# 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?

• Did you ask the person at your office about this? – chrisaycock Feb 9 '12 at 5:12
• Yes. The model uses a geometric brownian process and he was asking what the difference was between arithmetic brownian motion. This moved us to the conversation posed in my question. – strimp099 Feb 9 '12 at 15:22

## 1 Answer

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.