Foreign Exchange Option Valuation under Stochastic Volatility

Abstract

The case of pricing options under constant volatility has been common practise for decades. Yet market data proves that the volatility is a stochastic phenomenon, this is evident in longer duration instruments in which the volatility of underlying asset is dynamic and unpredictable. The methods of valuing options under stochastic volatility that have been extensively published focus mainly on stock markets and on options written on a single reference asset. This work probes the effect of valuing European call option written on a basket of currencies, under constant volatility and under stochastic volatility models. We apply a family of the stochastic models to investigate the relative performance of option prices. For the valuation of option under constant volatility, we derive a closed form analytic solution which relaxes some of the assumptions in the Black-Scholes model. The problem of two-dimensional random diffusion of exchange rates and volatilities is treated with present value scheme, mean reversion and non-mean reversion stochastic volatility models. A multi-factor Gaussian distribution function is applied on lognormal asset dynamics sampled from a normal distribution which we generate by the Box-Muller method and make inter dependent by Cholesky factor matrix decomposition. Furthermore, a Monte Carlo simulation method is adopted to approximate a general form of numeric solution The historic data considered dates from 31 December 1997 to 30 June 2008. The basket contains ZAR as base currency, USD, GBP, EUR and JPY are foreign currencies.