Option Valuation Using Asymptotic Expansion

Abstract

In this paper, the asymptotic expansion technique is proposed in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. We study the pricing of defaultable derivatives, such as bonds, bond options, and credit default swaps in the reduced form framework of intensity-based models. We use asymptotic expansions from which we derive approximations for the pricing functions of these derivatives. In particular, we assume an Ornstein-Uhlenbeck process for the interest rate, and a two-factor diffusion model for the intensity of default. The classical Black-Scholes formula gives the price of call options when the underlying is a geometric Brownian motion with a constant volatility. The underlying might be the price of a stock or an index and a constant volatility corresponds to a fixed standard deviation for the random fluctuations in the returns of the underlying. Modern market phenomena make it important to analyze the situation when this volatility is not fixed but rather is heterogeneous and varies with time. The volatility is modeled as stochastic process that varies on two characteristic time scales: fast mean reverting as evident by Fouque and el.(2000). Analysis of market data, however, shows the need for introducing also a slowly varying factor in the model for the stochastic volatility. In particular, the introduction of the slow factor gives a much better fit for options with longer maturities. The resulting approximation is still independent of the particular details of the volatility model and gives more flexibility in the parameterization of the implied volatility surface. The resulting approximation is still independent of the particular details of the volatility model and gives more flexibility in the parameterization of the implied volatility surface.