"In contrast to models that evolve the instantaneous short rate (Hull-White, Black-Karasinski models) or instantaneous forward rates (Heath-Jarrow-Morton model), which are not directly observable in the market, the objects modeled using LMM are market-observable quantities (LIBOR forward rates)."


Recall that the simple forward rate as at time t for lending/borrowing between time T and $T+\tau$ can be written in terms of the discount factors as follows:

$F(t,T, T+\tau)= \frac{1}{\tau}\left( \frac{B(t,T)}{B(t,T+\tau)}-1\right)$

Think of $\tau$ as 6 months or 3 months, and simple forward rate as LIBOR. You can also write it as follows:

$F(t,T, T+\tau)= \frac{1}{\tau}\left( \frac{B(t,T)-B(t,T+\tau)}{B(t,T+\tau)} \right)$

If you let $\tau$ tends to zero, then you get instantaneous forward rate.

$F(t,T, T)=-\frac{\partial \ln B(t,T) }{\partial T}$

It’s theoretical in the sense that you won’t borrow or lend for such a short period of time, but it is quite a useful construct.

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