Please help me solve $($c$)$ to $($e$)$A stock's current price is $$52.$ The annualized volatility is $18\%.$ The continuously compounded

risk$-$free rate is $1\%$ p$.$a$..$ The stock does not pay dividends. Consider a European call option

that has a strike price of $$50$ and expires in $6$ months.

$($a$)$ What is the call option premium based on Black$-$Scholes formula?

$($b$)$ What is the premium of a put option that is otherwise identical?

$($c$)$ How many shares of the stock would you purchase today to dynamically replicate the

payoff of the call at expiration?

$($d$)$ If the stock price changes to $$50$ immediately, what is the approximate change of the

call price based on delta?

$($e$)$ What is the approximate percentage change in the option premium for a $1\%$ change in

the stock, based on delta? Input the answer in percentage without the percent sign and

keep $2$ decimal places $($e$.$g$.8\mathrm{.}12\%$ as $8\mathrm{.}12).$

$($f$)$ If the Sharpe ratio of the stock is $0\mathrm{.}45.$ What is the Sharpe ratio of the call option?

$($g$)$ Suppose you are the broker that was forced to be the writer of the European call. You

use a replicating portfolio of stock and bond to delta$-$hedge the change in the call's

price $(S+B).$ You construct such a replicating portfolio today and hold on to it $($for

clarity, you do not rebalance it after today, just leave it unattended$).$ After one week,

suppose the stock price changes to $$53,$ what is your total profit in one week?