Could you please help me to understand meaning of instantaneous forward rate? I mean economic interpretation at basic level. What is it used for? How can i derive it from zero rate/price?



Given a forward rate, for example:

$ F(t, T, T+\delta)$

The instantaneous forward rate $f(t,T)$ fixed in $t$ is the limit when $\delta \rightarrow 0$ of your forward rate.

If the relation between forward rate and zero coupon bond is:

$F(t,T,T+\delta) = \frac{p(t,T) - p(t,T+\delta)}{\delta p(t,T+\delta)}$

We have,

\begin{equation} f(t,T) = \lim_{\delta\to0} \frac{p(t,T) - p(t,T+\delta)}{\delta p(t,T+\delta)} \end{equation}

\begin{equation} f(t,T) = -\frac{\partial \ln p(t,T)}{\partial T} \end{equation}

I hope this will help you,

  • $\begingroup$ Approximately, and using English, If an investor lends money from time t to time T, then the instantaneous forward rate is the annualized rate earned for extending the loan by one more "day" (or short time interval) beyond T. Am I right? $\endgroup$ – noob2 Dec 6 '18 at 15:53
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    $\begingroup$ An instantaneous forward rate (F) is the rate of return for an infinitesimal amount of time ($\delta$) measured as at some date (t) for a particular start-value date (T). In practice the shortest time one might be interested in is one day, in which case the rate might be determined by analysing subsequent discount factors. Truly infinitesimal forward rates might only be relevant for considering smoothness constraints of interest rate curves or for interest rate curve models for option pricing. For trading purposes or relative value assessment one-day forward rates is good enough imo. $\endgroup$ – Attack68 Dec 6 '18 at 17:48
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    $\begingroup$ Indeed Attack68 you're totally right. Instantaneous forward rates or, for example, short rates are not "tradable" instruments which can be directly observed. noob2 your example belongs to an IBOR rate (for example LIBOR) starting from now (=t) till maturity $T+\delta$: $L(t, T+\delta)$ which is different from the mentioned instantaneous forward rate $f(t,T)$ $\endgroup$ – Yassine Q. Dec 6 '18 at 18:00

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