Closed-form portfolio optimization under GARCH models. (2022) - Escobar-Anel, Marcos; Gollart, Maximilian; Zagst, Rudi
Abstract: This paper develops an approximate 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. Based on an approximation of the log returns from Campbell and Viceira (1999), 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 (Kraft, 2005), 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.The solution is extended to two dimensions under the multivariate affine GARCH in Escobar-Anel et al. (2020). Highlights: We rely on Campbell and Viceira (1999)'s methodology to obtain the first approximate closed-form solution to a dynamic portfolio optimization problem for a GARCH model (i.e. the HN-GARCH proposed by Heston and Nandi (2000)). We prove the convergence of the optimal HN-GARCH strategy in rebalancing frequency to the optimal strategy in Heston's model (as per Kraft (2005)). The convergence is shown numerically to be slightly non-linear for some risk aversion levels. We illustrate the impact of the various GARCH parameters on the optimal investment strategy, demonstrating the solution is quite robust against deviations from the true parameter values e.g. inaccurate estimations. The impact of the approximation from Campbell and Viceira (1999), i.e a failure of the self-financing condition, is shown to be negligible in terms of the wealth process and extra cash flows. We study the wealth-equivalent loss (WEL) incurred by an investor who trades daily, but uses popular closed-form continuous-time solutions (e.g. GBM or Heston model). The analysis demonstrates a good performance of Heston, except for high market prices of risk, and a poor performance of Merton's solution.