Pricing Geometric Averaging Asian Call Option Under Stochastic Volatility Model

摘要

Options give the holder the right to buy or sell the underlying asset without obligation. Under the assumptions that asset price is a geometric Brownian process and the price volatility is constant, BS model is an ideal pricing model for European options. Despite the success and popularity of BS model, studies in empirical finance reveal that the implied volatility obtained from financial market data is not a constant but shows the implied volatility "smile" phenomena, thus the assumption of constant volatility is unrealistic. More general non constant volatility models are needed to fix this problem. In particular, lots of attention has been paid to stochastic volatility models in which the volatility is randomly fluctuating driven by an additional Brownian motion. One of these approaches is dropping the assumption of constant volatility and assumes that the underlying asset is driven by a stochastic volatility. By assuming that volatility follows a stochastic process, studies show that stochastic volatility model can better explain the volatility "smile". This paper assumes that asset volatility follows a mean-reverting stochastic process, and studies the pricing problem of geometric average Asian call option which belongs to path-dependent options. By singular perturbation analysis, the corresponding partial differential equation of the stochastic volatility model is obtained, and analytical approximation formula for the geometric average Asian call option is derived.