Sparse Wavelet Methods for Option Pricing under Stochastic Volatility

Prices of European plain vanilla as well as barrier and compound options on one risky asset in a Black-Scholes market with stochastic volatility are expressed as solutions of degenerate parabolic partial differential equations in two spatial variables: the spot price S and the volatility process variable y. We present and analyze a pricing algorithm based on sparse wavelet space discretizations of order p greater than or equal to 1 in (S, y) and on hp-discontinuous Galerkin time-stepping with geometric step size reduction towards maturity T. Wavelet preconditioners adapted to the volatility models for a Generalized Minimum Residual method (GMRES) solver allow us to price contracts at all maturities 0 less than t less than or equal to T and all spot prices for a given strike K in essentially O(N) memory and work with accuracy of essentially O(N-p), a performance comparable to that of the best Fast Fourier Transform (FFT)-based pricing methods for constant volatility models (where essentially means up to powers of log N and (log h), respectively).