## BF vol quote

The

*BF vol quote***(also denoted as**

*σᵇᶠ***) 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.**

*BF*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,

**is not some sort of vol quote for the price of Iron Butterfly, since it is possible for two**

*σᵇᶠ***to have the same**

*Iron Butterflys***but different prices.**

*σᵇᶠ*For a given expiry

**and a given moneyness**

*T***(for example m = 25),**

*m***is defined as:**

*σᵇᶠ*σᵇᶠ = (σᶜ + σᵖ)/2 - σᵃᵗᵐ

where

**is the implied volatility of a European call that has delta**

*σᶜ*

*Δ = m/100***is the implied volatility of a European put that has delta**

*σᵖ*

*Δ = -m/100***is the implied volatility of an ATM (at-the-money) European call (or put) with the ATM condition defined as in ATM Def**

*σᵃᵗᵐ*For example, when T = 1 month and m = 25, one may refer to the respective

**as**

*σᵇᶠ*

*BF (25-Delta 1M)***represents the extra volatility which is added to the average of the two non-ATM volatilities compared to the ATM volatility.**

*σᵇᶠ*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

**on the volatility smile:**

*σᵇᶠ*The picture below shows the effect of

**on the probability density function of the terminal fx rate:**

*σᵇᶠ*Most currency markets exhibit a positive

*σᵇᶠ*