Understanding the Commodity Futures Term Structure Through Signatures
Hari P. Krishnan, Stephan Sturm
TL;DR
The paper addresses the interpretability gap of path-signature features for commodity futures term structures by embedding signatures in a perturbative Gibson-Schwartz framework around the convenience yield. It introduces a slow perturbation parameter $\delta$ and derives a rigorous expansion of the signature $S_{0,1}(\mathbb{F}^{\delta})$ in powers of $\sqrt{\delta}$ and $\delta$, with leading terms governed by the volatility $\gamma$ and higher-order terms by the mean-reversion $\kappa$ and level parameters $\theta$, $c$. The main result provides convergence bounds in a weighted signature norm: $\Vert S_{0,1}(\mathbb{F}^{\delta}) - S_{0,1}(\mathbb{F}^{(n)}) \Vert_{sig(p,2,w)} \leq \sqrt{\delta}^{\,n+1} \sqrt{p} K_2$, enabling controlled approximation accuracy. Empirically, the authors connect these theoretical findings to observed cross-commodity differentiation, arguing that the volatility of the convenience yield is the primary driver of cross-commodity separation in signature features, consistent with carry and convenience-yield studies $PSSW23$, $KMPV18$. Practically, the work advocates a hybrid pipeline combining interpretable model-based perturbations with data-driven path signatures to produce explainable features for cross-market classification and suggests extensions to multiscale perturbations and broader applications.
Abstract
Signature methods have been widely and effectively used as a tool for feature extraction in statistical learning methods, notably in mathematical finance. They lack, however, interpretability: in the general case, it is unclear why signatures actually work. The present article aims to address this issue directly, by introducing and developing the concept of signature perturbations. In particular, we construct a regular perturbation of the signature of the term structure of log prices for various commodities, in terms of the convenience yield. Our perturbation expansion and rigorous convergence estimates help explain the success of signature-based classification of commodities markets according to their term structure, with the volatility of the convenience yield as the major discriminant.