Asymptotic approximation of optimal portfolio for small time horizons

TL;DR

This paper derives a closed-form, small-time horizon approximation for optimal portfolio strategies in incomplete markets using HJB PDE analysis, providing rigorous bounds and a heuristic for finite horizons.

Contribution

It introduces a first-order approximation to the value function for portfolio optimization in incomplete markets, with a rigorous proof of its accuracy and a scheme for extending to finite horizons.

Findings

Derived a closed-form approximate trading strategy for small time horizons.

Proved the approximation's accuracy using martingale inequalities.

Suggested a heuristic extension to finite time horizons.

Abstract

We consider the problem of portfolio optimization in a simple incomplete market and under a general utility function. By working with the associated Hamilton-Jacobi-Bellman partial differential equation (HJB PDE), we obtain a closed-form formula for a trading strategy which approximates the optimal trading strategy when the time horizon is small. This strategy is generated by a first order approximation to the value function. The approximate value function is obtained by constructing classical sub- and super-solutions to the HJB PDE using a formal expansion in powers of horizon time. Martingale inequalities are used to sandwich the true value function between the constructed sub- and super-solutions. A rigorous proof of the accuracy of the approximation formulas is given. We end with a heuristic scheme for extending our small-time approximating formulas to approximating formulas in a finite time horizon.