
nuity of trajectories when Samuelson introduced in 1965 the geometric Brownian
* Fax: +33-1-34433001.
E-mail address: ******@ ().
0378-4266/02/$ - see front matter Ó 2002 Published by Elsevier Science .
PII: S0378-4266(02)00264-9
1298 H. Geman / Journal of Banking & Finance 26(2002) 1297–1316
motion, then used in the seminal papers by Black–Scholes (1973) and Merton (1973).
As documented in a