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

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 $\Sigma$.

Under the specified risk-neutral measure, the Itô product rule and $d(B_t^{-1})=-r_t{B}_t^{-1}dt$ giveThus discounted traded zero-coupon bonds have zero drift. Assume the usual integrability and admissibility conditions that make discounted replicating values true martingales, not merely local martingales. If $X$ is an attainable financial payoff with ${\mathbb{E}}_Q|B_T^{-1}X|<\infty$, its self-financing portfolio value $V_t$ has terminal value $V_T=X$ and satisfiesMultiplying by $B_t$ yieldsThis proves the requested conditional pricing formula. In a Brownian filtration, the Martingale representation theorem realizes the discounted conditional expectation as a stochastic integral. If an available bond has nonzero Itô diffusion exposure, its holdings can match that integral, with the remaining value in the bank account, giving replication under the usual square-integrability conditions.

These attainability or market completeness hypotheses matter for a completely arbitrary contingent claim: existence of a risk-neutral measure alone need not imply a unique price for every unspanned financial payoff, and a zero-drift local martingale alone need not satisfy the displayed terminal expectation identity. In the general path-dependent Heath-Jarrow-Morton model, write the value as $V_t$; notation $V(t,r_t)$ is justified only when the current short rate is a sufficient state for the financial payoff and dynamics.

Suppose the forward field is differentiable in maturity with the stochastic integrability needed for differentiation. As time increases, both arguments of $r_t=f(t,t)$ increase. The diagonal short-rate dynamics in the Heath-Jarrow-Morton model are thereforeUnder the HJM drift restriction from part (a), ${\alpha }^Q(t,t)=0$, givingMore explicitly the slope isIt depends on the forward curve and potentially the whole history, not necessarily on $r_t$ alone. Without the stated maturity regularity, the original forward equations do not by themselves guarantee this classical Itô process formula for the diagonal.

The short rate need not be Markov, even with one Brownian factor and deterministic volatility. An explicit example of One Brownian factor does not imply a Markov short rate takes $f_0(T)=0$ and $\sigma (t,T)=T-t$. Its no-arbitrage forward drift is ${\alpha }^Q(t,T)=(T-t{)}^3/2$, soThe continuous rate history determines $W_t$ from the left derivative of $I_t$. Hence for $h>0$,But $W_t$ is not determined by $I_t$: joint normality, $\mathrm{Var}{I}_t=t^3/3$, and $\mathrm{Cov}(W_t,I_t)=t^2/2$ giveThus the conditional mean of the future given the whole rate history cannot be a function of the current rate alone, contradicting the Markov property. Enlarging the state to $(I_t,W_t)$ gives a two-dimensional Markov process; special HJM specifications can instead close on a single short rate.