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In Bruno Remillard's text, "Statistical Methods for Financial Engineering," he states the following on p 148 after giving the general form of a bond price $P(t,T)$ under Vasicek's model:

Note that $P(t,T)$ depends only on $r(t)$ and the parameters of the model; however, the spot rate $r(t)$ is not observable.

Firstly, isn't Vasicek a type of short rate model, not the spot rate? If so, as I understand it the short rate is just the rate for short term borrowing. In that case, why can't we use, say, 1-week LIBOR rates as the short rate? Are those not observed? FRED actually offers a time series of such rates, so it seems that they are observable.

I've seen this mentioned elsewhere, too; that the short rate is not observable. What exactly does this mean?

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    $\begingroup$ In a continuous time model a 1-week rate is a long term rate! The spot rate is literally an instantaneous rate; where can I go to borrow for the next infinitesimal interval dt? ;-) $\endgroup$
    – nbbo2
    Nov 10, 2015 at 20:00
  • $\begingroup$ @noob2 Yeah, what's confused me is a number of sites suggest calibrating to this LIBOR rate for Vasicek. So, is this second post at this site wrong? quantnet.com/threads/vasicek-calibration.16527? $\endgroup$
    – bcf
    Nov 10, 2015 at 20:15
  • $\begingroup$ I agree with noob2. I think that the terms spot rate and short rate are used interchangeably in short rate models, both meaning instantenous rate. $\endgroup$
    – emot
    Jul 25, 2021 at 7:51

2 Answers 2

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I think that you are confusing instantenous sport rate $r(t)$ with continously compounded spot interest rate $R(t,T)$.

The instantenous sport rate $r(t)$ is just a rate over infinitesimal interval $dt$ and this rate is not observable, because the shortest rate traded is overnight rate i.e. 1 day rate.

The continuously-compounded spot interest rate $R(t,T)$ prevailing at time t for the maturity T is the constant rate at which an investment of $P(t, T)$ units of currency at time t accrues continuously to yield a unit amount of currency at maturity $T$ i.e. $P(T,T)=1$, in formulas:

$$R(t,T)=-\frac{lnP(t,T)}{T-t}$$

then of course

$$P(t,T)=e^{-R(t,T)(T-t)}$$

where $P(t,T)$ is zero-coupon bond or discount factor.

Notice that our yield curve is not continuous but a discrete construct. Typically it is built off of instruments with maturity 1D, 2D, 1W, 1M, 3M, 6M, 1Y, 2Y, ... etc. Therefore we say that $R(t,T)$ is observable. Even if we want to trade maturity that is not on our yield curve, let's say 3.5 months, then we can do it in practice, if we pay slight premium. But we can't trade $r(t)$, we can't trade loans/depos over 1 milisecond or a few seconds interval - it is not quoted hence not observable.

Notice that if you know parameters to Vasicek model (or any short rate model) and the instantenous rate $r(t)$ then you can calculate bond prices $P(t,T)$ for all $T$ and then calculate spot rates $R(t,T)$ for all maturities $T$. Therefore the pricing equation for bonds allows you to calculate $R(t,T)$ based on $r(t)$.

Notice that $$P(t,T)=E^Q[e^{-\int_t^T r(s) ds}]$$

then

$$R(t,T)=-\frac{ln(E^Q[e^{-\int_t^T r(s) ds}])}{T-t}$$

Ok, so how do we estimate $r(t)$?

Then we can ask, how we can use the pricing equation for bond when we don't know $r(t)$? In practice we just assume that the $r(t)$ is equal to the shortest rate $R(t,T)$ on the yield curve, 1 day or 1 week rate. i.e. $$r(t)=R(t,t+1/365)$$

Why? Because we can interpolate rate $r(t)$ from the yield curve. Notice that we know $P(t,t)$ - it is just equal to 1, also we know $P(t,t+1/365)$, based on these two discount factors we can interpolate $r(t)$ and it will be equal to $R(t,t+1/365)$ when we use log-linear interpolation. Please check my answer here link

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Spot rates cannot be directly observed --- Wikipedia

The way I understood this (in a more basic context) is simply that if

  • the 1-year forward rates during year 1,2,3 are $i_1,i_2,i_3$,
  • a bond coupon payment is to be made at the end of year 3, and
  • the spot rate for that payment is $i$ then $$(1+i_1)(1+i_2)(1+i_3)=(1+i)^3$$ so that we can calculate $i$, but not "observe" it prior to calculation.
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