Optimal strategies of investment in a linear stochastic model of market

TL;DR

This paper develops a continuous-time portfolio optimization framework with linear dependence of asset returns on economic factors, introducing a risk-sensitive functional to determine optimal investment strategies under different interest rate models.

Contribution

It introduces a new functional $Q_eta$ for portfolio optimization considering economic factors and compares its solutions with existing risk-sensitive control approaches.

Findings

Optimal trading positions derived from maximization of $Q_eta$.

Analysis of single-factor models with Vasicek and Cox-Ingersoll-Ross interest rate dynamics.

Comparison with Bielecki and Pliska's risk-sensitive control theory.

Abstract

We study the continuous time portfolio optimization model on the market where the mean returns of individual securities or asset categories are linearly dependent on underlying economic factors. We introduce the functional $Q_\gamma$ featuring the expected earnings yield of portfolio minus a penalty term proportional with a coefficient $\gamma$ to the variance when we keep the value of the factor levels fixed. The coefficient $\gamma$ plays the role of a risk-aversion parameter. We find the optimal trading positions that can be obtained as the solution to a maximization problem for $Q_\gamma$ at any moment of time. The single-factor case is analyzed in more details. We present a simple asset allocation example featuring an interest rate which affects a stock index and also serves as a second investment opportunity. We consider two possibilities: the interest rate for the bank account is governed by Vasicek-type and Cox-Ingersoll-Ross dynamics, respectively. Then we compare our results with the theory of Bielecki and Pliska where the authors employ the methods of the risk-sensitive control theory thereby using an infinite horizon objective featuring the long run expected growth rate, the asymptotic variance, and a risk-aversion parameter similar to $\gamma$.