TITLE:
Valuation of European and American Options under Variance Gamma Process
AUTHORS:
Ferry Jaya Permana, Dharma Lesmono, Erwinna Chendra
KEYWORDS:
Geometric Brownian Motion, European Option, American Option, Variance Gamma Process
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.2 No.11,
October
28,
2014
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.