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Rough volatility dynamics in commodity markets

Roberto Daluiso, Héctor Folgar-Cameán, Andrea Pallavicini, Carlos Vázquez

Abstract

In this paper, we develop a general rough volatility model for commodities that provides an automatic calibration of the initial term structure of the futures prices and an appropriate treatment of the Samuelson effect. After the theoretical analysis of this general model, we focus on the rBergomi and rHeston models and their calibration to market data of vanilla futures options on WTI Crude Oil. Finally, numerical results illustrate the performance of the proposed rough volatility models for commodities pricing.

Rough volatility dynamics in commodity markets

Abstract

In this paper, we develop a general rough volatility model for commodities that provides an automatic calibration of the initial term structure of the futures prices and an appropriate treatment of the Samuelson effect. After the theoretical analysis of this general model, we focus on the rBergomi and rHeston models and their calibration to market data of vanilla futures options on WTI Crude Oil. Finally, numerical results illustrate the performance of the proposed rough volatility models for commodities pricing.

Paper Structure

This paper contains 17 sections, 4 theorems, 38 equations, 18 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Commodity markets
  3. Rough volatility for commodities
  4. Fractional Brownian motion
  5. General rough volatility model for commodities
  6. Samuelson effect
  7. Simulation and calibration
  8. Initial forward variance curve model
  9. Mean reversion speed $a(t)$
  10. Calibration scheme
  11. Numerical results
  12. Estimation of the Hurst parameter
  13. Calibration data
  14. Calibration results: 14th March 2025
  15. Calibration results: 4th June 2025
  16. ...and 2 more sections

Key Result

Lemma 3.3

Given the SDE where $X_0$ is given and $a(t), b(t), g(t)$ are continuous functions on $t \in [0,T]$, then the solution is given by where

Figures (18)

  • Figure 1: ATM implied volatility with respect to the options maturity date of a simulated mid-curve option for different mean reversion speeds. The Samuelson effect is more pronounced as the mean reversion speed increases.
  • Figure 2: linear regression of $\log m(q,\Delta)$ on $\log \Delta$ for the WTI Crude Oil futures contracts listed in Table \ref{['tab:hurst']}.
  • Figure 3: Optimised time-dependent parameter of each model calibrated to the data from 14th March 2025. The vertical lines mark the maturity dates considered.
  • Figure 4: Calibration results of the rBergomi model (\ref{['eq:model_rb']}) for the data from 14th March 2025.
  • Figure 5: Calibration results of the rHeston model (\ref{['eq:model_rh']}) for the data from 14th March 2025.
  • ...and 13 more figures

Theorems & Definitions (13)

  • Definition 3.1: Fractional Brownian motion
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • Corollary 3.5: rBergomi
  • proof
  • Corollary 3.6: rHeston
  • proof
  • Remark 3.7
  • ...and 3 more