You are evaluating the following option: a European call option with a strike price of $$45,$ expiring in $6$ months, where the underlying stock price is $$40,$ the expected return of the stock is $7\%,$ and it has a volatility of $25\%($both annualized$).$ Assume the risk$-$free rate is $1\mathrm{.}5\%($annualized$),$ and the stock pays no$-$dividend. You can assume the option is for a single share of the underlying stock. $(26$ points$)$

$($a$)$ Calculate the value of the option using Black$-$Scholes. You can assume di $=$

$-0\mathrm{.}5355$ and d$2=0\mathrm{.}7122.(5$ points$)$

$($b$)$ Describe the transaction you would need to delta hedge a short position in the option using the underlying stock, including the number of shares you would need to use and the value of the position in the stock. $(4$ points$)$

$($c$)$ Suppose the historical proportional bid$-$ask spread of the stock was $1\mathrm{.}2\%.$ Calculate the total expenditure for entering into this hedge. $(5$ points$)$

$($d$)$ Assuming the returns are normally distributed, and given the information above, calculate the maximum you can expect to lose over the next $30-$days only on the hedge with $99\%$ confidences. You can assume there are $360$ days in the year. Note if you do not know the size of the hedge from part $($b$),$ you can assume a value.

$(5$ points$)$

$($e$)$ The option has a positive gamma. Will the maximum you can expect to lose on the option position over the next $30-$days with $99\%$ confidence be larger or smaller than that of the hedge? Explain your answer. $(7$ points$)$ this is a full task with side questions please answer the whole task