A Spectral Approach to Pricing of Arbitrage-Free SABR Discrete Barrier Options

Abstract

Market volatility smile risk in derivative pricing can be modelled by the Stochastic Alpha Beta Rho (SABR) model. Once calibrated to market data, prices of European and continuously monitored barrier options can be obtained using equivalent Black’s implied volatility approximations. However these prices are only accurate for options with short maturities. On the other-hand, discretely monitored barrier options cannot be priced using this approach and a numerical technique is required. A novel computational method based on a spectral discretisation of the pricing equation is proposed for the solution of these problems. The high accuracy of the method is first established for special cases of the SABR model where analytical solutions are available and the method is then applied to the pricing of discrete barriers under the arbitrage-free SABR model.