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Martingale property and moment explosions in signature volatility models

Eduardo Abi Jaber, Paul Gassiat, Dimitri Sotnikov

TL;DR

This work analyzes when a price process driven by a signature-based stochastic volatility is a true martingale and when its moments explode. By encoding the volatility as a finite linear combination of signature elements of a time-extended Brownian path, the authors derive a sharp criterion: for nontrivial truncation order $N\ge 2$, the price is a true martingale iff $N$ is odd and $\rho\,\sigma^{2\otimes N} \le 0$, with extensions to multidimensional drivers. They convert the martingale question into the explosion behavior of a signature SDE and develop two key bounds on signature words using shuffle algebra and Lyndon words, enabling precise control of explosion versus explosion-free regimes. The moment analysis, via the BD98 variational formula, yields a clear threshold: finite $m$-th moments occur if $|\rho| > \sqrt{1-1/m}$ and infinite moments if $|\rho| < \sqrt{1-1/m}$, with a nuanced critical case depending on coefficients and time horizon. Practically, these results guide truncation choices and parameter constraints when using signature-volatility models for approximation, ensuring martingale properties and stable moment behavior in applications such as derivative pricing and numerical methods.

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.

Martingale property and moment explosions in signature volatility models

TL;DR

This work analyzes when a price process driven by a signature-based stochastic volatility is a true martingale and when its moments explode. By encoding the volatility as a finite linear combination of signature elements of a time-extended Brownian path, the authors derive a sharp criterion: for nontrivial truncation order , the price is a true martingale iff is odd and , with extensions to multidimensional drivers. They convert the martingale question into the explosion behavior of a signature SDE and develop two key bounds on signature words using shuffle algebra and Lyndon words, enabling precise control of explosion versus explosion-free regimes. The moment analysis, via the BD98 variational formula, yields a clear threshold: finite -th moments occur if and infinite moments if , with a nuanced critical case depending on coefficients and time horizon. Practically, these results guide truncation choices and parameter constraints when using signature-volatility models for approximation, ensuring martingale properties and stable moment behavior in applications such as derivative pricing and numerical methods.

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.

Paper Structure

This paper contains 31 sections, 13 theorems, 167 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline.
  3. Tensor algebra
  4. Signatures
  5. Linear combinations of signature elements
  6. Main results
  7. Martingale property
  8. Moments
  9. Extension of the martingality result to multidimensional setting
  10. A general martingality criterion
  11. Signature SDE
  12. Martingality and explosion
  13. Key lemmas for bounds on the signatures
  14. Polynomial representations in the shuffle algebra
  15. Proof of Lemma \ref{['lemma:linear_from_upper_bound']}
  16. ...and 16 more sections

Key Result

Proposition 2.3

If $\bm{\ell}_1, \bm{\ell}_2 \in T^{}(\mathbb{R}^{d + 1})$, then

Figures (2)

  • Figure 1: The put (brown) and call (dark green) implied volatility smiles for $N = 4,\ \rho = 0.9$ (upper-left), $N = 4,\ \rho = -0.9$ (upper-right), $N = 5, \ \rho = 0.9$ (lower-left), and $N = 5,\ \rho = -0.9$ (lower-right). The confidence intervals with confidence level of $95\%$ do not intersect anywhere except for the forth plot $N = 5,\ \rho = -0.9$, corresponding to the martingale spot price.
  • Figure 2: Implied total variance $\sigma_{\mathrm{BS}}^2(T, k)T$ for $N = 3$ and correlation parameters $\rho = -0.7$ (brown) and $\rho = -0.8$ (dark green). Shaded regions represent $95\%$ confidence intervals. The dashed black lines show the asymptotic slopes $\beta_R$ as given by \ref{['eq:lee_wings']}.

Theorems & Definitions (41)

  • Definition 2.1: Shuffle product
  • Definition 2.2: Signature
  • Remark 2.2.1
  • Proposition 2.3: Shuffle property
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.1.1
  • Remark 3.1.2
  • Remark 3.1.3
  • ...and 31 more