Fast numerical valuation of American, exotic and complex options

The purpose of this paper is to present evidence in support of the hypothesis that fast, accurate and parametrically robust numerical valuation of a wide range of derivative securities can be achieved by use of direct numerical methods in the solution of the associated PDE problems. Specifically, linear programming methods for American vanilla and exotic options, and explicit methods for a three stochastic state variable problem (a multi-period terminable differential swap) are explored and promising numerical results are discussed. The resulting value surface gives, simultaneously, valuation for many maturities and underlying prices, and the parameters required for risk analysis.