Nonlinear Filtering of Stochastic Differential Equations with Jumps

Abstract

In this paper, we develop an approach for filtering state variables in the setting of continuous-time jump-diffusion models. Our method computes the filtering distribution of latent state variables conditional only on discretely observed observations in a manner consistent with the underlying continuous-time process. The algorithm is a combination of particle filtering methods and the "filling-in-the-missing-data" estimators which have recently become popular. We provide simulation evidence to verify that our method provides accurate inference. As an application, we apply the methodology to the multivariate jump models in Duffie, Pan and Singleton (2000) using daily S&P 500 returns from 1980-2000 and we investigate option pricing implications.

Keywords: Filtering, Stochastic Differential Equations, Jumps, Option Pricing, Volatility