Expert Opinions and Logarithmic Utility Maximization for Multivariate Stock Returns with Gaussian Drift

Abstract

This paper investigates optimal trading strategies in a financial market with multidimensional stock returns where the drift is an unobservable multivariate Ornstein-Uhlenbeck process. Information about the drift is obtained by observing stock returns and expert opinions. The latter provide unbiased estimates on the current state of the drift at discrete points in time. The optimal trading strategy of investors maximizing expected logarithmic utility of terminal wealth depends on the filter which is the conditional expectation of the drift given the available information. We state filtering equations to describe its dynamics for different information settings. Between expert opinions this is the Kalman filter. The conditional covariance matrices of the filter follow ordinary differential equations of Riccati type. We rely on basic theory about matrix Riccati equations to investigate their properties. Firstly, we consider the asymptotic behaviour of the covariance matrices for an increasing number of expert opinions on a finite time horizon. Secondly, we state conditions for the convergence of the covariance matrices on an infinite time horizon with regularly arriving expert opinions. Finally, we derive the optimal trading strategy of an investor. The optimal expected logarithmic utility of terminal wealth, the value function, is a functional of the conditional covariance matrices. Hence, our analysis of the covariance matrices allows us to deduce properties of the value function.