Deriving Zero-Coupon Bond Yields from a Short-Rate Model

# Deriving interest rate term structure in a short rate model


# 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 $R(t,T)$?

## Answer by g g (score 3, accepted)

https://quant.stackexchange.com/a/45076

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

- 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 $T$

- The price of a security paying $X$ at time $T$ discounted to $t=\tau$ is then $E\left[X \exp(-\int_\tau^T r(u)du)\right|\mathscr{F}_\tau]$

- 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".

## Answer by rvignolo (score 1)

https://quant.stackexchange.com/a/57838

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 $T$.