Small-Horizon Approximations for Optimal Portfolio Strategies

Summary

This work derives an approximate portfolio strategy for an investor in a simple incomplete market with a general utility function. It uses the Hamilton-Jacobi-Bellman equation for the value function and develops a first-order expansion in the length of the investment horizon. The resulting expression provides a closed-form approximation to the optimal trading strategy when the horizon is short.

To support the approximation, the authors construct sub- and super-solutions to the HJB equation and use martingale inequalities to bound the true value function between them. They provide a rigorous accuracy proof and outline a heuristic way to extend the approach to finite horizons. The document does not specify market data, asset classes, or numerical performance, so the result is a theoretical approximation method rather than an empirically evaluated strategy.

Key ideas

• A first-order expansion of the value function yields a closed-form strategy for short horizons.
• The optimization problem allows general utility in a simple incomplete market.
• Constructed sub- and super-solutions bound the HJB value function.
• Martingale inequalities support a rigorous proof of approximation accuracy.
• Extension from short to finite horizons is presented as a heuristic scheme.

Tags

# Asymptotic approximation of optimal portfolio for small time horizons


# Asymptotic approximation of optimal portfolio for small time horizons









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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