# FX convention and volatility calibration

In general, call/put options are quoted with respect to their Black-Scholes volatility.

In the FX market we define the risk reversal volatility as $$\sigma_{25-RR} = \sigma_{25-Call} - \sigma_{25-Put}$$ Question : is this the value to input in a Black-Scholes formula to get the price of a risk reversal option ? More precisely is any one of these equations holds ? $$PriceOfRR = CallBSPrice(\sigma_{25-RR})$$ or $$PriceOfRR = PutBSPrice(\sigma_{25-RR})$$

I am kind of confused because this does not seem to be correct since a flat BS volatility cannot price a risk reversal all the time since we would need atleast two points from the volatility smile in order to have a correct price, and even in the case where the previous equations have a solution there is no need that the solution have to be the difference between the implied volatility of the call and the put, it could be anything and only be found numerically.

If not, does anyone know how this volatility $$\sigma_{25-RR}$$ is computed in FX market by the market maker ? as it is and important input to establish the market volatility surface.

Thanks!

A good reference book for FX conventions can be found from the book Foreign Exchange Option Pricing by Iain Clark. The 25% delta risk-reversal quote $$\sigma_{25-RR}$$ satisfies the system of equations \begin{align*} \begin{cases} \Delta_{call}(k_{25-call}, \sigma_{25-call}) &\!\!\!= 0.25\\ \Delta_{put}(k_{25-put}, \sigma_{25-put}) &\!\!\!= -0.25\\ \sigma_{25-call} - \sigma_{25-put} &\!\!\!= \sigma_{25-RR}. \end{cases} \end{align*} This system of equations is not solvable by itself, as there are 4 unknowns but 3 equations. You need, for example, the smile butterfly volatility quote defined by \begin{align*} \sigma_{25-SF} = \frac{\sigma_{25-call} + \sigma_{25-put}}{2}-\sigma_{ATM}, \end{align*} where $$\sigma_{ATM}$$ is the at-the-money volatility quote.
• The value of $\sigma_{ATM}$ is a FX market observable value, In contrast with the Equity market where it is implied from the black scholes formula knowing the strike of the option. My other part of the question was how is such value $\sigma_{ATM}$ is fixed by the market maker ? Is there any financial consideration made for the value displayed ? – DeepInTheQF May 28 '20 at 14:52
• I do not know the notation for smile risk reversal and a market risk reversal. For smile butterfly, the quote satisfies the equation in my answer, but the market butterfly convention is much more complicated - it will be related to a market strangle, which is similar to the risk-reversal. The $\sigma_{ATM}$ quote is also complicated; it will be based on the spot, forward, or delta-neutral. See the book I recommended. – Gordon May 28 '20 at 14:58