Problem $3(40$p$)($Delta Hedging$)$

Assume we are in a Black$-$Scholes world where on day $t=0$ a stock is trading at $S_0=$$$35$ per share. The stock price volatility is $=25\%$ and it pays a continuous dividend yield $D=2\%.$ Suppose the writer of a European option sells a call option with strike $K=$$$33$ on $1000$ shares with time to expiration $T=180$ days.

Given risk$-$free interest rate $r=5\%$ per annum, calculate the following

$($a$)(10$p$)$ The call price and the corresponding delta at day $0.$

$($b$)(10p)$ The writer's trading strategy to maintain a delta$-$hedged portfolio on day $0.$ How much money does the writer need to borrow$/$put in a risk$-$free money market on day $0$ in order to maintain a delta$-$hedged portfolio?

$($c$)(10$p$)$ The writer's profit if the stock price increases to $$35\mathrm{.}50$ on day $1.$ Calculate also the cost to keep the portfolio delta neutral.

$($d$)(10$p$)$ The writer's profit if the stock price falls to $$34\mathrm{.}80$ on day $2.$ Calculate also the cost to keep the portfolio delta neutral.

Note if $ZN(0,1),$ then the cumulative standard normal distribution function in the range $[0,x],x>0$ can be approximated by