Linear and nonlinear filtering in mathematical finance: a review

The problem of estimating unobserved latent variables from observed market data arises frequently in mathematical finance. Kalman filter, first proposed in Kalman (1960), and its generalizations have been the main tools for estimating the unobserved variables from the observed ones in econometrics and in engineering for several decades and their use is now becoming common in finance. Kalman filter is a conditional moment estimator for linear Gaussian systems. It is used in calibration of time series models, forecasting of variables and also in data smoothing applications. The purpose of this paper is to provide an introductory and accessible exposition of applications of filtering in finance to operational researchers. It gives a brief overview of Kalman filtering theory and presents recent empirical results on two applications in finance. Some recent developments in approximate nonlinear filtering are also reviewed. The rest of the paper is organized as follows. In section 2, basic linear Gaussian filtering methodology is described, along with its application to maximum likelihood-based calibration to time series models. Sections 3.1 and 3.2 present two case studies for applications of Kalman filtering in mathematical finance. Section 4 outlines some recent approaches for approximate filtering in nonlinear time series and also presents a brief overview of an empirical application on calibration and forecasting using a nonlinear interest rate model. Finally, section 5 summarises the contributions discussed in the paper and outlines promising directions for future research.