# instantaneous forward rates vs forward LIBOR rates

HJM describes the behavior of instantaneous forward rates while BGM describes the behavior of forward Libor rates. From concept perspective, I understand forward libor rate are like forward Libor rate with different tenor, e.g 3M. They are directly tradable in the market with quotes? But what is the instantenous forward rates?

• Do you want the definition for instantaneous forward rates? – Gordon Feb 24 '16 at 15:29
• I found the definition of instantaneous forward rates over the internet. But I am still confused about how is instantaneous forward rates different from forward libor rates? – Quant2015 Feb 24 '16 at 16:03

The forward Libor rate at time $t$ is the forward rate over a certain accrual period $[T, T+\Delta]$, where $\Delta$, in years, can be 3 months or 6 months, and is defined by \begin{align*} L(t, T, T+\Delta) = \frac{1}{\Delta}\left(\frac{P(t, T)}{P(t, T+\Delta)}-1 \right), \end{align*} where $P(t, u)$ is the price at time $t$ of a zero coupon bond with unit face value and maturity $u$.