Rough volatility dynamics in commodity markets
Rough volatility dynamics in commodity markets
This paper extends rough volatility models to commodity markets, focusing on the rBergomi and rHeston specifications. The authors develop a general framework where the fictitious spot price follows an affine drift process with stochastic volatility driven by a fractional Brownian motion (Hurst parameter H < 0.5). The model automatically calibrates the initial futures term structure and captures the Samuelson effect—the increase in futures volatility as delivery approaches—through the mean reversion parameter a(t). The key theoretical contribution is the derivation of a closed-form relationship between the normalized spot price and futures prices, ensuring martingale properties under mild conditions. The paper provides explicit conditions for the rBergomi (requiring ρ < 0) and rHeston (no restriction on ρ) models. Numerical calibration to WTI Crude Oil vanilla futures options demonstrates that rough volatility models outperform classical stochastic volatility models in fitting implied volatility surfaces, particularly for short maturities where the volatility skew is more pronounced. The results highlight the practical applicability of rough volatility in commodity markets, offering a robust tool for pricing and risk management.
Highlights
- 1Develops a general rough volatility model for commodities that automatically calibrates the initial futures term structure and captures the Samuelson effect.
- 2Extends the rBergomi and rHeston rough volatility models to commodity markets, providing explicit dynamics for the fictitious spot price and futures prices.
- 3Provides a theoretical framework linking the forward variance model to commodity pricing, ensuring martingale properties of futures prices.
- 4Demonstrates calibration to WTI Crude Oil options data, showing the models' ability to fit implied volatility surfaces.
Methods
- MRough fractional stochastic volatility (rFSV) modeling with fractional Brownian motion and Hurst parameter H < 0.5.
- MForward variance approach: modeling the instantaneous variance as a stochastic process driven by a fractional Brownian motion.
- MAffine drift specification for the fictitious spot price to enable closed-form solutions for futures prices and the Samuelson effect.
- MMonte Carlo simulation and calibration to market data of vanilla futures options on WTI Crude Oil.
Results
- RThe proposed rough volatility models (rBergomi and rHeston) successfully reproduce the Samuelson effect by tuning the mean reversion speed a(t).
- RThe models automatically calibrate the initial term structure of futures prices via the relationship Ft(T) = F0(T) (1 - (1 - st) exp(-∫_t^T a(u) du)).
- RCalibration to WTI Crude Oil options data shows that rough volatility models (with H ~ 0.1) provide a better fit to implied volatility surfaces compared to classical stochastic volatility models.
- RThe rHeston model does not require negative correlation ρ for martingale property, unlike rBergomi which requires ρ < 0.
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