Paper: arXiv 2609.33913

Authors: Florian Bourgey, Jim Gatheral

Abstract

Building on the forest expansion of Alòs, Gatheral and Radoičić and on the explicit Bergomi-Guyon smile expansion derived by Bourgey and Gatheral (2026), we derive fixed-point approximations, in terms of the implied total variance at a small number of magic strikes, for the fair values of power payoff contracts; the variance and gamma contracts arise as the endpoints of this one-parameter family, priced by a single set of formulae. We prove that such approximations exist at every order: at order $n$, $2\lceil n/2\rceil+1$ magic strikes suffice. Numerical tests under the Heston and rough Bergomi models demonstrate good accuracy even for strongly skewed smiles, and the method applies directly to interpolated market smiles without strike inversion. For the volatility contract, we show that the maturity-T smile does not determine the leading correction to the Rolloos-Arslan approximation.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly mathematical approach to pricing variance and gamma contracts, building on advanced expansions. It demonstrates strong empirical rigor through numerical tests on established models and market data, and offers a novel fixed-point approximation method. The provided GitHub repository significantly enhances reproducibility.

Research Flowchart

  flowchart TD
    A[Research Goal: Price Power Payoff Contracts, esp. Variance & Gamma, using Magic Strikes] --> B{Methodology: Fixed-Point Approximations via Smile Expansions};
    B --> C[Inputs: Alòs, Gatheral & Radoičić Forest Expansion + Bourgey & Gatheral Smile Expansion];
    C --> D{Computational Process: Derivation of Formulae & Numerical Tests};
    D --> E[Data Sources: Heston Model & Rough Bergomi Model (for Numerical Tests)];
    E --> F{Key Findings: Accurate Pricing for Power Payoff Contracts (Variance & Gamma) using 2⌈n/2⌉+1 Magic Strikes};
    F --> G[Outcome: Valid for Skewed Smiles, Applies to Market Data, Volatility Contract Nuance Identified].