Skip to main content

Oil futures volatility smiles in 2020: Why the bachelier smile is flatter


In this paper, we consider the response of the oil-futures 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 long-ago 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 short-dated 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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. See Courtault et al. (2010) for the centenary anniversary review.


  • Avellaneda, M., & Stoikov, S. (2008). High-frequency trading in a limit order book. Quantitative Finance, 8(3), 217–224.

    Article  Google Scholar 

  • Carlson, B. (1972). The logarithmic mean. The American Mathematical Monthly, 79(6), 615–618.

    Article  Google Scholar 

  • Carmona, D. (2003). Pricing and hedging spread options. SIAM Review, 45, n4.

    Article  Google Scholar 

  • Carr, P., & Madan, D. (2005). A note on sufficient conditions for no arbitrage. Finance Research Letters, 2, 125–130.

    Article  Google Scholar 

  • Courtault, K., Bru, C., & Lebon, L. M. (2000). Louis Bachelier On the centenary of Théorie de la Spéculation. Mathematical Finance, 10(3), 562.

    Article  Google Scholar 

  • Dumas, B., Fleming, J., & Whaley, R. E. (1998). Implied volatility functions: Empirical tests. The Journal of Finance, 8(6), 2059–2106.

    Article  Google Scholar 

  • Grunspan, (2011a). “A Note on the Equivalence between the Normal and the Lognormal Implied Volatility : A Model Free Approach”,

  • Grunspan, (2011b). “Asymptotic Expansions of the Lognormal Implied Volatility: A Model Free Approach”, SSRN 1963266.

  • Roper, M., & Rutkowski, M. (2009). On the relationship between the call price surface and the implied volatility surface close to expiry. International Journal of Theoretical and Applied Finance, 12(4), 427–441.

    Article  Google Scholar 

  • Terakodo, S. (2019). “On Option Pricing Formula Based on the Bachelier model” , SSRN 3428994

  • Zhao, B., & Hodges, S. D. (2013). Parametric modeling of implied smile functions: A generalized SVI model. Review Derivative and Research, 16, 53–77.

    Article  Google Scholar 

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Roza Galeeva.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix: Cleaning the options data

Appendix: Cleaning the options data

Following the set-up 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 call-put parity.) Interest rates are assumed to be zero. Theoretically those call option prices should satisfy no-arbitrage conditions, including:

  1. 1.

    Monotonicity: for fixed maturity j call option is decreasing function of strike:

    $$\begin{aligned} 0 \le \frac{C_{i-1,j}-C_{i,j}}{K_i-K_{i-1}} \le 1 , i > 0 \end{aligned}$$
  2. 2.

    Convexity: for fixed maturity j butterfly spreads are non-negative:

    $$\begin{aligned} C_{i-1,j} - \frac{K_{i+1}-K_{i-1}}{K_{i+1}-K_i} C_{i,j} + \frac{K_i-K_{i-1}}{K_{i+1}-K_i} C_{i+1,j} \ge 0 \end{aligned}$$
  3. 3.

    Calendar spreads: for each strike \(K_i\) and each maturity \(T_j\) calendar spreads are non-negative

    $$\begin{aligned} C_{i,j+1}-C_{i,j} \ge 0 \end{aligned}$$

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:

$$\begin{aligned} \min _{\{{\tilde{C}}_{i,j}, {\tilde{P}}_{i,j}\}} \sum _{i=1}^{N_j} \left( ({\tilde{C}}_{i,j}- C_{i,j})^2 + ({\tilde{P}}_{i,j}- P_{i,j})^2 \right) \end{aligned}$$
Table 4 Statistics on arbitrage occurrences in quotes on 3nb contract

where \({\tilde{C}}_{i,j}\), \({\tilde{P}}_{i,j}\) satisfy no-arbitrage 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 no-arbitrage 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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Galeeva, R., Ronn, E. Oil futures volatility smiles in 2020: Why the bachelier smile is flatter. Rev Deriv Res 25, 173–187 (2022).

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Oil market volatility “smile”
  • Bachelier vs. Black option models

JEL Classification:

  • G12 – Asset Pricing
  • G13 – Contingent Pricing