# Synthetic equity index futures calendar spread using options

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?

• 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. – Alex C Dec 14 '18 at 1:48
• thanks, then for the synthetic future, should it be using ATMF or ATM? if ATMF is used, then still the synthetic spread is 0? – tudali0928 Dec 14 '18 at 2:10
• With the synthetic you will profit or lose from movement of the spread just like with actual futures. – Alex C Dec 14 '18 at 2:49
• @AlexC you should post it as answer, this is correct – Ezy Dec 14 '18 at 10:39