Paper: arXiv 2602.17090

Authors: Takuji Arai

Abstract

We explore local risk-minimization, a quadratic hedging method for incomplete markets, in exponential additive models. The objectives are to derive explicit mathematical expressions and to conduct numerical experiments. While local risk-minimization is well studied for Lévy processes, little is known for the additive process case because, unlike Lévy processes, the Lévy measure for an additive process depends on time, which significantly complicates the mathematical framework. This paper shall provide a set of necessary conditions for deriving expressions for LRM strategies in exponential additive models, as integrability conditions on the Lévy measure, which allow us to confirm whether these conditions are satisfied for given concrete models. In the final section, we introduce the variance-gamma scaled self-decomposable process, a Sato process that generalizes the variance-gamma process, as a primary example, and perform numerical experiments.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a sophisticated theoretical framework extending local risk-minimization to non-stationary exponential additive processes using Malliavin-Skorohod calculus. While mathematically dense and rigorous, it lacks substantial real-market backtesting, relying primarily on stylized numerical experiments for validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Derive Local Risk-Minimization<br>LRM for Exponential Additive Models] --> B[Define Time-Dependent Lévy Measure<br>& Integrability Conditions]
    C[Inputs: Market Parameters & Contract Payoffs] --> D[Derive Explicit LRM Strategies<br>via Föllmer-Schweizer Decomposition]
    B --> D
    D --> E[Introduce Sato Process:<br>VG-Scaled Self-Decomposable Model]
    E --> F[Conduct Numerical Experiments<br>& Evaluate Quadratic Hedging]
    F --> G[Outcomes: Explicit LRM Conditions &<br>Validated Hedging in Time-Inhomogeneous Markets]