Estimating option prices using multilevel particle filters

Abstract

Option valuation problems are often solved using standard Monte Carlo (MC) methods. These techniques can often be enhanced using several strategies especially when one discretizes the dynamics of the underlying asset, of which we assume follows a diffusion process. We consider the combination of two methodologies in this direction. The first is the well-known multilevel Monte Carlo (MLMC) method [7], which is known to reduce the computational effort to achieve a given level of mean square error (MSE) relative to MC in some cases. Sequential Monte Carlo (SMC) (or the particle filter (PF)) methods (e.g. [6]) have also been shown to be beneficial in many option pricing problems potentially reducing variances by large magnitudes (relative to MC) [11, 17]. We propose a multilevel particle filter (MLPF) as an alternative approach to price options. We show via numerical simulations that under suitable assumptions regarding the discretization of the SDE driven by Brownian motion the cost to obtain O $(\epsilon^2)$ MSE scales like O $(\epsilon^{-2.5})$ for our method, as compared with the standard particle filter O $(\epsilon^{-3})$.