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I understand it is possible to synthetic a future using long call and short put ATM options which has the same expiry as the futures. Can we do the following to synthetic a future calendar spread?

$F_x$ and $F_y$ are future prices expiring on month $x$ and $y$ respectively,

$F_x - F_y$ is synthetic using $(\mathrm{Call}_x - \mathrm{Put}_x) - (\mathrm{Call}_y - \mathrm{Put}_y)$

Once thing confuses me is $F_x - F_y$ is a calendar spread which is usually non-zero. However, $\mathrm{Call}_x - \mathrm{Put}_x$ (or $\mathrm{Call}_y - \mathrm{Put}_y$) is 0 due to call put parity when strike price is ATM.

What's wrong with my reasoning?

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  • $\begingroup$ The strike for which Call-Put = 0 is not ATM but ATMF (at the money forward, i.e. based on where the future, not the spot, is trading) which will be different for month x and month y. $\endgroup$ – Alex C Dec 14 '18 at 1:48
  • $\begingroup$ thanks, then for the synthetic future, should it be using ATMF or ATM? if ATMF is used, then still the synthetic spread is 0? $\endgroup$ – tudali0928 Dec 14 '18 at 2:10
  • $\begingroup$ With the synthetic you will profit or lose from movement of the spread just like with actual futures. $\endgroup$ – Alex C Dec 14 '18 at 2:49
  • $\begingroup$ @AlexC you should post it as answer, this is correct $\endgroup$ – Ezy Dec 14 '18 at 10:39
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The option strikes do not have to be ATMF to create a synthetic future. The requirement is that they must be the same strike for the Put and the Call; and have the same expiry as the maturity of the future. Additionally, to be strict, they should be European exercise.

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Strictly speaking the strike for which Call-Put = 0 is not ATM but ATMF (at the money forward, i.e. based on where the future, not the spot, is trading) which will be different for month x and month y.

Once you set up your synthetic spread this way, you will profit or lose from movement of the spread just like with actual futures.

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