A forward contract on a non$-$dividend$-$paying stock $($St$)0<=$t$<=$T is a derivative dependent on the stock. Recall that the value of a long position in a forward contract $($initiated at time $0$ with arbitrage$-$free forward price F$0($T$)=$ S$0$e$^$rT $)$ at a general time t in $[0,$T$]$ is given by

f long $=($Ft$($T$)-$ F$0($T$))*$ e$^-$r$($T$-$t$)=$ St $-$ F$0($T$)*$ e$^-$r$($T$-$t$)($T in $[0,$T$]$

where Ft$($T$)=$Ste$^$r$($T$-$t$)$ is the arbitrage$-$free forward price at time t$.$

Assume the Black$-$Scholes$-$Merton model for the stock price process $($St$)0<=$t$<=$T$.$

$($a$)$ Show that the value process $($price process$)($ft long$)0<=$t$<=$T satisfies the Black$-$Scholes$-$Merton partial differential equation. Specify a suitable boundary condition.

$($b$)$ What is the delta hedging strategy for the short position in the forward contract? Discuss