## ATM-RR-BF

Subtype of Vol InputThis type is used to describe the volatility surface of the fx rate in the fx options market.

As is common for all option types, the volatility depends on both Maturity and strike, with the fx options twist that the strike is specified in ATM (At-The-Money) and delta (also known as "moneyness") terms.

The ATM and Delta strikes may be defined in several ways, listed respectively in ATM Def and Delta Def

For a given set of expiries and an agreed delta

**, the following three numbers are provided:**

*δ***= the volatility for a call (or put) ATM option**

*σᵃᵗᵐ***= the RR vol quote with respect to the agreed delta level**

*σʳʳ***= the BF vol quote with respect to the agreed delta level**

*σᵇᶠ*Technically these numbers are supplied by means of a Set object that contains ATM vols, RR vol quotes and BF vol quotes for various expiries, arranged in the following columns:

#Expiry

#ATM

#RR

#BF

For the given expiries, these quotes are used to calculate the volatility

**of the call option of which the delta equals**

*σᶜ***and the volatility**

*δ***of the put option of which the delta equals**

*σᵖ***, using the following equations:**

*-δ*σᶜ = σᵃᵗᵐ + σᵇᶠ + ½ σʳʳ

σᵖ = σᵃᵗᵐ + σᵇᶠ - ½ σʳʳ

Knowing these vols, the corresponding strikes

**can be calculated by solving the Black Scholes FX formula with respect to strike and the following three pairs can be constructed:**

*Kᵃᵗᵐ , Kᶜ , Kᵖ*(Kᵃᵗᵐ , σᵃᵗᵐ) , (Kᶜ , σᶜ) , (Kᵖ , σᵖ)

These pairs represent the volatility information for each respective expiry in the regular strike terms that are used in other option markets, such as equity options.

The final construction of the volatility smile across all strikes requires an interpolation procedure, which follows the specifications supplied as an object of type Fx Vol Spec