# What's the difference between instantaneous forward rates and observable forward rates?

Source:

"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.