$3\mathrm{.}3.$ Forward rate evolution by parallel shifts from an initial curve $f(0,T)=$

$h(T)$ is defined as

$f(t,T)=h(T-t)+x(t),$ for any $0tT,$

where $h(t)$ is a smooth deterministic function and where $x(t)$ is an It$$

process such that $dx(t)=(t)dt+(t)d{W}^{**}(t)$ with $W^{**}(t)$ a Brownian

motion under the martingale probability. Show that in the HJM frame$-$

work, the relationship between the drift and diffusion coefficients for the

forward rate implies that for each $t0$

$(t)=a,$

$(t)=b,$

$h(t)=-\frac{a^2}{2}{t}^2+bt+c,$

where $a,b,c$ are some constants. Hence or otherwise argue that forward

rate evolution by parallel shifts is incompatible with the HJM model.

$(25$ marks$)$