  In this paper, we investigate an optimal investment problem associated with
proportional portfolio insurance (PPI) strategies in the presence of jumps in
the underlying dynamics. PPI strategies enable investors to mitigate downside
risk while still retaining the potential for upside gains. This is achieved by
maintaining an 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 diffusion modeling
frameworks with continuous trading, see e.g., Cont and Tankov (2009), real
market applications exhibit a significant non-negligible risk, known as gap
risk, which increases with the multiplier value. The goal of this paper is 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 loss-averse agent
who aims at maximizing the expected utility of the terminal wealth exceeding a
minimum guarantee. Technically, we model agent's preferences with an S-shaped
utility functions to accommodate the possibility that gap risk occurs, and
address the optimization problem via a generalization of the martingale
approach that turns to be valid under market incompleteness in a jump-diffusion
framework.
