Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 38 / 3 / c / Solution

To avoid confusing the continuous-time bank account with the coefficient of $r_t$, denote the latter by $b(\tau )$ and the other coefficient by $a(\tau )$, where $\tau =T-t$. Apply the Itô formula to $D_t[a(\tau )+b(\tau )r_t]$. Since the expression is affine in $r$, its second rate derivative is zero. Its drift isThe quadratic terms cancel. The remaining expression isIt vanishes when $a^{\prime }=0$ and $b^{\prime }=-a-b$. For a unit bond payoff choose terminal conditions $a(0)=1$, $b(0)=0$, givingThe resulting local martingale is $D_t[1-(1-e^{-\tau })r_t]$. The allowed bound $0\le r_t\le 1$ puts it between zero and one, so the bounded local martingale criterion makes it a true martingale. At maturity it equals $D_T$. Comparing with part (a) therefore givesIn particular $e^{-(T-t)}\le P(t,T)\le 1$, and $P(T,T)=1$. This is linear bond pricing in a bounded short-rate diffusion; choosing zero coefficients would produce a local martingale but would not price the required terminal payoff.