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In futures there exist exchange traded calendar spread contracts, which trade as a single unit (think May/June Crude Oil). The bid ask spread for the spread contracts is the same as that of the outrights, which typically makes trading the spread contract (if you would like to trade a spread) cheaper than trading the outright contracts individually.

Is there a way to take this into account in a portfolio optimization problem. For example, perhaps without transaction costs my mean variance optimization says to buy 55 June (M) crude oil contracts and Sell 45 July (N) crude oil contracts. This amounts to buying 55 spread (M/N) contracts and buying 10 July (N) contracts. However, because trading the M/N spread contracts are 50% cheaper than trading the outright contracts, the optimal solution knowing the transaction cost difference may be just to buy 50 M/N spread contracts.

Is there a way to formalize this? I know the case in which you have individual assets people typically use the 3/2, volume based transaction model by Chriss. But in cases in which these spread contracts exist is there a formal way of modelling the transaction costs?

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I think the assumption you made is incorrect. Buying 55 spreads + buy 10 July is totally different scenario comparing buying 50 spreads from the risk perspective as you are net long 10 June contracts in the former strategies.

I guess what you are thinking is what is most cost-effective execution strategy for a given trading strategy. That will be an apple-to-apple comparison. For the case you used, what is the best execution strategy? Should be (buy 55spreads + buy 10 July), (buy 45spreads + buy 10 June), or other ways? There are always tradeoffs. Buy 55 June, sell 45 july will be more costly, but the execution efficiency will be higher.

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