What does instantaneous forward mean?

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?

Thanks

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,

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

$$$$f(t,T) = -\frac{\partial \ln p(t,T)}{\partial T}$$$$

• 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? – noob2 Dec 6 '18 at 15:53
• 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. – Attack68 Dec 6 '18 at 17:48
• 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 $f(t,T)$ – Yassine Q. Dec 6 '18 at 18:00

1. Observable instruments, spot rates, and forward rates

First remember that something observable means that you can observe/find the rate in the market by looking at traded rate instruments or fixings.

1.1. Observed spot rates

For simplicity, assume Zero Coupon Bonds (ZCBs) are traded with time left to maturity of 10Y, 15Y and 20Y. Hence, by observing these instruments, we directly deduce the spot rates $$R(0, 10Y), R(0,15Y), R(0,20Y)$$ I have assumed that today is time zero.

1.2. Observed forward rates

By no arbitrage argument, we can directly back out the observed forward rates between these times: $$R(0; 10Y, 15Y), R(0; 10Y, 20Y), R(0; 15Y, 20Y).$$

This is about as far we get by observing traded instruments in the market. The above rates are all model independent since we have observed them.

2. Non observable spot rates and forward rates

How do we calculate spot rates $$R(0, T)$$ and forward rates $$R(0; T, S)$$ for times $$T$$and/or $$S$$ other than the ones we have observed in the market? We can build a model that interpolates the known spot rates into a spot rate curve. From this curve we can get any arbitrary spot rate, and hence also any arbitrary forward rate implied from the spot rates. Now these rates are model dependent in the sense that they are as good as the model/interpolation we make.

2.1. Instantaneous forward rates

Using our constructed curve model, we can get any forward rate $$R(0; T, T+\delta)$$, i.e. the forward rate today, between time $$T$$ and time $$T+\delta$$. If we let $$\delta$$ go to zero, we get the instantaneous forward rate $$f(0; T) := R(0; T, T)$$, which is the forward rate between $$T$$ and an infinitesimal time forward. You can integrate up this instantaneous forward rate between two time points to get the forward rate back:

$$R(0; T, T+\delta) = \frac{1}{\delta}\int_T^{T+\delta}f(0; s)ds$$

So in a sense, the instantaneous forward rate describes the slope/derivative of the spot curve at one specific time point. Or you can think of the forward rate as an average of the instantaneous forward rate when using continuously compounded rates.

2.2. Lending/borrowing at the instantaneous forward rate

Assuming that we can borrow and lend at these rates, the rate $$R(0,10Y,15Y)$$ is the rate you get between time year 10 and 15 if you agree on it today. However, the rate is only for the full period and rates are not the same for smaller periods in between due to the spot curve not being flat. You can equivalently today make an agreement to lend/deposit money between, let's say year 10 and 14, at rate $$R(0,10Y,15Y)$$, and another agreement where you lend between year 14 and 15 at rate $$R(0,14Y,15Y)$$. The two rates are different, but you will be indifferent between making the 10Y-15Y agreement or making the two 10Y-14Y, 14Y-15Y agreements. By the same token you can make a lot of small lending agreement in sequence, starting and ending between 10Y and 15Y, each agreement with its own forward rate. If you make infinite many small agreements, each from a time and an infinitesimal time forward, the rate for each agreement will be the instantaneous forward rate. However, if you calculate the mean by integrating up these instantaneous rates, you will get the rate for the full period 10Y-15Y.

3.1. Simple spot rate $$L(t,T)$$

The price of a ZCB is $$p(t,T) = \frac{1}{1+L(t,T)\cdot (T-t)},$$ where L simply compounded spot rate.

3.2 Simply compounded forward rate $$F(t;T,T+\delta)$$

The relation between $$F(t, T, T+\delta)$$ and the two zero coupon bonds is $$p(t,T) \cdot [1 + F(t, T, T+\delta)\cdot \delta]^{-1} = p(t,T+\delta) \\ \Leftrightarrow \\ F(t, T, T+\delta) = \frac{p(t,T)-p(t,T+\delta)}{\delta \cdot p(t,T+\delta)} \\ f(t, T) := F(t,T,T) = \lim_{\delta \to 0}F(t, T, T+\delta) = -\frac{\partial \ln p(t,T)}{\partial T}$$

3.3. Continuously compounded spot rate $$R(t,T)$$

The price of a ZCB is $$p(t,T) = e^{-R(t,T)\cdot (T-t)},$$ where R(t,T) is the continuously compounded rate

3.4. Continuously compounded forward rate $$R(t;T, T+\delta)$$

The relation between $$R(t, T, T+\delta)$$ and the two zero coupon bonds is $$p(t,T) \cdot e^{-R(t, T, T+\delta)\cdot \delta} = p(t,T+\delta) \\ \Leftrightarrow \\ R(t, T, T+\delta) = -\frac{\ln p(t,T+\delta)-\ln p(t,T)}{\delta} \\ f(t, T) := R(t,T,T) = \lim_{\delta \to 0}R(t, T, T+\delta) = -\frac{\partial \ln p(t,T)}{\partial T}$$ Here $$R$$ looks identical to the definition of a derivative, so if we let $$\delta \to 0$$, we get the same instantaneous forward rate.

• This suggests that $R(t,T)(T-t) = \int_t^Tf(t,s)ds$ - is that correct? And how does the so-called short-rate $r(s)$ fit into all of this? Thanks – Confounded Dec 5 '19 at 14:05
• If $t$ denotes the time today, i.e all market rates are known at time $t$, then your suggestion is correct. Regarding the short rate $r(t)$, it is the instantaneous forward rate taken at time $t$ (today). That is, $r(t) \equiv f(t, t)$. – Pontus Hultkrantz Dec 28 '19 at 17:00