Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 211 / 2 / b

For fixed maturity $T$, define $b_t^T={\int }_t^T\sigma (t,u)du$ and $I_t^T={\int }_t^{T}f(t,u)du$. The stochastic Fubini theorem and the moving lower endpoint giveThe factor $1/2$ is the integral over one of the two triangles in the square $[t,T{]}^2$. Applying the Itô formula to $P(t,T)=e^{-I_t^T}$, its quadratic-variation correction cancels that factor:Set $D_t=e^{-{\int }_0^t{r}_{s}ds}$, the reciprocal of the continuous-time bank account. The Itô product rule then givesIf $|\sigma |\le M$, then $|b_t^T|\le MT$ on this finite horizon. The Novikov condition holds, so this stochastic exponential is a true martingale, not merely a local martingale. The discounted price is thereforeThe authoritative PDF discounts to $t$ in this part. The TeX transcription's upper endpoint $T$ would include future short rates and is incorrect here.