Numerical Pricing of Equity Barrier Options with Local Volatility

Abstract

This thesis is about the pricing of equity barrier options under local volatility. We study Dupire's nonparametric local volatility model, which can be defined in terms of call option prices or in terms of implied volatilities. No-arbitrage conditions are derived for the call option surface, and equivalent conditions for the total variance surface. Dupire's model is implemented based on a Stochastic Volatility Inspired parameterization of the implied volatility surface. The Dupire-SVI model can accurately reproduce the implied volatility smile. Furthermore, we show how to incorporate dividends into our local volatility model. Lastly, we discuss option pricing by solving forward-backward stochastic differential equations with the BCOS method, a Fourier cosine expansion method. We propose a novel pricing method for barrier options by applying the BCOS method to reflected forward-backward stochastic differential equations. We compare the BCOS results to those of the Crank-Nicolson scheme.