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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 use 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. $\endgroup$
    – user34971
    Mar 22, 2019 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, 2019 at 20:11

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There is the following paper by Roger Lee and Dan Wang on Displaced Lognormal Diffusion implied volatility. They give some no arbitrage bounds. It could be more useful for interest rates than normal volatilities as you may actually want to bound your rates from below.

Lee & Wang, Displaced Lognormal Volatility Skews, 2009

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  • $\begingroup$ Thank you for the reference. I am not sure how wise it is to bound the interest rates from below: the reason log-normal volatilities have been used for interest rates is that people assumed that interest rates cannot be negative. They turned out to be wrong. So if we simply assume a new lower bound than who guarantees that it will not be reached? $\endgroup$
    – Cettt
    Feb 21, 2020 at 11:18
  • $\begingroup$ With a displaced diffusion you can have a bound that is less than zero, e.g. -10%. You are flexible in choosing the level of the lower bound. In the extreme case you could even let the lower bound approach negative infinity, in which case you approach the normal model. $\endgroup$
    – user34971
    Feb 21, 2020 at 11:22
  • $\begingroup$ I know, it's just the fact that you have to assume lower bound is what bugs me. $\endgroup$
    – Cettt
    Feb 21, 2020 at 11:25

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