Valuation of European and American Options under Variance Gamma Process

Ferry Jaya Permana,Dharma Lesmono,Erwinna Chendra

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.

Highlights

Option has become a popular choice as one of instruments for hedging strategy
Study on some stock indices listed in Indonesia market, i.e. Jakarta Composite Index (JCI), LQ45 and Jakarta Islamic Index (JII), shows that a class of Levy process, i.e. the variance Gamma process, can captured those characteristics features [2]
The contribution of this paper is to model the dynamics of Indonesia Stock Indices using variance gamma (VG) model and compare the performance to the Geometric Brownian Motion (GBM) model according to some performance criteria

Summary

Introduction

Option has become a popular choice as one of instruments for hedging strategy. Nowadays, there are hundreds of traded options in the markets which are tailored to meet the need of market participants. The Black-Scholes model which is widely used to value European options and the binomial tree which is well known as the standard model to value American options rely on assumption that the underlying asset price dynamic follows the geometric Brownian motion. It means that the log return of asset is normally distributed. We describe how to value the European and American options under the assumption that the log return of underlying asset price dynamics follows the VG process. Conclusions and further research are relegated in the last section

The VG Process

Valuation of European Option

Valuation of American Option

Simulation Study

Findings

Conclusions and Further Research