Martingale property and moment explosions in signature volatility models

2025-03-21

AI Summary

This paper investigates linear volatility models driven by time-extended Brownian motion signatures, focusing on the true martingale property and existence of higher-order moments for the log-price process. Methodologically, it integrates signature analysis, explosion-time theory for stochastic differential equations (SDEs), martingale characterization criteria, and asymptotic estimation of higher-order moments. The main contribution is the first derivation of necessary and sufficient conditions linking the parity of the volatility’s polynomial degree, the sign of key parameters, and the true martingale property: the price process is a true martingale if and only if the degree is odd and the dominant parameter is negative. Additionally, explicit parameter thresholds for the existence of each moment order are precisely characterized. These results resolve the long-standing moment explosion problem in this class of models and provide foundational theoretical robustness for signature-driven volatility models in financial applications—particularly option pricing.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• Characterizing martingale property in signature volatility models with Brownian motion
• Establishing conditions for moment explosions in signature stochastic differential equations
• Determining odd-order signature combinations to preserve martingale property

Innovation

• Signature volatility models use Brownian motion extensions
• Martingale property requires odd-order linear forms
• Negative correlation parameter ensures moment existence