Question $2.$ Assume that the Black$-$Scholes model holds for a non$-$dividend

paying stock. You are given :

Current stock price is $50$

$$ The stock's volatility $$ is $0\mathrm{.}3$

The annual continuously compounding risk$-$free interest rate is $0\mathrm{.}03$

Current time is $t=0.$ A derivative with a one$-$year time$-$to$-$maturity has the

following payoff function :

If $S_{1}50$ then the payoff is $0$ ;

if $S_1-50S_1>5550$ then the payoff $is{S}_1-50$

$-if{S}_1>55$ then the payoff $is5$