Sparse wavelet methods for option pricing under stochastic volatility

Abstract

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 solution of degenerate parabolic partial differential equations with 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 $p\geq 1$ in the spot price $S$ or the log-returns $x=\log S$ and in $y$, the volatility driving process, and on $hp$-discontinuous Galerkin time-stepping with geometric step size reduction towards maturity $T$. Wavelet preconditioners adapted to the volatility modelsfor a GMRES solver allow to price contracts at all maturities $0