On the optimal design of a new class of proportional portfolio insurance strategies in a jump-diffusion framework

In this paper, we investigate an optimal investment problem associated with proportional portfolio insurance (PPI) strategies when there are jumps in the dynamics of the underlying assets. PPI strategies enable investors to mitigate downside risk while still retaining the potential for upside gains. This is achieved by maintaining exposure to risky assets proportional to the difference between the portfolio value and the present value of the guaranteed amount. While PPI strategies are known to be free of downside risk in a diffusion modeling framework with continuous trading (see, e.g. Cont, R., & Tankov, P. (2009). Constant proportion portfolio insurance in the presence of jumps in asset prices. Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 19(3), 379-401), real market applications exhibit a significant, non-negligible risk, known as gap risk, which increases with the multiplier value. This paper aims to determine the optimal PPI strategy in a setting where gap risk may occur due to downward jumps in the asset price dynamics. We consider a risk-averse agent who aims to maximize the expected utility of the terminal wealth exceeding a minimum guarantee. We address the optimization problem via a generalization of the martingale approach that proves to be valid under market incompleteness in a jump-diffusion framework. We also discuss a few interesting cases numerically for specific jump size distributions.