Abstract
In this paper, we consider the response of the oilfutures option market to the onset of severe conditions in the aftermath of Feb. 15, 2020. Motivated in part by the decline of the WTI futures contract into negative territory on April 20, 2020, for the derivative market on oil futures we consider an analytical contrast between the traditional Black model and its longago predecessor, the Bachelier model. Under 2020 crash conditions, the Bachelier model performs better than Black, displaying a significantly flatter vol smile. Based in part on previous published research for shortdated maturities , the rationale for this difference is built on the contrast between between implied Black and Bachelier volatilities. Other than for extreme strikes and high Black vols, we show that the rapport works well in a wider range of maturities and volatilities. Using options data over the year 2020, we explore a notion of normalized strike to measure quantitatively the vol skew.
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Notes
See Courtault et al. (2010) for the centenary anniversary review.
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Acknowledgements
The authors acknowledge with thanks useful discussions with Alexander Eydeland, assistance from Glenn Andrews and research assistance provided at the University of Texas at Austin, while remaining solely responsible for any errors therein. Portions of this work were presented in March 2021 at the University of Texas at Austin, at the June 2020 RiskMathics Risk Management & Trading Conference, and at the Jan. 2021 joint 28th Annual Conference on Pacific Basin Finance, Economics, Accounting and Management and the 14th NCTU International Finance Conference.
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Appendix: Cleaning the options data
Appendix: Cleaning the options data
Following the setup in Carr and Madan (2005), for simplicity we consider only call options \(C_{ij}\) with strikes \(K_i\), \(K_0< K_1<K_2< \ldots < K_{N_j} \) with maturity \(T_j\). (Puts can be transformed to calls by callput parity.) Interest rates are assumed to be zero. Theoretically those call option prices should satisfy noarbitrage conditions, including:

1.
Monotonicity: for fixed maturity j call option is decreasing function of strike:
$$\begin{aligned} 0 \le \frac{C_{i1,j}C_{i,j}}{K_iK_{i1}} \le 1 , i > 0 \end{aligned}$$(8) 
2.
Convexity: for fixed maturity j butterfly spreads are nonnegative:
$$\begin{aligned} C_{i1,j}  \frac{K_{i+1}K_{i1}}{K_{i+1}K_i} C_{i,j} + \frac{K_iK_{i1}}{K_{i+1}K_i} C_{i+1,j} \ge 0 \end{aligned}$$(9) 
3.
Calendar spreads: for each strike \(K_i\) and each maturity \(T_j\) calendar spreads are nonnegative
$$\begin{aligned} C_{i,j+1}C_{i,j} \ge 0 \end{aligned}$$(10)
Since we treated each maturity separately, we were concerned with conditions 8 and 9. It is well known options data occasionally display violations of arbitrage conditions. Instead of deleting those strikes with arbitrage and loosing the valuable information, we follow the procedure in Zhao and Hodges, (2013): For each fixed maturity j we search for closest quotes calls \(\tilde{C_{i,j}} \) and puts \(\tilde{P_{i,j} }\) as a solution to the constrained optimization problem:
where \({\tilde{C}}_{i,j}\), \({\tilde{P}}_{i,j}\) satisfy noarbitrage constrains 8 and 9. Note, that for majority of strikes there is no change, and we need to change (slightly) the quotes only at a small portion of strikes. We analyzed quotes on several nearby contracts, as 1nb, 3nb, 6nb, and 12nb. We make the following three observations:

The most violations of noarbitrage were observed for options on third nearby contract. There were none for 12th nearby contract

There were no contraventions of the monotonicity condtion 8, but there were several negative butterflies, with typical value of negative butterfly \(=0.02\).

Unsurprisingly, the biggest number of arbitrage appearances occurs in the period March  July 20, with average daily number of 30 (third nearby)
In the table below we give statistics for options on 3nb contract for chosen days in 2020. We give the total number of quotes, the number of observed arbitrage occurrences, the average change in options quotes and the maximum change in options quotes.
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Galeeva, R., Ronn, E. Oil futures volatility smiles in 2020: Why the bachelier smile is flatter. Rev Deriv Res 25, 173–187 (2022). https://doi.org/10.1007/s1114702209185z
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DOI: https://doi.org/10.1007/s1114702209185z
Keywords
 Oil market volatility “smile”
 Bachelier vs. Black option models
JEL Classification:
 G12 – Asset Pricing
 G13 – Contingent Pricing