The additive Bachelier model with an application to the oil option market in the Covid period

Abstract

In April 2020, the Chicago Mercantile Exchange temporarily switched the pricing formula for West Texas Intermediate oil market options from the Black model to the Bachelier model. In this context, we introduce an additive Bachelier model that provides a simple closed-form solution and a good description of the implied volatility surface. This new additive model exhibits several notable mathematical and financial properties. It ensures the no-arbitrage condition, a critical requirement in highly volatile markets, while also enabling a parsimonious synthesis of the volatility surface. The model features only three parameters, each with a clear financial interpretation: the volatility term structure, the vol-of-vol, and a parameter for modelling skew. Model calibration can follow a cascade procedure: first, it accurately replicates the term structures of forwards and At-The-Money volatilities observed in the market; second, it fits the smile of the volatility surface. The proposed model also supports efficient pricing of path-dependent exotic options via Monte Carlo simulation, using a straightforward and computationally efficient approach. Overall, this model provides a robust and parsimonious description of the oil option market during the exceptionally volatile first period of the Covid-19 pandemic.

Figures (10)

• Figure 1: Spread of the discount-rates (from option prices) on the OIS curve, for all value dates considered in the analysis. The vertical lines correspond to the expiries of the first two options. In blue (dots and continuous line, left axis), we can see the spread corresponding to the first maturity: we observe that it rapidly increases when approaching options' first expiry, reaching some hundreds of basis points. In green (squares and dashed line, right axis), we see the average of the spread corresponding to the other expiries: we can observe that it is of a few basis points (an average of $2$ bps), i.e. its order of magnitude is much smaller than the spread for the first expiry.
• Figure 2: Parameter $\eta$ of the additive Bachelier calibrated for the different value dates considered in the analysis. The vertical lines correspond to the expiries of the first two options. We observe that its value does not change significantly for the value dates, and it moves slowly over time.
• Figure 3: Parameter $k$ of the additive Bachelier calibrated for the different value dates considered in the analysis. The vertical lines correspond to the expiries of the first two options. We observe that its value moves slowly over time.
• Figure 4: For each value date $t_0$, RMSE between market prices and model prices using the parameters $\eta_{t_0},\,k_{t_0}$ calibrated for the value date $t_0$, $RMSE_{t_0}(\eta_{t_0},k_{t_0})$, and the RMSE between market prices and model prices obtained using the previous day parameters $\eta_{t_0-1},\,k_{t_0-1}$$RMSE_{t_0}(\eta_{t_0-1},k_{t_0-1})$. The difference between the two errors is negligible for almost all dates (of the order of $1$ cent or below), confirming the stability over two consecutive value dates of the additive Bachelier model.
• Figure 5: Difference of RMSE over a $5$-day window using fixed versus optimal parameters. We compare the $5$-day RMSE between market prices and model prices using the parameters $\eta_s,\,k_s$ for the each value date $s\in\left[t_0-4,\,t_0\right]$, $RMSE_{t_0-4:t_0}(\eta_{t_0-4:t_0}, k_{t_0-4:t_0})$, against the $RMSE_{t_0-4:t_0}(\eta_{t_0-5}, k_{t_0-5})$ obtained using the fixed parameters $\eta_{t_0-5},\,k_{t_0-5}$, calibrated before the beginning of the window. We observe a contained increase in the calibration error of about $1$ cent or below, with the only exception of the $27^{\mathrm{th}}$ of May 2020, when it is less than $2.5$cents. This fact highlights the robustness of the additive Bachelier model’s parameters.

Theorems & Definitions (28)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• Proposition 2.7
• Corollary 2.8
• Proposition 2.9
• Proposition 2.10