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There exist formulas to convert between normal and lognormal interest rate volatilities. In the most simple form the approximation for ATM volatilities would be $\sigma_{LogNorm}=\frac{\sigma_{Norm}}{\text{|forward rate|}}$. Such conversion makes it possible to calculate $\sigma_{LogNorm}$ also for negative rates, for which the normal volatility exists but the lognormal Black volatility doesn't.

The question is: Could these converted lognormal volatilities be used in a lognormal Libor Market Model (LMM) for modelling negative interest rates? Or is there something fundamentally wrong with such approach?

I believe the correct/standard approach would be to use a shifted lognormal LMM, but I currently only have access to a lognormal BGM model.

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I've not seen the abs function on the forward rate here before, the approximation comes from matching variances of a Black (Lognormal) and Bachelier (Normal) SDE. The Black SDE doesn't have this restriction on absolute forward rate, so this requires some further clarification.

It seems that the LMM dynamics for which your $\sigma_{LogNorm}$ is valid would be $$dF_t = |F_t|\sigma_{LogNorm}dW_t$$ rather than the usual description. So it would not be a consistent volatility to use in your model when rates are negative.

To see how to implement the displaced lognormal solution please see this post. The other solution is to avoid using lognormal dynamics altogether and use a Normal LMM.

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