Pricing Commodity Spread Options with Negative Values

Summary

The document considers a calendar spread option on nearby and deferred Brent contracts, where the spread may be zero or negative. This makes a standard Black–Scholes treatment unsuitable, since its lognormal price assumption does not accommodate negative underlying values. The discussion presents two alternatives: model the spread directly with a normal distribution and use a Bachelier-style approach, or use established spread-option formulas such as Margrabe for a zero strike and Kirk for a nonzero strike.

The analytical spread models require an estimate of the correlation between the two commodity prices. When market quotes do not provide implied correlations, one response suggests examining the history of correlations and choosing values from the high or low end of their observed range according to whether a bid or ask is being estimated. The other response suggests estimating the spread’s dollar volatility from historical data if a normal approximation is defensible. These are modeling suggestions, not a comparison backed by pricing results; the suitability of the distribution and historical estimates remains dependent on the commodity and market conditions.

Key ideas

• A commodity calendar spread can be zero or negative, which conflicts with Black–Scholes assumptions.
• A direct normal model can turn spread-option valuation into a Bachelier-style problem.
• Margrabe and Kirk formulas address different strike cases for spread options and require correlation.
• Historical correlation ranges can serve as a proxy when implied correlations are unavailable.
• The normal approximation and historical inputs need to be assessed for the specific commodity.

Tags

# Pricing options with 0 or negative underlying values


# Pricing options with 0 or negative underlying values












I am trying to calculate the value of an option whose underlying is the calendar spread between two months for a commodity (front month Brent vs 2nd month), usually known as a calendar spread option.

I am avoiding a CSO model as I do not know where to find implied correlation marks. However, when using the BS model I am running into issues as this spread can be negative when the market is in contango, or zero.

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/43856

You could just consider the calendar spread as a single variable. Depending on the commodity you might be able to convince yourself that it is approximated well by a normal distribution, in which case you can estimate the dollar standard deviation from historical data and then you have a simple Bachelier type modeling problem.

## Answer by ZRH (score 0)

https://quant.stackexchange.com/a/43855

There are essentially two analytical models for pricing spread options: i) the Margrabe model for the exchange of two assets (i.e. an option on the spread with strike zero), ii) the Kirk model for options on spreads with non-zero strike.

For both of them, you will need to specify a correlation coefficient $\rho$. Using BS is not possible, since it does not allow for negative prices, which is possible in spread options. Furthermore, commodity price spreads will be distributed in a significantly non-lognormal fashion, which any BS-type model is not able to capture.

Where there are no prices to infer implied correlations, I usually look at how historical correlations evolved over time. Depending on whether bid or ask prices need to be determined, i will use levels at the high or low end of the range. Hope that helps...

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