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Sparse Wavelet Methods for Option Pricing under Stochastic Volatility

by N. Hilber and A.-M. Matache and Ch. Schwab

(Report number 2004-07)

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

Keywords:

@Techreport{HMS04_334,
  author = {N. Hilber and A.-M. Matache and Ch. Schwab},
  title = {Sparse Wavelet Methods for Option Pricing under Stochastic Volatility},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2004-07},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2004/2004-07.pdf },
  year = {2004}
}

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