Portfolio optimization in a market with hidden Gaussian drift and expert opinions

Overview

Abstract

This paper investigates the optimal selection of portfolios for power utility maximizing investors in 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 fixed and known dates or randomly over time.  Applying Kalman filter techniques we derive estimates of the hidden drift which are described by the conditional mean and covariance of the drift given the observations. The utility maximization problem is solved with dynamic programming methods.

For expert opinions that arrive on fixed dates, the corresponding dynamic programming equation (DPE) can be solved in closed form, and the value function and the optimal trading strategy for an investor can be derived. They make it possible to quantify the monetary value of the information provided by the expert opinions. We illustrate our theoretical findings with results from numerical experiments.

If the arrival dates are random and modeled as the jump times of a homogeneous Poisson process, the DPE is a partial integro-differential equation and degenerate in the diffusion part of the differential operator. We therefore adopt a regularization approach and add a Brownian perturbation to the state process, scaled by a small parameter that approaches zero. We prove that the value functions of the regularized problems converge to the value function of the original problem. This enables the construction of ε-optimal strategies.