## Iron Butterfly

The Iron Butterfly option strategy with strikes K₁, K and K₂ with K₁ < K < K₂ and expiry T is a package consisting of the following vanilla options, all with expiry T:
1) Long put with strike = K₁
2) Short put with strike = K
3) Short call with strike = K
4) Long call with strike = K₂

At the time of inception, it generally holds that K = Kᵃᵗᵐ, where Kᵃᵗᵐ is the the ATM (at-the-money) strike, which means that the long options are OTM (out-of-the-money).
Furthermore, in a regular Iron Butterfly, the distances K - K₁ and K₂ - K are equal in moneyness terms (see below).
The reverse Iron Butterfly is produced by interchanging long with short.

The Iron Butterfly is particular important in the forex options market because of its use in quoting the fx volatility smile as described in
Fx Vol Spec

As an example, consider a Iron Butterfly on the fx rate USD/JPY with K₁ = 90, K = 100, K₂ = 110 and assume for simplicity that Kᵃᵗᵐ = 100
The payoff at expiry T is a function of the fx rate observed at T and given by the diagram below:

This picture makes it clear that this product pays off when the fx rate at T is close to Kᵃᵗᵐ, which is equivalent to the realized vol being low.
In other words, being long on an Iron Butterfly means being short on fx vol and vice versa.

In the fx market, strikes are typically quoted in moneyness m terms, with m = 100|Δ| and Δ being the delta defined as in
Delta Def
In moneyness terms, a strike such as the K₁ above, would be represented by the corresponding moneyness value m.
If - for example - the delta for the put with strike K₁ = 90 were the number Δ = -0.25, then m = 25 and the put option would be referred as delta-25 put
Then the construction of a regular Iron Butterfly would involve a delta-25 call, which would be a call option with a strike K'₂ chosen so that its delta equaled 0.25.
Note that generally K'₂ ≠ 110, but the distances Kᵃᵗᵐ - K₁ = 100 - 90 and K'₂ - Kᵃᵗᵐ = K'₂ - 100 are considered equal when measured in moneyness terms, since both moneyness values equal 25.