Three options are roughly enough - Variance-optimal semi-static hedging in the rough Heston model

Paper: arXiv 2610.13543

Authors: Paolo Di Tella, Martin Keller-Ressel, Beatrice Ongarato

Abstract

This paper studies semi-static variance-optimal hedging of variance swaps in the rough Heston model under a risk-neutral measure. We combine dynamic trading in the underlying asset with static positions in a finite basket of European options, with the aim of minimizing the variance of the residual hedging error. Exploiting the Markovian lift provided by the forward variance curve, the Fourier representations of the claims and the Laplace transform characterization of the log-price, we derive semi-explicit formulas for the optimal strategies and the hedging error. We then study the optimal claim selection problem, investigating numerically how the choice of static instruments depends on the Hurst index. Under standard parametrization, we discover that “three options are roughly enough”, i.e. adding three static vanilla options reduces the hedging error by more than 95% relative to the purely dynamic hedge for all considered roughness regimes H= 0.1, 0.3, 0.5.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly mathematical approach to a practical hedging problem, deriving semi-explicit formulas for optimal strategies. While it includes numerical analysis and investigates optimal claim selection, the empirical validation is primarily numerical rather than based on real-world backtesting. The novelty lies in extending semi-static variance-optimal hedging to the rough Heston model and quantifying the impact of the Hurst index.

Research Flowchart

  flowchart TD
    A[Research Goal: Variance-Optimal Semi-Static Hedging of Variance Swaps in Rough Heston] --> B(Methodology: Combine Dynamic Trading & Static Options)
    B --> C{Inputs: Rough Heston Model, Risk-Neutral Measure, European Options Basket}
    C --> D[Computational Processes: Markovian Lift, Fourier Representations, Laplace Transforms]
    D --> E(Derive Semi-Explicit Formulas for Optimal Strategies & Hedging Error)
    E --> F{Study Optimal Claim Selection & Hurst Index Impact (H=0.1, 0.3, 0.5)}
    F --> G[Key Finding: "Three Options Are Roughly Enough" - >95% Error Reduction]