Closed-form portfolio optimization under GARCH models

TL;DR

This paper derives a closed-form formula for optimal portfolio allocation when asset volatility follows a GARCH(1,1) process, providing a practical solution that converges to continuous-time models and demonstrates robustness in daily trading scenarios.

Contribution

It introduces the first closed-form optimal portfolio strategy under GARCH models, linking discrete-time GARCH dynamics with continuous-time stochastic volatility solutions.

Findings

Optimal strategy is independent of asset development.

Solution converges to continuous-time Heston model under certain conditions.

Robustness of solutions in daily trading scenarios with good wealth performance.

Abstract

This paper develops the first closed-form optimal portfolio allocation formula for a spot asset whose variance follows a GARCH(1,1) process. We consider an investor with constant relative risk aversion (CRRA) utility who wants to maximize the expected utility from terminal wealth under a Heston and Nandi (2000) GARCH (HN-GARCH) model. We obtain closed formulas for the optimal investment strategy, the value function and the optimal terminal wealth. We find the optimal strategy is independent of the development of the risky asset, and the solution converges to that of a continuous-time Heston stochastic volatility model, albeit under additional conditions. For a daily trading scenario, the optimal solutions are quite robust to variations in the parameters, while the numerical wealth equivalent loss (WEL) analysis shows good performance of the Heston solution, with a quite inferior performance of the Merton solution.