Why Converted Lognormal Volatility Fails for Negative-Rate LMMs
Summary
The discussion examines whether normal rate volatilities converted into lognormal volatilities can support a standard lognormal Libor Market Model when forward rates are negative. A simple at-the-money conversion divides normal volatility by the absolute forward rate, producing a positive volatility value even when the rate is below zero.
The answer explains that this conversion corresponds to dynamics whose diffusion scales with the absolute value of the forward rate. Standard lognormal dynamics instead scale with the rate itself, so the converted volatility is inconsistent with the usual model for negative rates. The discussion recommends a displaced lognormal model or a normal LMM as alternatives. It offers a conceptual model-consistency argument, but no derivation, calibration, or empirical comparison; model choice depends on available implementation and desired rate dynamics.
Key ideas
- Converting normal volatility using the absolute forward rate implies diffusion proportional to the absolute rate.
- That diffusion does not match standard lognormal LMM dynamics for negative rates.
- A displaced lognormal LMM is presented as one alternative for modeling negative rates.
- A normal LMM avoids the lognormal rate restriction by using normal dynamics.
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# Using converted lognormal volatilities for negative rates in a lognormal Libor Market Model (LMM)
# Using converted lognormal volatilities for negative rates in a lognormal Libor Market Model (LMM)
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.
## Answer by BrownianBread (score 3)
https://quant.stackexchange.com/a/66622
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.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.