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usually the term implied volatility refers to Black-Scholes implied volatility (also Log-Normal volatility): it is defined as a quantity which when plugged in the Black-Scholes formula returns the right price. In an article by Roper (2010) certain conditions are given which guarantee that the Black-Scholes implied Volatility Surface is consistent with No Arbitrage.

When working with interest rates it has become convenient to work with Bachelier implied volatility (or normal implied volatility) instead of Black-Scholes implied volatility. The Bachelier implied volatility is the parameter which when plugged into to the Bachelier option pricing formula returns the correct price. I was wondering whether there is an article which characterizes No Arbitrage conditions for Normal Implied Volatility. I think that such conditions would differ from the conditions given in Roper, since he uses the Black-Scholes formula to transform the Black-Scholes implied volatility surface to the option price surface and checks for No-Arbitrage there.

If not, I think a simple approach to obtain such conditions would be to transform the No-Arbitrage conditions for (Call)-option prices into conditions for implied volatility.

References

Roper, M. (2010). Arbitrage free implied volatility surfaces. preprint.

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    $\begingroup$ Good question, I have been wondering the same thing. $\endgroup$ – ilovevolatility Mar 22 at 8:12
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    $\begingroup$ The no arbitrage conditions are applied to the option prices, regardless of the underlying volatility forms. For example, the convexity with respect to the strike should be satisfied. $\endgroup$ – Gordon Apr 27 at 20:11

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