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## Thursday, May 28, 2009

### FX vol fitting, according to a practitioner online

-- Based on http://quantdev.net/market/bootstrapping/4-buildinganfxvolsurface --
The market provide fairly liquid quotes on volatility out to about 2 years for only 3 types of instruments --

Strangle
Risk Reversal

The quotes are provided in terms of volatility for a specific delta. For example: a quote will be given for the volatility for a 25 delta Strangle, or a 10 delta Risk Reversal for a specific maturity. In order to construct our volatility surface we need quotes for an ATM Straddle, a 25 delta Strangle and a 25 delta Risk Reversal, and a 10 delta Strangle and a 10 delta Risk Reversal with a range of maturities.

We can imply the volatility for the specific deltas at a particular maturity by using our quotes. The 50 delta implied vol is simply the volatility of the ATM Straddle. An ATM Straddle is a call and a put with the same strike and maturity, and chosen so the delta of the straddle is zero.

The 25 delta call implied volatility is the ATM Straddle volatility + (25 delta Risk Reversal volatility / 2) + 25 delta Strangle.

On a smile curve, the x-axis would include {10delta, 25 delta, 50 delta, 25 delta, 10 delta} in symmetry. The volatility calculated for the 25Δ call (OTM) is the same as that for a 75Δ put (ITM), so for call values you go along the curve from End A to End B, and for put values you would go along the curve from End B to End A.

Based on http://quantdev.net/market/bootstrapping/4-buildinganfxvolsurface says but I doubt "Usually you would then turn the vol curve for each maturity so it is denominated in strike rather than delta, as it then becomes much easier to use in practice given that you know the strike for the option you want to price, but not the delta."