Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 5 / c

For $V(t,r)$ sufficiently smooth, Itô formula under the physical measure givesHence $m=V_t+b{V}_r+{\sigma }^2{V}_{rr}/2$ and $s=\sigma V_r$. Substitute these into the common risk-price condition $m-rV=\theta s$ from part (b), obtaining the Cox-Ross short-rate pricing equationFor a terminal payoff $H(r_T)$ the final condition is $V(T,r)=H(r)$. Writing prices as functions of $(t,r)$ requires the relevant Markov state to be sufficient for the claim; a path-dependent payoff generally requires additional state variables. The negative sign in $b-\sigma \theta$ is the adjustment from physical to risk-neutral rate drift.