## Iron Butterfly

The

**option strategy with strikes**

*Iron Butterfly***,**

*K₁***and**

*K***with**

*K₂***and expiry**

*K₁ < K < K₂***is a package consisting of the following vanilla options, all with expiry**

*T***:**

*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

**, where**

*K = Kᵃᵗᵐ***is the the ATM (at-the-money) strike, which means that the long options are OTM (out-of-the-money).**

*Kᵃᵗᵐ*Furthermore, in a regular

**, the distances**

*Iron Butterfly***and**

*K - K₁***are equal in moneyness terms (see below).**

*K₂ - K*The reverse

**is produced by interchanging long with short.**

*Iron Butterfly*The

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

*Iron Butterfly*As an example, consider a

**on the fx rate USD/JPY with**

*Iron Butterfly***and assume for simplicity that**

*K₁ = 90, K = 100, K₂ = 110*

*Kᵃᵗᵐ = 100*The payoff at expiry

**is a function of the fx rate observed at**

*T***and given by the diagram below:**

*T*This picture makes it clear that this product pays off when the fx rate at

**is close to**

*T***, which is equivalent to the realized vol being low.**

*Kᵃᵗᵐ*In other words, being long on an

**means being short on fx vol and vice versa.**

*Iron Butterfly*In the fx market, strikes are typically quoted in moneyness

**terms, with**

*m***and**

*m = 100|Δ|***being the delta defined as in Delta Def**

*Δ*In moneyness terms, a strike such as the

**above, would be represented by the corresponding moneyness value**

*K₁***.**

*m*If - for example - the delta for the put with strike

**were the number**

*K₁ = 90***, then**

*Δ = -0.25***and the put option would be referred as**

*m = 25*

*delta-25 put*Then the construction of a regular

**would involve a**

*Iron Butterfly***, which would be a call option with a strike**

*delta-25 call***chosen so that its delta equaled 0.25.**

*K'₂*Note that generally

**, but the distances**

*K'₂ ≠ 110***and**

*Kᵃᵗᵐ - K₁ = 100 - 90***are considered equal when measured in moneyness terms, since both moneyness values equal 25.**

*K'₂ - Kᵃᵗᵐ = K'₂ - 100*