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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: enter image description here

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?

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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.

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  • $\begingroup$ Thanks for your answer Gordon, so the Butterfly vol formula is : Butterfly(∆)=0.5[CallVol(∆)+PutVol(∆)]−ATMVol ? $\endgroup$ – Baptiste Feb 19 '16 at 15:31
  • $\begingroup$ @Baptiste. Yes, for smile butterfly. The other is the market butterfly, which is more complicated, and you can check the two references I listed if you are interested in. $\endgroup$ – Gordon Feb 19 '16 at 15:35
  • $\begingroup$ I found the paper by Uwe Wystup, I will have a look at it soon. I am not sure to understand the difference between smile butterfly and market butterfly: if I understand correctly, smile butterfly is the average of the volatilities at the strikes of the OTM options minus the ATM vol, while the market butterfly strategy (ie long call k1, long call K3 and 2 * short K2) ? Also, you said in your answer that my "confusion is caused by the misuse of notations". Why did you mean by that? $\endgroup$ – Baptiste Feb 19 '16 at 16:01
  • $\begingroup$ @Baptiste: The strangle in your equation is actually the butterfly, that is a misuse of notation. For the market butterfly, it is hard to explain in short, but the book and paper should have some explanations. $\endgroup$ – Gordon Feb 19 '16 at 18:08

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