Building Calendar Spreads from Synthetic Futures with Options

Summary

The document explains how to represent a calendar spread between two futures expiries using options. For each maturity, a long call and short put at the same strike and expiry form a synthetic futures position; combining the two maturities with opposite signs creates the spread. The key condition is matching call and put strikes within each expiry, rather than assuming the strike must be at the spot price.

The discussion resolves an apparent conflict with put-call parity: the call-minus-put value is zero at the at-the-money-forward strike, and that forward level can differ across expiries. A calendar spread can therefore have a nonzero value even when each leg is set relative to its own forward. The answer notes that European exercise is the strict setting for the parity relationship. It offers a conceptual explanation, not market data or a worked valuation, and practical implementation still depends on contract details and the relevant option pricing assumptions.

Key ideas

• A call and put with the same strike and expiry can form a synthetic futures position.
• A calendar spread can be constructed by combining synthetic futures at different expiries.
• The relevant zero-value strike under put-call parity is at the money forward, which can vary by expiry.
• European exercise is the strict condition cited for the parity relationship.

Tags

# Synthetic equity index futures calendar spread using options


# 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?

## Answer by AlRacoon (score 2)

https://quant.stackexchange.com/a/43480

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.

## Answer by Alex C (score 0)

https://quant.stackexchange.com/a/43078

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.