Implied roughness in the term structure of oil market volatility

Abstract

This paper analyses the attributes and the significance of the roughness of oil market volatility. We employ unspanned stochastic volatility models driven by rough Brownian motions that yield semi-analytical prices for future options entailing efficient calibration applications. By performing a Monte Carlo simulation study, we show that the semi-analytical pricing performs well thus establishing its efficiency for calibration applications. Thus we calibrate option prices written on oil futures and provide empirical evidence of the roughness in oil volatility. Introducing just one additional parameter, the Hurst parameter, indicating the roughness of the volatility improves the calibration by almost a factor of 10. The calibrated option-implied Hurst parameter varies over time, but the entire set of parameters becomes more stable than in the non-rough case corresponding to a fixed Hurst parameter 1/2. These results underscore the importance to model the time dependency of the roughness of oil market volatility.

Keywords:

• Fractional Brownian motion
• Rough volatility
• Oil volatility
• Calibrated Hurst parameter
• Options

JEL Classifications:

• C32
• C63
• G13
• Q40

Open Scholarship

This article has earned the Center for Open Science badge for Open Materials. The materials are openly accessible at https://github.com/mathlion2023/Rough-forward-stochastic-volatility-model.

Acknowledgments

We thank the editors, anonymous reviewers, Erik Schlögl, Jan Obloj, and the participants of the Interdisciplinary Statistics Colloquium at the Universities of Giessen and Marburg, (Giessen, Germany), the 10$\mathrm{th}$ International Congress of Industrial and Applied Mathematics 2023 (Tokyo, Japan), the 6$\mathrm{th}$ Quantitative Finance and Risk Analysis (QFRA) Symposium 2023 (Crete, Greece), and the participants of the National Institute for Theoretical and Computational Sciences (NITheCS) colloquium, Mathematics in Finance conference (MIF2023, Kruger National Park, South Africa).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Due to time-inhomogeneity of the fractional differential equation, standard numerical schemes are inefficient, for example Adam method (see Diethelm and Freed 2004).

2 To speed up the calibration process, we use previously calibrated parameters as initial inputs of the current calibration.

3 Bergomi model is a lognormal stochastic volatility model, with the log of the forward variance being the sum of Ornstein–Uhlenbeck stochastic process, see Bergomi (2005).

4 This differentiates our approach to existing literature that requires high frequency datasets to estimate volatility and subsequently the associated Hurst parameter, see Gatheral et al. (2018), Cont and Purba (2023), and references therein.

5 This period has been selected as an illustrative period covering times of high volatility in the oil markets markedly around the GFC but also periods of low volatility (e.g. 2007 and 2009). In particular, during high oil volatility environments, models typically cannot provide good fit, thus it is useful to demonstrate outperformance of the proposed models during such periods.

6 In our calibration application, local and global optimisations are considered. Global optimisation is performed with Adaptive Simulated Annealing (ASA) (Ingber 1996) and the Matlab Builtin function genetic algorithm (GA). Local optimisation is handled by fmincon, because our preliminary result shows no substantial difference in the convergence to the minimum point. Also, fmincon is faster. For the initial parameter set, we first ran the optimisation 10 times and take the best parameters as initial guest. Moving forward, for daily recalibration, we used yesterday's calibrated parameters as initial guest to today's calibration.

7 The code used to produce the results is available in the github link below https://github.com/mathlion2023/Rough-forward-stochastic-volatility-model. The data of this study have been purchased from the NYMEX and they cannot become publicly available.

8 In Matlab, we ran the Monte Carlo parfor over all simulations and have activated all cores using parallel pool (parpool) for computational speed enhancement.

9 For model calibration, we use the initial parameter values of the classical model as the ones given in table 2. For the rough forward stochastic volatility model, in addition to the parameters in table 2, we start with a Hurst parameter value of H = 0.5. We ran the calibration 10 times and choose the best calibrated parameters for day 1. We proceed with daily recalibration by setting the initial parameters to be the previous day calibrated parameters.

Additional information

Funding

Nikitopoulos acknowledges financial support from Australian Research Council DP 130103315 grant. Mesias Alfeus acknowledges financial support from the Department of Science and Innovation (DSI) and the NITheCS Implementation of Industry Directed Research Programme (IDRP) [DSI/CON C3025, 2023].