Finding Strikes from Nonmonotonic Premium-Adjusted FX Deltas

Summary

The document concerns calibration of an FX volatility surface from quoted deltas, focusing on converting premium-adjusted forward deltas into strikes. It gives the premium-adjusted delta expression, which includes the strike-to-forward ratio and a normal cumulative distribution term. Unlike the non-premium-adjusted formulas the author has handled, this delta is not monotonic in strike, so a quoted delta may correspond to more than one strike. The question asks why a source recommends selecting the solution on the right-hand side of the delta maximum.

The text raises the root-selection issue but provides no answer or derivation of the recommended convention. It therefore identifies a calibration ambiguity rather than explaining how to resolve it. Any practical strike search would need to account for the specified delta convention and determine which branch market quotes intend; the document does not provide additional market conventions or a worked numerical example.

Key ideas

• Premium-adjusted forward delta depends on both strike relative to forward and a normal distribution term.
• Premium adjustment can make delta nonmonotonic in strike.
• A quoted delta may therefore admit multiple candidate strikes.
• The document asks about selecting the root beyond the delta maximum but does not supply the rationale.

Tags

# Conversion of a premium-adjusted delta to a strike


# Conversion of a premium-adjusted delta to a strike












I am trying to compute the calibration of an FX market volatility surface, and especially I want to retrieve the strikes from the deltas quoted.

I don't have any trouble reverse-engineering the formulas for non premium-adjusted deltas. However, for many currency pairs, the convention is to use premium-adjusted deltas, which are not monotonic in strike. For example, for a premium-adjusted forward delta (Source: Clark - FX Option Pricing - Chap3 p47): $$ \Delta_{F;\%} = \omega\frac{K}{F_{0,T}}N(\omega d_{2}) $$

Following this source, p14:

> One common solution to this problem is to search for strikes corresponding to deltas which are on the right hand side of the delta maximum.

What is the argument or the intuition behind this statement?

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