BF vol quote
The BF vol quote σᵇᶠ (also denoted as BF) is used in fx markets in conjunction with the RR vol quote as a market input towards the construction of the volatility smile for various expiries.
Its name stems from the fact it is defined through the implied volatilities of the options involved in an Iron Butterfly
In spite of the its vol-like symbol, σᵇᶠ is not some sort of vol quote for the price of Iron Butterfly, since it is possible for two Iron Butterflys to have the same σᵇᶠ but different prices.
For a given expiry T and a given moneyness m (for example m = 25), σᵇᶠ is defined as:
σᵇᶠ = (σᶜ + σᵖ)/2 - σᵃᵗᵐ
where
σᶜ is the implied volatility of a European call that has delta Δ = m/100
σᵖ is the implied volatility of a European put that has delta Δ = -m/100
σᵃᵗᵐ is the implied volatility of an ATM (at-the-money) European call (or put) with the ATM condition defined as in ATM Def
For example, when T = 1 month and m = 25, one may refer to the respective σᵇᶠ as BF (25-Delta 1M)
σᵇᶠ represents the extra volatility which is added to the average of the two non-ATM volatilities compared to the ATM volatility.
It measures the curvature of the volatility smile and also the curtosis of the terminal fx rate probability distribution.
The picture below shows the effect of σᵇᶠ on the volatility smile:
The picture below shows the effect of σᵇᶠ on the probability density function of the terminal fx rate:
Most currency markets exhibit a positive σᵇᶠ