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FX Delta Conventions and Put-Call Volatility Smile Symmetry

Article Quant Q&A · Author: Mauro Meneses Ramirez
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Summary

The document explains when an FX volatility smile quote for a put can be matched to a call quote with the complementary delta. Under the forward delta convention, put and call deltas at the same strike sum to one, so a 95-delta put and a 5-delta call correspond to the same strike and volatility. This relationship does not automatically hold under other delta conventions.

The answer notes that FX markets use multiple delta conventions and that the convention can vary by market and tenor. It gives EURUSD as an example, with spot delta described as typical through one year and forward delta used for longer tenors. The rule is therefore conditional: traders must identify the quote convention and translate delta to strike accordingly. The document provides no market data or detailed strike-construction procedure.

Key ideas

  • Complementary put and call deltas identify the same strike under forward delta.
  • The put-call delta relationship depends on the convention used to define delta.
  • FX delta conventions can vary across markets and maturities.
  • A volatility quote should be mapped to its strike using the applicable convention.

Tags

ForexVolatilityDerivatives pricing
Full text
# On a FX volatility smile, Is a-delta put volatility equal to (1-a)-delta call volatility?


# On a FX volatility smile, Is a-delta put volatility equal to (1-a)-delta call volatility?












On a FX volatility smile in terms of delta, If I want 95-delta put vol, Is this volatility equal to a 5-delta call vol, and viceversa?

Thanks.

## Answer by Antoine Conze (score 2, accepted)

https://quant.stackexchange.com/a/39033

It depends on the delta convention. The parity $$ \Delta_{\text{Put}(\text{strike } K)} = 1 - \Delta_{\text{Call}(\text{strike } K)} $$ works only when the delta convention is forward delta, so the 95-delta put vol is equal to the 5-delta call vol only in this case.

See Reiswich & Wystup FX Volatility Smile Construction for a list of all delta conventions and how to build the corresponding strikes. For most markets (e.g. EURUSD, ...) the delta convention is spot delta until 1Y, and forward delta over 1Y.

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.