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Recovering FX Smile Volatilities from Risk Reversals and Butterflies

Article Quant Q&A · Author: Baptiste
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Summary

The document addresses how to infer call and put implied volatilities for an EUR/USD smile when market quotes include at-the-money volatility, risk reversals, and butterfly volatilities at selected deltas, but no separately labeled strangle quotes. The key point is a convention issue: the average of call and put volatility relative to at-the-money volatility is identified as the smile butterfly in the convention used by the question. With that interpretation, the butterfly supplies the average-volatility component, while the risk reversal expresses the call-minus-put difference; together with the at-the-money level, these quantities determine the two wing volatilities.

The response attributes the apparent missing input to confusion over terminology and notation. It also notes that other treatments of FX smile construction exist, so the precise definitions used by a data source matter. The discussion is conceptual and gives no worked numerical example or full surface-fitting procedure. It explains how to interpret the quoted inputs, not how to interpolate across deltas or maturities or ensure an arbitrage-consistent surface.

Key ideas

  • A risk reversal represents the difference between call and put implied volatilities under the stated convention.
  • The average call and put volatility relative to at-the-money volatility is treated as the smile butterfly.
  • At-the-money volatility, butterfly, and risk reversal together provide the inputs to recover wing volatilities.
  • Quote definitions can vary, so the market convention should be checked before applying the relationships.
  • The discussion does not cover interpolation or construction of a complete volatility surface.

Tags

ForexOptionsVolatilityDerivatives pricing
Full text
# Find call and put volatilities using ATM, Risk reversal and Butterflies volatilities


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












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 by Gordon (score 5, accepted)

https://quant.stackexchange.com/a/24401

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.