I have often seen a statement that we can model only a short rate process $r(t)$ and then use it to derive a term structure $R(t,T)$ for every $t$. Could someone please elaborate? Say, I’ve simulated $r(t)$ up to time $t$, what would I use to derive $R(t,T)$?


This is indeed a standard result. You can convince yourself by noticing

  1. The bank account grows from 1 at $t=\tau$ to $E\left[\exp(\int_\tau^T r(u)du)|\mathscr{F}_\tau\right]$ at time $T$
  2. The price of a security paying $X$ at time $T$ discounted to $t=\tau$ is then $E\left[X \exp(-\int_\tau^T r(u)du)\right|\mathscr{F}_\tau]$
  3. Hence the price of a credit risk-free zero coupon bond, which pays 1 at $T$ is $$ B(\tau,T)=E\left[\exp(-\int_0^T r(u)du|\mathscr{F}_\tau\right],$$ which will define the yield curve at $t=\tau$.

So the only challenge remaining is to go from $r|\mathscr{F}_\tau$ to $\int_\tau^T r(u)du|\mathscr{F}_\tau$. This can be either done by approximations of the integral (e.g. by Riemann sums) or in Gaussian models by avoiding discretisation (and its errors) using that $r|\mathscr{F}_\tau$ and $\int_\tau^T r(u)du|\mathscr{F}_\tau$ are joint Gaussian and simulating joint normal distributions. Every textbook on short rate models will probably explain this. Look for example in Chapter 3 of Glassermann's "Monte Carlo models in financial engineering".

  • $\begingroup$ Thanks for your answer. I am still a bit confused. I get all your points and agree with formula 3, but what I have in the end is simulated continuous short rate $r(t)$ up to time $t$ and now I need to derive $R(t,T)$. How would I practically proceed? I am not interested in the bond price as the expectation, but rather I want to get a distribution of bond prices, computed on each path. So, in my case $B(t,T)$ is a random variable, I have simulated $r(t)$ and now I need the term structure $R(t,T)$ to compute $B(t,T)$ for each scenario. $\endgroup$ – tosik Apr 13 at 20:00
  • $\begingroup$ Bond prices - like all prices in stochastic models - are expectations! But you are right future bond prices are random variables. I edited to include the conditioning information. For more concrete formulas you need to specify a concrete model. For example in the Hull-White model bond prices are $B(\tau,T)=\exp(-A(\tau,T) r(\tau) + C(\tau,T))$ where $A(\tau,T)$ and $C(\tau,T)$ are defined in terms of initial market conditions. $\endgroup$ – g g Apr 13 at 23:16

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