Absolute Price Variability for Commodity Spread Options
Summary
The discussion explains why a commodity spread option may use variability measured in absolute price changes rather than return volatility. For a spread defined as the difference between two asset prices, the value can cross zero. That makes a geometric Brownian motion model, which assumes positive values, unsuitable for the spread itself.
One alternative is to model the spread directly with an arithmetic Brownian motion or another tractable process. This can simplify pricing compared with modeling the difference of two geometric Brownian motions. The direct spread approach can also make it easier to specify skew for the spread than to combine the skew properties of both underlying assets. Under this framing, “variability” may be a clearer term than volatility. The answer provides conceptual modeling guidance, not a full pricing derivation, calibration procedure, or evidence that a particular process fits a given spread.
Key ideas
- A price spread can be negative, so a geometric Brownian motion for the spread is inappropriate.
- Modeling the spread directly with an arithmetic process can simplify analysis.
- The difference of two geometric Brownian motions is less tractable than a single geometric process.
- Specifying skew for the spread directly can be easier than combining the underlyings’ skew.
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# Price volatility instead of return volatility for spread option parameter # Price volatility instead of return volatility for spread option parameter I overheard someone at work today talking about a commodity spread option pricing model and he was asking our quant if he should use price volatility instead of return volatility as the volatility input parameter. He was referring to volatility calculated on absolute price change as opposed to relative price change. I have never heard of a model that would theoretically accept such a volatility parameter. Are there models that explicitly do? Are there other reasons why one might use volatility calculated on absolute price change as opposed to relative price change? ## Answer by Brian B (score 4, accepted) https://quant.stackexchange.com/a/2950 Spreads between asset prices $A_1$ and $A_2$ are indeed the key here. Since spreads can go negative, one certainly cannot model them with a geometric brownian motion. The natural quant inclination is to model spreads as the difference between two GBMs. Unfortunately, a difference of GBMs is not nearly so mathematically tractable as a GBM itself. In contrast, if we consider the spread $S=A_1-A_2$ as an arithmetic brownian motion or other tractable process, life gets much easier. In particular an ABM is actually easier to deal with mathematically than a GBM. The attractiveness of using $S$ is especially high once you consider skew, since taking a difference of $A_1$ and $A_2$ in a way consistent with their skew is rather difficult, while just postulating a skew for $S$ is as simple to treat as any other skew. It is common practice, when using ABM, to avoid the term volatility and use some other parameter name like variability.
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