Paper: arXiv 2403.00139

Authors: Tim Leung, Matthew Lorig, Yoshihiro Shirai

Abstract

This paper analyzes a problem of optimal static hedging using derivatives in incomplete markets. The investor is assumed to have a risk exposure to two underlying assets. The hedging instruments are vanilla options written on a single underlying asset. The hedging problem is formulated as a utility maximization problem whereby the form of the optimal static hedge is determined. Among our results, a semi-analytical solution for the optimizer is found through variational methods for exponential, power/logarithmic, and quadratic utility. When vanilla options are available for each underlying asset, the optimal solution is related to the fixed points of a Lipschitz map. In the case of exponential utility, there is only one such fixed point, and subsequent iterations of the map converge to it.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using variational calculus and fixed-point theorems to solve optimal static derivative hedging under incomplete markets. However, it is purely mathematical and lacks empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A[Goal: Optimal Static Hedging in Incomplete Markets] --> B[Input: Dual Asset Exposure & Single-Asset Vanilla Options]
    B --> C[Formulate Utility Maximization Problem]
    C --> D[Apply Variational Methods for Semi-Analytical Solutions]
    D --> E{Option Availability}
    E -- Single Asset --> F[Analytical Solutions for Exponential, Power, Log & Quadratic Utility]
    E -- Both Assets --> G[Fixed-Point Iteration of Lipschitz Map for Exponential Utility]
    F --> H[Outcome: Optimal Static Derivative Positions Determined]
    G --> H