A multigrid preconditioner for an adaptive Black-Scholes solver

Abstract

This paper is concerned with the efficient solution of the linear systems of equations that arise from an adaptive space-implicit time discretisation of the Black-Scholes equation. These nonsymmetric systems are very large and sparse, so an iterative method will usually be the method of choice. However, such a method may require a large number of iterations to converge, particularly when the timestep used is large (which is often the case towards the end of a simulation which uses adaptive timestepping). An appropriate preconditioner is therefore desirable. In this paper we show that a very simple multigrid algorithm with standard components works well as a preconditioner for these problems. We analyse the eigenvalue spectrum of the multigrid iteration matrix for uniform grid problems and illustrate the method’s efficiency in practice by considering the results of numerical experiments on both uniform grids and those which use adaptivity in space

An Iterative Method for Pricing American Options Under Jump-Diffusion Models

“…The PFAS method was used to price American options under stochastic volatility by Clarke and Parrott in [9], and Oosterlee in [13]. Some alternative approaches employing multigrid methods for option pricing have been considered in [35], [36], [37]. Reisinger and Wittum described a projected multigrid (PMG) method for LCPs which resembles more closely a classical multigrid method for linear problems in [38].…”

Adaptive finite differences and IMEX time-stepping to price options under Bates model

“…Several different efficient solution and approximation methods have been proposed for these problems. These include multigrid methods [7], [22], preconditioned iterations [40], [25], and directional operator splitting methods [13], [16]. Here we employ a direct solver based on LU decomposition.…”