# Hierarchical Graph Learning for Calendar Spread Strategies in Commodity Futures Markets

arXiv: https://arxiv.org/abs/2606.25811 · HTML: https://mlsanity.com/paper/2606.25811/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 48 references that are in the map.
- First posted on arXiv 24 Jun 2026.

## Builds on (most cited first, 7 of 7)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [When Do Neural Nets Outperform Boosted Trees on Tabular Data?](https://mlsanity.com/paper/2305.02997/) (2023)
- [Deep Learning Statistical Arbitrage](https://mlsanity.com/paper/2106.04028/) (2021)
- [Index Tracking via Learning to Predict Market Sensitivities](https://mlsanity.com/paper/2209.00780/) (2022)

## Abstract (arXiv)

Commodity futures can be represented hierarchically, with underlying assets at the upper level and individual futures contracts at the lower level. Entities at each level can be connected by edges reflecting inherent correlations, with cross-level edges capturing contract-to-underlying asset connections. Building on our observations of these structures, we propose a hierarchical graph learning approach for calendar spread (CS) strategies in commodity futures markets, addressing two significant gaps in the machine-learning literature: (i) the absence of learning-based methods for CS strategies in futures markets, and (ii) the lack of consideration of maturity-dependent interrelationships across commodity futures. We first establish the efficacy of CS strategies by analytically showing that CS strategies can possess higher risk-adjusted returns, measured by the information ratio, and lower risk, measured by variance and delta, than long-only strategies. We then introduce a method to convert learning-based predictions into CS positions. Next, we develop a hierarchical graph learning method that predicts futures price movements by utilizing the maturity-dependent interrelationships, thereby yielding a CS trading algorithm. Empirical results on commodity futures markets traded on the Chicago Mercantile Exchange Group demonstrate that our method outperforms benchmark models in both prediction and trading performance. We find that maturity-dependent interrelationships across commodity futures are instrumental in prediction and that CS trading based on hierarchical graph learning is effective for statistical arbitrage.