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?

  • $\begingroup$ Do you want the definition for instantaneous forward rates? $\endgroup$ – Gordon Feb 24 '16 at 15:29
  • $\begingroup$ 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? $\endgroup$ – 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 $u$.

The instantaneous forward rate is the forward Libor rate over an infinitesimal accrual period. That is, \begin{align*} f(t, T) &=\lim_{\Delta\rightarrow 0}L(t, T, T+\Delta)\\ &=\lim_{\Delta\rightarrow 0}\frac{P(t, T) - P(t, T+\Delta)}{\Delta}\frac{1}{P(t, T+\Delta)}\\ &=-\frac{\partial P(t, T)}{\partial T}\frac{1}{P(t, T)}\\ &=-\frac{\partial }{\partial T}\ln P(t, T). \end{align*}

  • $\begingroup$ Thank you Gordon! So this instantaneous forward rate is basically not directly observable in the markets at all? $\endgroup$ – Quant2015 Feb 24 '16 at 16:35
  • $\begingroup$ @Quant2015: No, it is not. $\endgroup$ – Gordon Feb 24 '16 at 16:36

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