Do jumps matter in discrete-time portfolio optimization?

Abstract

This paper studies a discrete-time portfolio optimization problem, wherein the underlying risky asset follows a Lévy GARCH model. Besides a Gaussian noise, the framework allows for various jump increments, including infinite-activity jumps. Using a dynamic programming approach and exploiting the affine nature of the model, we derive a single equation satisfied by the optimal strategy, and we show numerically that this equation leads to a unique solution in all special cases. In our numerical study, we focus on the impact of jumps and evaluate the difference to investors employing a Gaussian HN-GARCH model without jumps or a homoscedastic variant. We find that both jump-free models yield insignificant values for the wealth-equivalent loss when re-calibrated to simulated returns from the jump models. The low wealth-equivalent loss values remain consistent for modified parameters in the jump models, indicating extreme market situations. We therefore conclude, in support of practitioners’ preferences, that simpler models can successfully mimic the strategy and performance of discrete-time conditional heteroscedastic jump models.