Numerical Methods For The Valuation Of American Options

In this paper, we present a fast solver for computing Black-Scholes equation of Amer-ican options. Perfectly matched layer(PML) method is proposed to truncate the unboundeddomain into bounded computational domain, and under some weak assumptions on the PMLmedium parameter, it is shown that the solution of truncated PML problem converges to thesolution of unbounded Black-Scholes equation in concerned domain and exponentially de-cays in perfectly matched layer. The treatment of the free boundary is solved by front-fixingmethod based on some properties of optimal exercise boundary. Finite element method anddiscontinuous Galerkin method are used to solve the resulting problem. Numerical simula-tions are presented to test the performance of proposed algorithm. Comparisons to resultsobtained by previous approaches indicate its high accuracy and efciency.