It was a historic moment to be remembered when crude oil prices plunged into negative price territory on Monday, April 20, 2020. Imagine’s Data, Technical, Development and Professional Services teams stood ready to assist users in ensuring they could continue to correctly calculate risk metrics across commodity trading books.
While negative prices are not a new thing in financial markets, it was the first time for crude oil exhibited such behavior. Standard pricing models (Black Scholes) for commodity futures options are not designed to handle negative underlying prices. As such, Imagine applied an ‘early roll’ of the front month contract of May crude to the June expiry as the negative settle price was confirmed.
The market responded by announcing a switch to the Bachelier model in these circumstances, which is based on the assumption that the underlying price follows a normal price distribution. Imagine already employs this model for pricing commodity spread options, and will be extending the model to support standard commodity futures options in an upcoming release. In addition, Imagine will provide a separate process that assumes normal price returns instead of lognormal returns for selected commodities to cope with the negative prices for risk engine calculations (term volatility and VaR calculations). Clients will have the flexibility to choose the type of process type used for selected commodities. based on their risk calculation preferences.
In parallel, Imagine is implementing a Gaussian Copula model that inherently switches from a lognormal to a normal price process; a perfect solution should negative oil prices recur.
The occurrence of negative commodity prices for the oil contracts – at this point only occurring in the front month – violates the usual mathematical underpinnings of the standard mathematical models and requires a rapid adaptation to this unprecedented situation. Fortunately, Imagine has had ample experience over the last several years handling negative interest rates within a risk management context. There are methodologies that can swiftly be applied to ensure that Imagine not only calculates the exposures correctly, but that in the future when looking back on this time (by which we mean the time series of prices), we will be able to incorporate effectively the negative prices into calculations such as HVaR (Historical VaR) and term volatilities.
There are three stages to our response – the immediate, the short term, and the permanent, covering the following issues:
- How will we handle negative historical prices in HVaR and how will those prices affect calculations of realized volatilities and correlations?
- What adjustments are we making to the roll curve when the price of the front month contract is negative?
- What will we do if the first two months’ prices are negative, or if the price switches back to being positive again?
- How will we handle low/negative prices for commodity futures options?
- Will we be able to accommodate negative strikes?
Negative Historical Prices
As mentioned above, we already have ample experience with negative interest rates, and will apply the same methodology here ─ we store negative interest rates in our time series and will do the same for negative commodity prices. However, calculating volatilities, correlations, and HVaR relies on historical returns, not prices; therefore we modified our returns calculation by introducing a “Gaussian copula” that seamlessly switches from lognormal to normal as rates drop below the threshold of 1.0 (contact us for an appendix that illustrates this methodology with some slides from previous discussions of this topic). The only modification needed here is to specify the threshold, which we receive from the exchanges. For example, the CME has decided that for oil prices below $8.0 per barrel, they will be switching to a normal (as opposed to lognormal) option pricing model. Actually, they have stated that for prices between $8.00 and $11.00 they may be switching. We take that to mean a definitive switch if below $8.00. We will discuss option pricing models later in this article.
This modification is a permanent solution, and will be implemented by the end of May.
Adjustments to the Roll Curve and the Specification of the “Spot” Month
There are two modes for specifying the forward structure of the roll curve: roll yield and roll spread. The first relies upon lognormality, and as such cannot be used when there are negative spreads. Fortunately, the spread formulation works perfectly well with negative prices and so we can set the affected roll curves to use the spread methodology. We will make an additional roll curve with this setting available – with the extension “RCPS” – which can then be chosen selectively by each user.
As an immediate solution, we will continue to designate the spot month as the nearest non-negative contract. This is an immediate solution, but the permanent solution is to implement the full model alluded to below, in which case the price of the front month could be negative and yet also serve as the spot. It is unclear at this moment if that is desirable, but at least there will be an option to do so.
Commodity Futures Options for Low/Negative Prices
Our immediate and short-term response is to provide the normal model (Bachelier), which is valid for negative prices as well as negative strikes. We already use this model for options on spread futures, and so it is a straightforward modification for us to make. The Bachelier model will be made available over the next few weeks.
To support this model we will also be constructing new implied volatility surfaces to which the commodity futures options can be pointed. It is a strength of our system that the user can choose the commodity futures options for which they want to use the Bachelier model, and those for which they prefer to use the standard lognormal model; the user is not restricted to making a single choice for the entire option series.
The permanent solution will be to introduce a commodity futures option model that transitions from normal to lognormal on its own, such as the Gaussian copula. This development is currently underway, with plans for a roll out within the next thirty to forty-five days. We will provide a white paper to support this model when it is released.
About the Author
Bill Daher has been with Imagine for ten years and is currently a team leader in Consulting for Imagine’s APAC region. Bill has been heavily involved in the architecture of quantitative solutions in Imagine. He holds a Masters of Actuarial Studies from Macquarie University in Sydney, Australia.
Contact Bill by email or phone: +1 646 827 4427