## Risk Reversal

The long

**option strategy with strikes**

*Risk Reversal***and**

*K₁***with**

*K₂***and expiry**

*K₁ < K₂***is a package consisting of the following vanilla options:**

*T*1) Short put with strike =

**and expiry**

*K₁*

*T*2) Long call with strike =

**and expiry**

*K₂*

*T*At the time of inception, it generally holds that

**, where**

*K₁ < Kᵃᵗᵐ < K₂***is the the ATM (at-the-money) strike, which means that both options are OTM (out-of-the-money).**

*Kᵃᵗᵐ*Furthermore, in a regular

**, at inception the distances**

*Risk Reversal***and**

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

*K₂ - Kᵃᵗᵐ*The short

**consists of a long put combined with a short call.**

*Risk Reversal*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**

*Risk Reversal*As an example, consider a long

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

*Risk Reversal*

*K₁ = 90, K₂ = 110*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 why a long

**is also known as a bullish**

*Risk Reversal***, since an investor in such a product would obviously expect an upside rather than a downside market outcome for the fx rate at**

*Risk Reversal*

*T*The short

**has the inverse payoff profile and is also known as a bearish**

*Risk Reversal*

*Risk Reversal*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**

*Risk Reversal***, 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*