# How to price a calendar spread option?

How do you price calendar spread options, that is, options on the same underlying and the same strike but different times to maturity?

Clarification: I'm interested in the pricing of a a CSO (calender spread option) as defined by CME Group. Following their definition, the payout of the option at expiry is (price of long nearby futures contract (e.g. September 2008) - price of deferred futures contract (e.g. September 2009) ). (Thanks to user508 for the link and clarification below).

This means the question is still open, and I would be really interested in an answer.

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First, a quote from Hull:

"A calendar spread could be created by selling a call option with a certain strike price and buying a longer maturity call option with the same strike price"

That means that you can decompose the calendar spread option as a combination of vanilla plain options, which you can price with B-S, trees etc.

For instance, you succeed to represent your calendar spread option in the way the quote above directs, with $C_1$ being a call option with a shorter maturity $T_1$ to be selled and $C_2$ being a call option with a longer maturity $T_2$ to be bought.

Then, Price(calender spread option) = Price($C_2$) - Price($C_1$)

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Brian,

I think there is a misunderstanding here. A calendar spread option is not a spread option as you mean it (an option on the spread between two underlying). As Tai explains, rightly, it is a strategy where you are long an option with a given maturity and short another option with a different maturity (same underlying, same strike). http://en.wikipedia.org/wiki/Calendar_spread

To answer Tai's question: there is no difficulty here. Pricing is linear: price the long option (use Black-Scholes for example), price the short option and take the difference long minus short. Just make sure you've got the right implied volatility for each.

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You seem to have a comment embedded in your answer. You should comment on Brian's post separately via add comment (he likely won't see your post here otherwise) and then post your answer only here. – chrisaycock Feb 1 '12 at 14:16
I happened to see this anyway, and upvoted Olivier's answer here. – Brian B Feb 3 '12 at 13:57
A calendar spread option is not the same thing as a calendar spread of options. cmegroup.com/trading/interest-rates/stir/… – user508 Feb 4 '12 at 22:58

This is a bit of an old question, but I thought I'd contribute to add more weight to to what some people have been saying.

A CSO (calendar spread option) is NOT a calendar spread of options. If you read it carefully, you can see the Hull quote Max Li posted is talking about a calendar spread, not a CSO.

A CSO needs to be priced the same way as a spread option. The two underlying futures can be thought of as seperate instruments although with a very high correlation before the first future expires.

This means that any spread option model (such as Kirk's approximation) can be used to price the CSO if you are careful, although the same issues apply as with any spread option model, such as you must be very careful when picking the correlation to use.

The link edouard gave originally, Analytic Approximations for Spread Options goes through Kirk's approximation in some detail.

It is also possible to look at the spread itself as normally distributed but I can't find any useful links with a quick search.

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I think the definition that the CME has is not what is mentioned here. The pricing is actually occurring on the spread value (between two DIFFERENT underlying futures). Therefore, you can have negative strikes and negative underlying values as inputs into you pricing model equation.

I don't think it's as simple as pricing two separate expirations with same strike and underlying.

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Notice that in their paper Analytic Approximations for Spread Options, the authors define a spread option as follows:

A spread option is an option whose pay-off depends on the price spread between two correlated underlying assets.

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