Valuation of European and American Options under Variance Gamma Process

Ferry Jaya Permana, Dharma Lesmono, Erwinna Chendra^*

Department of Mathematics, Universitas Katolik Parahyangan, Bandung, Indonesia.

DOI: 10.4236/jamp.2014.211114                

Abstract

Geometric Brownian Motion (GBM) is widely used to model the asset price dynamics. Option price models such as the Black-Sholes and the binomial tree models rely on the assumption that the underlying asset price dynamics follow the GBM. Modeling the asset price dynamics by using the GBM implies that the log return of assets at particular time is normally distributed. Many studies on real data in the markets showed that the GBM fails to capture the characteristic features of asset price dynamics that exhibit heavy tails and excess kurtosis. In our study, a class of Levy process, which is called a variance gamma (VG) process, performs much better than GBM model for modeling the dynamics of those stock indices. However, valuation of financial instruments, e.g. options, under the VG process has not been well developed. Here, we propose a new approach to the valuation of European option. It is based on the conditional distribution of the VG process. We also apply the path simulation model to value American options by assuming the underlying asset log return follow the VG process. Such a model is similar with that proposed by Tiley [1]. Simulation study shows that the proposed method performs well in term of the option price.

Keywords

Geometric Brownian Motion, European Option, American Option, Variance Gamma Process

Conflicts of Interest

The authors declare no conflicts of interest.

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