Is anybody using normal Libor Market Model (LMM) (as opposed to shifted lognormal LMM)? It could be one of the approaches to dealing with negative rates. If you do, have you encountered any theoretical or practical difficulties with it? Thank you.
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Yes I used one in the early 2000s. At the time, US interest rates were quite high (5 or 6pct) and the market skew was such that -100bp receivers were more expensive than +100 payers. The lognormal model is very inappropriate for this skew regime , but the normal model is much closer, having symmetric skew. As a byproduct of this we noticed that the model put nonzero values on zero floors, but it was not the main purpose.
When rates rallied close to zero following the 2008 crisis , we had to migrate away from the normal model because the market skew became heavily payers-over-receivers, due to a heavy right hand tail. That is the main theoretical problem. When rates are low , the normal model miscalibrates the skew. When rates are low it also places too high a value on zero floors compared to the market , in my experience.
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$\begingroup$ Thank you. On an even more basic level, isn't the marginal distribution of forward bond price $P(t,T_{k-1},T_k)=(1+ F(t,T_{k-1},T_k) \tau_k))^{-1}$ weird? Assuming $T_k$-forward measure, for simplicity, it is reciprocal normal (!), negative valued etc. $\endgroup$– ir7Commented Jan 29, 2018 at 14:13
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$\begingroup$ If I understand you correctly, it isn't too weird, because the probability density of rates below say -3% is negligible at reasonable market volatilities. $\endgroup$– dm63Commented Jan 30, 2018 at 1:59