Paper: arXiv 2503.17103

Authors: Eduardo Abi Jaber, Paul Gassiat, Dimitri Sotnikov

Abstract

We study the martingale property and moment explosions of a signature volatility model, where the volatility process of the log-price is given by a linear form of the signature of a time-extended Brownian motion. Excluding trivial cases, we demonstrate that the price process is a true martingale if and only if the order of the linear form is odd and a correlation parameter is negative. The proof involves a fine analysis of the explosion time of a signature stochastic differential equation. This result is of key practical relevance, as it highlights that, when used for approximation purposes, the linear combination of signature elements must be taken of odd order to preserve the martingale property. Once martingality is established, we also characterize the existence of higher moments of the price process in terms of a condition on a correlation parameter.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a highly rigorous theoretical analysis of the martingale property and moment explosions in signature volatility models using advanced stochastic analysis and tensor algebras. However, as a purely theoretical contribution, it lacks empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze Martingale Property & Moment Explosions in Signature Volatility Models] --> B[Input: Log-Price Model driven by Signature of Time-Extended Brownian Motion]
    B --> C[Methodology: Fine Analysis of Explosion Times in Signature SDEs]
    C --> D[Evaluate Conditions for True Martingale vs. Local Martingale]
    D --> E[Key Outcome 1: True Martingale iff Order of Linear Form is Odd AND Correlation < 0]
    D --> F[Key Outcome 2: Existence of Higher Price Moments Characterized by Correlation Parameter]
    E --> G[Practical Finding: Truncate Signature at Odd Order for Valid Asset Pricing]