Skip to content
All library documents

Building FX Risk Reversals from Delta Volatility Quotes

Article Quant Q&A · Author: DeepInTheQF
Read the original

Summary

The document explains what an FX 25-delta risk-reversal volatility quote represents and why it cannot be used directly as the volatility input for a Black–Scholes price of a risk-reversal option. The quote is the difference between the implied volatilities of the 25-delta call and put, each associated with its own strike and delta convention. It is therefore a description of the volatility smile, not a standalone option volatility.

To determine the call and put volatilities, the answer gives equations for their respective delta constraints and for the quoted volatility difference. Those conditions leave the system underdetermined: a further quote, such as the 25-delta smile butterfly relative to at-the-money volatility, is needed to identify the two volatilities. The discussion clarifies quote construction, rather than providing a direct option-pricing recipe; the exact conventions and inputs used in a market calibration still matter.

Key ideas

  • A 25-delta FX risk reversal quotes the difference between call and put implied volatilities.
  • The risk-reversal quote alone does not specify a Black–Scholes volatility for pricing the combined position.
  • The call and put volatilities are tied to separate 25-delta conditions.
  • A butterfly quote together with at-the-money volatility can provide the additional information needed to determine the two smile volatilities.

Tags

ForexOptionsVolatilityDerivatives pricing
Full text
# FX convention and volatility calibration


# 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!

## Answer by Gordon (score 2)

https://quant.stackexchange.com/a/54488

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.

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.