Asymptotic filter behavior for high-frequency expert opinions in a market with Gaussian drift

Abstract

This paper investigates a financial market where stock returns depend on a hidden Gaussian mean reverting drift process. Information on the drift is obtained from returns and expert opinions in the form of noisy signals about the current state of the drift arriving at the jump times of a homogeneous Poisson process. Drift estimates are based on Kalman filter techniques and described by the conditional mean and covariance matrix of the drift given the observations. We study the filter asymptotics for increasing arrival intensity of expert opinions and prove that the conditional mean is a consistent drift estimator; it converges in the mean-square sense to the hidden drift. Thus, in the limit as the arrival intensity goes to infinity investors have full information about the drift.

Keywords:

• Bayesian updating
• Black-Litterman model
• expert opinions
• Ornstein-Uhlenbeck process
• partial information
• Kalman filter

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 93E11
• secondary 60F17
• 60G35

Acknowledgments

The authors thank Dorothee Westphal and Jörn Sass (TU Kaiserslautern) for valuable discussions that improved this paper.