A Krylov Subspace Method for Option Pricing

We consider the pricing of financial contracts that are based on two or three underlyings and are modelled using time dependent linear parabolic partial differential equations (PDEs). To provide accurate and efficient numerical approximations to the financial contract's value, we decompose the numerical solution into two parts. The first part involves the spatial discretization, using finite difference methods of the governing PDE. From this a large system of ordinary differential equations (ODEs) with a special affine structure is obtained. On the second part, we develop highly efficient numerical methods to approximate the exact solution of the system of ODEs with a high level of accuracy. We compare our approach, which is based of Krylov subspace methods, with the Crank-Nicolson and Alternating Direct Implicit (ADI) methods on Rainbow and Basket options, European call options using Heston stochastic volatility, and power reverse dual currency (PRDC) swaps with Bermudan and knockout features