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 $R(t,T)$?
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2 Answers
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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".
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$\begingroup$ 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. $\endgroup$– tosikApr 13, 2019 at 20:00
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$\begingroup$ 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 gApr 13, 2019 at 23:16
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$\begingroup$ I see a typo in (3). Integral domain is from $\tau$ (not $0$) to $T$. $\endgroup$– ir7Sep 6, 2020 at 19:00
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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 $T$.
