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

Abstract

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.

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

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

  (8)

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

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

  (9)

• 3. Calendar spreads: for each strike $K_i$ and each maturity $T_j$ calendar spreads are non-negative

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

  (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:

$$\begin{aligned} \min _{\{{\tilde{C}}_{i,j}, {\tilde{P}}_{i,j}\}} \sum _{i=1}^{N_j} \left( ({\tilde{C}}_{i,j}- C_{i,j})^2 + ({\tilde{P}}_{i,j}- P_{i,j})^2 \right) \end{aligned}$$

(11)

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.