# Spot Interest Rate at time $t$

I know that the general model for the dynamics of the spot interest rate is

$$dr(t)=\mu(r,t)dt+\sigma(r,t)dB(t)$$

My question is, if $P(t,T)$ is the bond value at time $t$, how would I derive $dP$?

Assume the SDE $dr(t)=\mu(r(t),t)dt + \sigma(r(t),t) dB(t)$ is under the risk neutral measure and that is has a solution.

By construction $P(t,T) = E[e^{-\int_t^T r(u) du}]$ under the risk neutral measure.

Because of the model Markov property you know $P(t,T) = P(r(t), t, T)$ where $P(r, t, T)$ is a function of current short rate, time and maturity, therefore $$dP(t,T) = \frac{\partial P}{\partial r} dr(t) + \frac{\partial P}{\partial t} dt + \frac{1}{2}\frac{\partial^2 P}{\partial r^2} \sigma(r(t), t)^2 dt$$ Also under the risk neutral measure the drift for $dP(t,T)/P(t,T)$ is $r(t) dt$, hence the above equation simplifies to $$dP(t,T) = P(t,T) r(t) dt + \frac{\partial P}{\partial r} \sigma(r(t), t) dB(t)$$ If you want to go further you need to be able to compute $E[e^{-\int_t^T r(u) du}]$ explicitly, which is possible for some models such as Hull & White.

• A minor concern is the Markov property assumption. – Gordon Dec 19 '17 at 21:06