### FX Volatility Smile conventions - Risk Reversal and Strangle

In the FX market, volatility smile is quoted using ATM volatility, and 25-Delta Risk Reversal and 25-Delta Strangle. The ATM volatility, as its names implies, gives the volatility corresponding to the ATM strike, which, as we know from the discussion in the previous sections, depends on the delta conventions. The Risk Reversal is the difference between the volatility of a call option with a delta of 0.25 and the volatility of a put option with delta of -0.25, again the delta conventions vary. The strangle refers to the market strangle, which is slightly more complicated (not conceptually, but in terms of computations involved to translate the quote into strike and volatility), but here we assume that we are given the smile strangle, which is the difference between the average of the 25-delta call and 25-delta put volatilities and the ATM volatility.

$$ ATM Volatility= \sigma_{ATM} $$ $$ 25Delta Risk Reversal=RR_{25Delta}=\sigma_{25Delta-Call}-\sigma_{25Delta-Put} $$ $$ 25Delta Smile Strangle=SS_{25Delta}=\frac{\sigma_{25Delta-Call}+\sigma_{25Delta-Put}}{2}-\sigma_{ATM} $$

We can solve the last two equations for \( \sigma_{25Delta-Call}\) and \( \sigma_{25Delta-Put}\). Firstly,

$$ RR_{25Delta}+ 2 SS_{25Delta}=\sigma_{25Delta-Call}-\sigma_{25Delta-Put}+2 \frac{\sigma_{25Delta-Call}+\sigma_{25Delta-Put}}{2}-2\sigma_{ATM} $$ $$ RR_{25Delta}+ 2 SS_{25Delta}=2 \sigma_{25Delta-Call}-2\sigma_{ATM} $$ $$\sigma_{25Delta-Call}= \sigma_{ATM} + 0.5 RR_{25Delta}+ SS_{25Delta}$$Similarly,

$$ RR_{25Delta}- 2 SS_{25Delta}=\sigma_{25Delta-Call}-\sigma_{25Delta-Put}-2 \frac{\sigma_{25Delta-Call}+\sigma_{25Delta-Put}}{2}+2\sigma_{ATM} $$ $$ RR_{25Delta}- 2 SS_{25Delta}=-2 \sigma_{25Delta-Put}+2\sigma_{ATM} $$ $$\sigma_{25Delta-Put}= \sigma_{ATM} - 0.5 RR_{25Delta}+ SS_{25Delta}$$Thus, given the ATM volatility, and 25-Delta Risk Reversal and 25-Delta Strangle quotes, we can calculate the 25-Delta call and put volatilities. We can then calculate the strike levels for the three volatilities, giving us three pairs of strikes and volatilities. We can then fit a smile function, such as quadratic, to these quotes, which we can then use to get volatility for any strike.