Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 23 / 5 / a

The instantaneous forward rate is defined by $f(t,T)=-{\partial }_T\log P(t,T)$, with $P(t,t)=1$. Integrating in maturity gives the zero-coupon bond price and its diagonal short rate:The solution of the continuous-time bank account equation $d{B}_t=r_t{B}_{t}dt$, $B_0=1$, is $B_t=\exp ({\int }_0^t{r}_{s}ds)$; the integration variable is time. Its discounted bond value isAll these are adapted processes when the Heath-Jarrow-Morton model coefficients are adapted and the integrals exist.

For market completeness, stochastic integration over maturity givesThe Itô formula for the exponential then yieldsConsequently under a risk-neutral measure the no-arbitrage drift restriction isThe last equality needs sufficient maturity regularity and has a positive sign despite the negative definition of $\Sigma$.