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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.

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    $\begingroup$ Correct me if I am wrong, but standard deviation of x and 1/x are not same: If x is lognormal, then SD(x) = SD(1/x). $\endgroup$
    – byouness
    May 13, 2018 at 20:48

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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.

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