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arXiv · 1611.09300

Asymptotic approximation of optimal portfolio for small time horizons

Rohini Kumar·Hussein Nasralah
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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.

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Rohini Kumar, Hussein Nasralah. 2016-11-28. Asymptotic approximation of optimal portfolio for small time horizons. https://arxiv.org/abs/1611.09300

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