Asymptotic methods for option pricing

Abstract

We study an asymptotic analysis of derivative prices when the underlying asset is driven by stochastic volatility model. The classical Black-Scholes model gives the analytic solution of European options price when the underlying asset is a geometric Brownian motion with constant volatility. Many papers suggest that the volatility should be driven by a stochastic process to reflect the market price movement. An empirical analysis of high-frequency index data confirms that volatility is fast mean-reverting to its long-term mean. This motivates an asymptotic analysis of partial differential equation satisfied by derivative prices. In this paper we study a corrected pricing formula and an implied volatility formula. We also produce volatility smile from market data.