Linear bond pricing in a bounded short-rate diffusion
= Linear bond pricing in a bounded short-rate diffusion
{title2=$P(t,T)=1-(1-e^{-(T-t)})r_t$}
In the model $dr_t=r_t(r_t-1)(dt+dW_t)$ under the pricing measure, with $0\leq r_t\leq1$, the discounted affine expression $D_t[A(T-t)+B(T-t)r_t]$ has zero drift when $A'=0$ and $B'=-A-B$. Terminal conditions $A(0)=1$, $B(0)=0$ yield the displayed unit bond price. The expression is bounded, so the <bounded local martingale criterion> justifies pricing by <conditional expectation>.