$(29$ points$)$ The payoff function of a $-$option at maturity $T$ is given below,

$C(S_T,T)=\{\begin{array}{cc}0,& S_{T}K+\end{array}$

where $K$ is the strike price and $S_t$ is the current price of a risky stock. The stock does not pay dividends. $K$ and $$ are constant parameters.

$(1).$ In the Black$-$Scholes framework, price the $-$option by partial differential equation approach. You need to show clear derivation steps.

$(2).$ Please evaluate the price of the $-$option when $0,\underset{0}{lim}C(S_t,t)=?$ You need to show clear derivation steps.