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FX Implied Volatility Under Currency Inversion

Article Quant Q&A · Author: bf52020
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Summary

The note asks whether an implied volatility surface for INR/USD can be reused for the inverse USD/INR rate. It distinguishes the standard deviations of an exchange rate and its reciprocal, which are generally not equal, from the relationship between option-implied volatilities under a consistent pricing setup.

The answer states that USD/INR implied volatility at strike 1/K and maturity T matches INR/USD implied volatility at strike K and the same maturity when each is priced under its corresponding currency forward measure. The explanation relies on the payoff equivalence between a call on INR/USD settled in USD and a put on USD/INR settled in INR. This is a strike-mapped option-pricing relationship, not a claim that reciprocal spot returns have identical standard deviations. The note gives no derivation or numerical example, and the measure and settlement conventions matter when applying the result.

Key ideas

  • Reciprocal exchange rates do not generally have the same standard deviation.
  • The volatility comparison maps strike K to reciprocal strike 1/K at the same maturity.
  • The stated equality uses different currency forward measures for the two rate orientations.
  • A call on one exchange-rate orientation can correspond to a put on its inverse with currency-specific payoff settlement.

Tags

ForexOptionsVolatilityDerivatives pricing
Full text
# Implied volatility of inverse quote


# Implied volatility of inverse quote












Suppose I have a quote of INR/USD and the implied vol surface is also given. Is it theoritically correct to use to same implied vol for analysis of the inverse quote, i.e. USD/INR.

Correct me if I am wrong, but standard deviation of x and 1/x are not same.

## Answer by Antoine Conze (score 3, accepted)

https://quant.stackexchange.com/a/39741

The implied vol at strike $1/K$ and maturity $T$ for USDINR under the INR $T$-forward measure is the same than the implied vol at strike $K$ and maturity $T$ for INRUSD under the USD $T$-forward measure.

This is a consequence of a Call on INRUSD with payoff in USD being equivalent to a Put on USDINR with payoff in INR.

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.