Find call and put volatilities using ATM, Risk reversal and Butterflies volatilities

Question

I have to plot the implied volatility surface for EUR/USD.

So, my goal is to produce something like that, from put delta 10 to call delta 10:

Searching for informations, I found that I could find call et put volatilities using

Strangle(∆) = 0,5[Call Vol(∆) + Put Vol(∆)] - ATM Vol

Risk Reversal(∆) = Call Vol(∆) - Put Vol(∆)

Hence,

Call Vol(∆) = Strangle(∆) + 0,5RR(∆) + ATM Vol

Put Vol(∆) = Call Vol(∆) - RR(∆)

However, in my exercise, I have only ATM, 25∆ risk reversal, 10∆ risk reversal, 25∆ butterfly and 10∆ butterfly volatility quotations. So absolutely no strangle data.

With the data I have, is there any way to find the volatilities for both call et put?

Answer 1

The strangle vol defined in your formula \begin{align*} Strangle(∆) = 0.5[Call Vol(∆) + Put Vol(∆)] - ATM Vol \end{align*} is the smile butterfly volatility. Then you have the volatility quote. Your confusion is caused by the misuse of notations.

Note that, other treatments are also available. See for example, FX Volatility Smile Construction by UWe Wystup (this link may not work directly, but it can be searched.) Another good source is the book Foreign Exchange Option Pricing by Iain J. Clark.