Financial Derivative and Energy Market Valuation: Theory and Implementation in MATLAB

Chapter 7: Kalman Filter

7.1 Introduction

Several estimation procedures have been presented in the preceding chapters that were based on solving for an exact solution of the price. Known empirical data was arranged into a form to calculate the minimum variance to yield the optimal unknown parameters of the model. In this chapter, a powerful estimation technique known as the Kalman filter is discussed. The basic Kalman filter is a recursive least squares fitting to a linear model (Watson and Augustine, 1983). The Kalman filter minimizes the mean square error of the estimated parameters assuming that the models are linear functions of the underlying variables and that all sources of noise are additive Gaussian.

The success of the Kalman filter is the ability to find an optimal recursive solution with very little computational burden. This chapter begins with a derivation of the Kalman filter, followed by a simple implementation of the model. Chapter 8 (Futures and Forwards) explores futures contracts and, building off the understanding of this chapter, shows how the Kalman filter is excellent for estimating and predicting the cross-sectional behavior of futures contracts. Chapter 9 discusses nonlinear and non-Gaussian variants of the Kalman filter.

7.2 Kalman Filter Derivation

The Kalman filter recursively calculates a vector of unobserved variables from a vector of observed variables ...