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

ArXiv ID: 2407.21148 “View on arXiv”

Authors: Unknown

Abstract

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.

Keywords: Portfolio insurance, Jump-diffusion, Gap risk, S-shaped utility, Martingale approach, Portfolio Management

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats
  • Why: The paper relies heavily on advanced mathematical techniques including jump-diffusion SDEs, martingale methods in incomplete markets, and nonlinear PDEs, resulting in a high math complexity score. However, while it includes numerical analysis and discusses market applications, the lack of backtested performance metrics, implementation details, or real data validation places it in the low empirical rigor category.
  flowchart TD
    A["Research Goal:<br>Find Optimal PPI Strategy<br>with Jump-Diffusion Risk"] --> B["Model Setup:<br>Jump-Diffusion Asset Price &<br>S-shaped Utility Function"]
    B --> C["Methodology:<br>Generalized Martingale Approach<br>under Market Incompleteness"]
    C --> D["Computational Process:<br>Solve Optimization via<br>Lagrange Multiplier & Characteristic Function"]
    D --> E["Key Findings:<br>Optimal Multiplier Formula<br>impacted by Jump Intensity<br>and Gap Risk Parameters"]