The Variance Gamma Process and Option Pricing

Abstract

A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time change. These additional parameters provide control over the skewness and kurtosis of the return distribution. Closed forms are obtained for the return density and the prices of European options. The statistical and risk neutral densities are estimated for data on the S&P500 Index and the prices of options on this Index. It is observed that the statistical density is symmetric with some kurtosis, while the risk neutral density is negatively skewed with a larger kurtosis. The additional parameters also correct for pricing biases of the Black Scholes model that is a parametric special case of the option pricing model developed here.

Pricing of Discretely Sampled Asian Options Under Levy Processes

“…With ν = 0 and σ = 0, we obtain the Variance Gamma Process (VGP), which was introduced to Finance by Madan and co-authors [53,52,51].…”

Bayesian Learning of Impacts of Self-Exciting Jumps in Returns and Volatility

“…The dynamics (1) indicates two distinct types of shocks to asset returns: small continuous shocks, captured by a Brownian motion, and large discontinuous shocks, modeled in this paper by the Variance Gamma process of Madan, Carr, and Chang (1998), a stochastic process in the class of infinite activity Lévy processes. The jump component is important for generating the return non-normality and capturing extreme events.…”

A multivariate Lévy process model with linear correlation

“…(Variance gamma process) The variance gamma process of [17] is a Lévy process obtained by evaluating Brownian motion with drift at a random time given by a gamma process; …”

“…Particularly, in mathematical finance, there exists a vast literature on Lévy process modeling for a single asset (see, for example, Carr et al [6], Kou [15], Madan et al [17], Prause [22], Rydberg [23], Schoutens and Teugels [25]). In the recent structured products market, it has become quite usual that the coupons are determined by more than a single index, for example, two FX rates of USDJPY and AUDJPY.…”