ABSTRACT
Abstract The existence of an implied volatility indicates that the BlackScholes assumption that assets have a constant volatility should be relaxed. The simplest extension of the log-normal Black-Scholes model is to assume that assets still follow a one-dimensional Itoˆ diffusion process but with a volatility function σloc(t, f) depending on the underlying forward f and the time t. As shown by Dupire [81], prices of European call-put options determine the diffusion term σloc(t, f) uniquely. In this chapter, we show how to apply the theorem (4.4) to find an asymptotic solution to the backward Kolmogorov (Black-Scholes) equation and derive an asymptotic implied volatility in the context of local volatility models (LVM). Before moving on to the general case in the second section, we consider in the first section a specific separable local volatility function: σloc(t, f) = A(t)C(f). Throughout this chapter, deterministic interest rates are assumed for the sake of simplicity.
