Analytical Pricing Formulas in Local Volatility Models for Barrier Options

Abstract

Local Volatility is widely used to model the volatility of an asset when pricing derivatives of this asset. Although it has imperfections, local volatility is a fairly accurate and flexible model. Still, for most options no closed solution comparable with Black-Scholes exists. Analytical approximations give fast, but accurate pricing. We analyse a method that uses a parametrization of the SDE to expand the local volatility price expectation as a sum of its Black-Scholes price and Greeks. A second approach aims for an expansion of the transition density or fundamental solution. We show that for Barrier options these methods give results accurate enough for practical use, while maintaining very fast.