Paper: arXiv 2610.28071

Authors: Malthe Rauh Johansen, Nina Lange, David Skovmand

Abstract

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.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 9.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a mathematically sophisticated arithmetic multi-factor stochastic volatility model with semi-closed-form solutions. It demonstrates strong empirical rigor through large-scale estimation using an unscented Kalman filter on extensive WTI futures and options data, including novel insights from calendar spread options. The work offers significant novelty by applying arithmetic models to crude oil and incorporating calendar spread options in a dynamic estimation framework.

Research Flowchart

  flowchart TD
    A[Research Goal: Develop and estimate an arithmetic multi-factor stochastic volatility model for commodity futures and options] --> B{Methodology: Arithmetic Multi-Factor Stochastic Volatility Model};
    B --> C{Data Input: WTI Crude Oil Futures, Vanilla Options, Calendar Spread Options (2013-2024)};
    C --> D[Computational Process: Unscented Kalman Filter];
    D --> E[Key Findings];
    E --> F1[F1: Three stochastic volatility factors capture volatility term structure level and slope];
    E --> F2[F2: Hedges based on these factors remove almost all daily risk of vanilla option portfolios];
    E --> F3[F3: Calendar spread options provide critical dependence structure information; models without them overprice spreads due to overstating decorrelation];
    E --> F4[F4: Arithmetic model fits futures/vanilla options better and hedges slightly better than geometric counterpart].