## Risk Reversal

The long Risk Reversal option strategy with strikes K₁ and K₂ with K₁ < K₂ and expiry T is a package consisting of the following vanilla options:
1) Short put with strike = K₁ and expiry T
2) Long call with strike = K₂ and expiry T

At the time of inception, it generally holds that K₁ < Kᵃᵗᵐ < K₂, where Kᵃᵗᵐ is the the ATM (at-the-money) strike, which means that both options are OTM (out-of-the-money).
Furthermore, in a regular Risk Reversal, at inception the distances Kᵃᵗᵐ - K₁ and K₂ - Kᵃᵗᵐ are equal in moneyness terms (see below).
The short Risk Reversal consists of a long put combined with a short call.

The Risk Reversal 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 long Risk Reversal on the fx rate USD/JPY with K₁ = 90, K₂ = 110
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 why a long Risk Reversal 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 T
The short Risk Reversal has the inverse payoff profile and is also known as a bearish Risk Reversal

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 Risk Reversal 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.