Assume a continuously compounding dollar interest rate of $5\%$ for all maturities, whenever applicable.

If you go short a $2-$year forward contract on a security with a delivery price $(K)$ of $$100,$

$($a$)$ What is your payoff at expiry if the security price at expiry $(S_T)$ is $$90,$$$100,$$$110\mathrm{.}52,$ and $$120,$ respectively? $($Reminder: Four answers to the four different scenarios.$)$

$($b$)$ Plot your payoff at expiry as a function of the security price at expiry.

$($c$)$ If the forward is on a stock with a current spot price $(S_t)$ of $$50,$ the present value of the dividend payments over the next two years is $$5.$ There are no other costs or benefits in carrying the stock except interest cost. What should be the current forward price $(F)$ on the stock with a $2-$year maturity?

$($d$)$ Based on your calculated forward price, what is the current value of your short position on the $2-$year forward with a delivery price of $$100?$

$($e$)$ If the market quote for the $2-$year forward is $$50($with zero bid$-$ask spread$),$ is there an arbitrage opportunity? How can you set up trades to exploit the opportunity?