Malthe Rauh Johansen, Nina Lange, David Skovmand
arXiv 20 Oct 2026 · q-fin.PR
arXiv:2610.28071 · PDF · Extracted main text
We develop an arithmetic multi-factor stochastic volatility model for commodity futures and options. The arithmetic specification yields a low-dimensional Markovian representation of the futures curve and semi-closed-form pricing formulas for vanilla and basket options, including calendar spread options. These properties make the model suitable for large-scale empirical estimation. We estimate the model using an unscented Kalman filter on a panel of WTI crude oil futures, vanilla options, and calendar spread options spanning 2013-2024. We find that three stochastic volatility factors are needed to capture both the level and slope of the volatility term structure, and that hedges based on these factors remove almost all of the daily risk of a portfolio of vanilla options. Calendar spread options provide information about the dependence structure of futures prices that is not captured by futures and vanilla options alone. A model estimated without calendar spread options systematically overprices them by overstating the decorrelation between adjacent futures contracts. Including these options in the estimation brings the model's decorrelation close to the observed decorrelation, which removes the overpricing and improves their hedges, at only a small loss of accuracy in the pricing and hedging of vanilla options. Finally, we show that the arithmetic model fits futures and vanilla options better than its geometric counterpart and hedges the vanilla options slightly better.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Carl Chiarella and Boda Kang and Christina Sklibosios Nikitopoulos a… (2013) Humps in the volatility structure of the crude oil futures market: New evidence | 1.000 | 11 | 4 | 100% |
| 2 | Trolle, Anders B. and Schwartz, Eduardo S (2009) Unspanned stochastic volatility and the pricing of commodity derivatives | 1.000 | 10 | 4 | 100% |
| 3 | Duffie, Darrell and Pan, Jun and Singleton, Kenneth (2000) Transform Analysis and Asset Pricing for Affine Jump-diffusions | 0.874 | 5 | 2 | 100% |
| 4 | Schneider, Lorenz and Tavin, Bertrand (2018) From the Samuelson volatility effect to a Samuelson correlation effect: An analysis of crude oil calendar spread options | 0.874 | 5 | 2 | 100% |
| 5 | Bouchouev, Ilia Virtual Barrels: Quantitative Trading in the Oil Market | 0.811 | 4 | 2 | 100% |
| 6 | Andersen, Leif (2010) Markov models for commodity futures: Theory and practice | 0.737 | 3 | 2 | 100% |
| 7 | Cheng, Benjamin and Nikitopoulos, Christina Sklibosios and Schlögl,… (2018) Pricing of long-dated commodity derivatives: Do stochastic interest rates matter? | 0.737 | 3 | 2 | 100% |
| 8 | Duffee, Gregory R (2002) Term Premia and Interest Rate Forecasts in Affine Models | 0.737 | 3 | 2 | 100% |
| 9 | Benth, Fred Espen and Piccirilli, Marco and Vargiolu, Tiziano (2019) Mean-Reverting Additive Energy Forward Curves in a Heath–Jarrow–Morton Framework | 0.644 | 2 | 2 | 100% |
| 10 | Borovkova, Svetlana and Geman, Hélyette (2006) Seasonal and Stochastic Effects in Commodity Forward Curves | 0.644 | 2 | 2 | 100% |
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