Consider an option whose payoff at maturity T is $\backslash$Phi $($ST $)=$ S

$4$

T

$,$ where St is the underlying

asset price which follows a Black$-$Scholes model, with drift $,$ volatility $\backslash$sigma $.$ The risk free rate

is r$.$

$($i$)$ Using the risk neutral valuation compute the option price at time t$.$

$($ii$)$ Verify that the option price obtained in $($i$)$ satisfies the Black$-$Scholes pricing PDE