Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 48 / 4 / c

Assume the forward field is sufficiently differentiable in maturity to differentiate its deterministic and stochastic integrals. Moving along the diagonal adds the maturity derivative to the fixed-maturity dynamics:The Heath-Jarrow-Morton model no-arbitrage drift is ${\alpha }_Q(t,T)=\sigma (t,T){\int }_t^T\sigma (t,u)du$, so ${\alpha }_Q(t,t)=0$. Equivalently, from the integrated forward equation,That drift can depend on the entire previous forward curve or driving path, rather than on the current scalar rate.

The short rate need not be Markov. For a concrete One Brownian factor does not imply a Markov short rate example, take the deterministic volatility $\sigma (t,T)=T-t$. Its admissible risk-neutral drift is $(T-t{)}^3/2$, andSubtracting the deterministic part leaves $I_t={\int }_0^t{W}_{s}ds$. The past of $I$ reveals its derivative $W_t$, and $\mathbb{E}[I_{t+h}-I_t\mid {\mathcal{F}}_t^I]=h{W}_t$. Yet $W_t$ is not determined by $I_t$: the Gaussian covariance calculation givesConsequently the conditional law of the future depends on more than $I_t$, proving failure of the Markov property for the scalar short rate. The enlarged pair $(I_t,W_t)$ is Markov. Special maturity-volatility structures, such as those producing a Vasicek-type short-rate diffusion, can instead admit a scalar Markov realization.