# Deriving interest rate term structure in a short rate model

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 $$B(\tau,T)=E\left[\exp(-\int_\tau^T r(u)du|\mathscr{F}_\tau\right],$$ 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".

• 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. – tosik Apr 13 '19 at 20:00
• 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. – g g Apr 13 '19 at 23:16
• I see a typo in (3). Integral domain is from $\tau$ (not $0$) to $T$. – ir7 Sep 6 '20 at 19:00
• Thanks - corrected! – g g Sep 6 '20 at 20:22

A short rate model provides an analytical solution for the zero coupon bond $$P(t, T)$$, given by the following expectation:

$$P(t, T) = E_t^Q \left[ \exp \left( - \int_t^T r(s) ds \right) \right].$$

For example, depending on notation, when $$r(t)$$ follows a short rate model, the previous equation yields to:

$$P(t, T) = \exp(A(t, T) - B(t, T) \cdot r(t))$$

where $$A(t, T)$$ and $$B(t, T)$$ are the solution of a system of ODEs (called Riccati system). For many models, $$A(t, T)$$ and $$B(t, T)$$ have analytical solutions that depend on the parameters of the stochastic differential equation of the short rate $$dr(t)$$. For other models, both surfaces must be obtained by numerical methods.

Once you have $$A(t, T)$$ and $$B(t, T)$$ (from the ODE system) and $$r(t)$$ (from the Monte Carlo simulation), you can compute $$P(t, T)$$ for any pair $$(t, T)$$. Then, I am assuming that you are calling $$R(t, T)$$ to the continuously-compounded spot interest rate, such that:

$$R(t, T) = - \frac{\log P(t, T)}{\tau(t, T)}$$

where $$\tau(t, T)$$ denotes the year fraction between $$t$$ and $$T$$.