$4\mathrm{.}2.$ Suppose that default$-$free bond prices grow according to a constant rate $r,$

$B(t,T)=e^{-r(T-t)}$

and consider a single default time $$ with exponential distribution such

that the default spread is $>0.$ A default digital put with maturity

$T>0$ pays one unit of cash at the default time $$ if a default occurs no

later than $T($otherwise it pays nothing$).$ Derive a formula for the time $0$

price of a default digital put, and compute this price when $r=5\%,T=0\mathrm{.}5$

$($half a year$)$ and $=2\%.(40$ marks$)$