Interpreting the Delta Difference in FX Volatility Smile Construction

Summary

The document presents a question about replicating an FX volatility-smile construction method, focusing on how to interpret a delta-related quantity defined using call and put deltas at the strike associated with a specified put delta. The questioner is working with risk reversal and strangle quotes at a 25-delta level and is unsure which volatility should be used when determining that strike.

The central issue is the apparent circularity: the strike depends on an implied volatility, but the volatility at that delta seems to require the smile that the construction is intended to produce. The document provides no accepted answer or resolution, so it does not establish a calculation procedure or demonstrate results. It is useful as a statement of a practical implementation ambiguity in FX option quoting and smile construction. Readers would need the referenced paper’s definitions and a worked derivation to determine how the delta convention, volatility input, and strike calculation fit together.

Key ideas

• The question concerns a delta difference evaluated at a strike tied to a specified put delta.
• The author is constructing an FX volatility smile from risk reversal and strangle quotes.
• The uncertainty is which volatility input determines the delta-based strike during smile construction.
• The document raises the issue but supplies no answer or validated calculation method.

Tags

# Volatility smile construction for fx options confusion


# Volatility smile construction for fx options confusion












I am working through the paper "FX volatility smile construction" by Wystup and Reiswich (2010). I am trying to replicate the results they obtained with the model they define in the paper.

Unfortunately I am not understanding clearly how exactly the term $a$ on page 20 is supposed to be calculated. They define it as:

> Furthermore, we define a variable $a$ which is the difference of a call delta, corresponding to $a\ −\tilde\Delta$ put delta, and the $−\tilde\Delta$ put delta for any delta type and is given by $$ a := \Delta(K_{\tilde{\Delta}P},\sigma,1) - \Delta(K_{\tilde{\Delta}P},\sigma,-1).$$

From what I gather $K_{\tilde{\Delta}P}$ is supposed to be the strike at Delta Put 25 (in my case since I am working with RR and Strange taken at 25 delta), but to retrieve this information I would need the volatility that I have not yet implied for delta 25. Moreover that sigma volatility I suppose is not the one retrieved from the vol smile at 25 delta, otherwise that all structure could be avoided by expressing it as -0.25.

I might be completely blind to the easy and correct interpretation, but I need an external nudge. Thanks in any case for having read the question and sorry for the poor notation in the formula.

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