$1.$ You decide to create a bull spread on a stock, anticipating the stock price will increase. You sell a $6-$month call option for $$2\mathrm{.}83/$share with a strike price of $$45\mathrm{.}00,$ and you buy a call option for $$5\mathrm{.}11/$share with a strike price of $$40\mathrm{.}00.$ The current spot price is $$42\mathrm{.}00,$ and assume an interest rate of $3\%$ per annum. $(3$ points$)$

a$.$ What is the initial cost per share of entering this position?

b$.$ Draw a profit diagram of this position. Clearly indicate all breakeven points, intercepts, and the heights of all lines where applicable $($biggest loss, biggest gain$).$

c$.$ Indicate your payoff and profit given the following final stock prices:

Payoff Profit

St $=$ $$38\mathrm{.}75$

St $=$ $$41\mathrm{.}50$

St $=$ $$50\mathrm{.}42$