Incorrect characterization of spot rate?

Question

Is the t in the red boxed $R(t,T)$ supposed to be the same as the S in the green boxed $R(S,T)$?

Answer 1

Yes. The map $R(\cdot;S,T):\mathbb{R}^{2}\to\mathbb{R}$ completely describes the forward rate/spot rate term interest rate structure for each $t\geq0$. (You can think of it as the market interest rate surface for the rate $R$ at time $t$).

The notation $R(t;S,T)$ is meant to remind you that $R$ is a stochastic process for $t>0$, the periods of time where full information is unavailable (it is assumed $t=0$ is today, the point at which we have full information). More explicitly, $R$ depends on $\omega(t)$, where $\omega(t)\in(\Omega,\mathbb{P},\mathcal{F}_{t})$, a filtered probability space on which the stochastic model is supported.

Moreover, the value of $R$ for a fixed $t\geq0$ further depends on the period of time one is measuring the interest rates, and this is captured by the arguments $(S,T)$.

When the first argument $t$ is omitted, the time is implied (usually $t=0$ since this is when it makes most sense to discuss the term structure of interest rates, as we have full information at this point - any other times would be random model based).

Note that the use of $(t;S,T)$ as labels for the variables does not commit anyone to using these precise letters/notations when using the function - in fact, it would make the presentation considerably more complicated if forced to do so, as it is more convenient to use the letters which are most natural for whatever the context is. However, the author is committed to maintaining the order of arguments, else there would be ambiguity on the meaning of the variables.

It is helpful to think of some special cases, or maps, when other variables are fixed.

• If $S=t$ and both are fixed, then $T\mapsto R(t;S,T)$ is the spot rate at time $t$ (modeled or observed today depending on whether $t=0$ or $t>0$) for maturities of length $T-t$.
• If $S>t$ and both are fixed, then $T\mapsto R(t;S,T)$ is the forward rate calculated at time $t$, beginning at time $S$, and lasting to time $T$ for a tenor of length $T-S$.
• If $S$ and $T$ are fixed, then $t\mapsto R(t;S+t,T+t)$ is the random interest rate model of the spot ($S=0$) or forward ($S>0$) rate $R$ beginning at time $S+t$ and ending at time $T+t$ for a tenor of length $T-S$.
• If $t$, $S$, and $T$ are all fixed, then the auxiliary map $\alpha\mapsto R(t;S+\alpha,T+\alpha)$ corresponds to $T-S$ tenured forward rate at the future time $\alpha$, as calculated at the present time $t$ (note that "present" may not actually be today if $t>0$, but $S$ is always a "future" time with respect to $t$, that is $S\geq t$).

You can experiment with this more if you wish. Note the obvious domain restrictions (unless you want to consider past history of the process, but for most circumstances setting $t=0$ to be today is what most people have in mind)

$$0\leq t\leq S<T.$$ The domanin can be somewhat extended by using the convention $R(t;S,T)\equiv0$ wherever $S=T.$

Note Some authors may not explicit modify the arguments $S$ and $T$ according to $t\geq0$. For example, they may write $R(0;0,T)$ and $R(1;0,T)$, but mean that the former is the spot rate with maturity $T$ at time $t=0$ and the latter the spot rate with maturity $T$ but at time $t=1$. In our notation, this would be a domain violation and we would write $R(0;0,T)$ and $R(1;1,T+1)$. This is just a stylistic preference. I believe the former style is more prevalent, but the latter style we've adopted here is nicer mathematically. But then again, most authors adapt their notation to emphasize what is being studied (forward rates, spot rates, randomly modeled future term structure, etc.) when discussing interest rates anyway, so such issues rarely arise anyway.

Answer 2

I am note $100\%$ sure that I understand the question. But yes. More formally one could write $R(t,S,T)$ for the rate from $S$ to $T$ observed at $t$ and $R(t,t,T)$ for the spot.