 Solution

The unit zero-coupon bond pays one at maturity. Its risk-neutral valuation therefore isNo particular rate coefficients were specified, so this expectation is the general answer; evaluating it further requires a chosen one-factor short-rate model.

For the alternative derivation let $D_t=\exp (-{\int }_0^t{r}_{s}ds)$ be the bank discount factor. For an integrable discounted payoff, pricing giveswhich is a martingale by the tower property of conditional expectation. For a payoff determined by the Markov rate state, this conditional expectation is represented by the indicated pricing function. A path-dependent claim requires an appropriately enlarged state, as in part (c). Now $d{D}_t=-r_t{D}_{t}dt$ has finite variation. Applying Itô formula and the product rule under the risk-neutral measure, whose rate drift is $b-\sigma \theta$, yieldsUnder the smoothness and integrability conditions used above, the martingale has zero finite-variation drift. The bracket therefore vanishes, giving againFor the bond the terminal condition is $P(T,T)=1$. This argument also shows why stochastic discounting remains inside the conditional expectation.