This 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 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 and
For a given set of expiries and an agreed delta δ, the following three numbers are provided:
σᵃᵗᵐ = the volatility for a call (or put) ATM option
σʳʳ = the with respect to the agreed delta level
σᵇᶠ = the with respect to the agreed delta level
Technically these numbers are supplied by means of a object that contains ATM vols, and for various expiries, arranged in the following columns:
For the given expiries, these quotes are used to calculate the volatility σᶜ of the call option of which the delta equals δ and the volatility σᵖ of the put option of which the delta equals -δ, using the following equations:
σᶜ = σᵃᵗᵐ + σᵇᶠ + ½ σʳʳ
σᵖ = σᵃᵗᵐ + σᵇᶠ - ½ σʳʳ
Knowing these vols, the corresponding strikes Kᵃᵗᵐ , Kᶜ , Kᵖ can be calculated by solving the with respect to strike and the following three pairs can be constructed:
(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