Log-Utility Trading with Expert Views on Mean-Reverting Drift

Summary

The paper derives an optimal portfolio strategy for an investor maximizing expected logarithmic utility when multivariate stock-return drift is hidden and evolves as an Ornstein–Uhlenbeck process. The investor learns about drift from observed returns and from expert opinions, which are unbiased estimates arriving at discrete times. The strategy depends on the conditional expected drift given the information available to the investor.

Between expert updates, the estimate follows a Kalman filter, while its conditional covariance evolves through matrix Riccati equations. The analysis studies how those covariances behave as expert opinions become more frequent over a finite horizon and under regularly spaced updates over an infinite horizon. The value function depends on the covariance matrices, linking information quality to optimal expected utility. The document presents a mathematical model and convergence analysis, but gives no empirical evaluation; practical performance would depend on the assumed drift dynamics and the reliability and timing of expert estimates.

Key ideas

• The unobserved multivariate return drift is modeled as a mean-reverting Ornstein–Uhlenbeck process.
• Investors update drift estimates using returns and unbiased expert opinions at discrete times.
• A Kalman filter describes estimate dynamics between expert updates, with covariance governed by Riccati equations.
• The optimal log-utility strategy depends on the filtered conditional expectation of drift.
• The analysis connects covariance convergence to the value function but does not report empirical trading results.

Tags

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


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









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.