$3.$ Suppose you are a market$-$maker in the S&P $500$ index contracts. The S&P $500$ spot price is $$5,200,$ the risk$-$free rate is $4\mathrm{.}8\%.,$ and the dividend yeild on the index is $1\mathrm{.}5\%.$

a$.$ What is the no$-$arbitrage price of the S&P $500$ forward contract for delivery in $9$ months?

b$.$ Suppose a customer wishes to enter a long physical index position. If you take the opposite position, you will hedge your sesulting short position using the forward and borrowing or lending. Then, how much you need to lend or borrow?

$4.$ Suppose that silver costs $$24$ today per troy ounce and the continuously compounded interest rate is $3\mathrm{.}5\%.$ The silver price in $1$ year is uncertain and silver can be stored at no cost and does not earn any convenience yield.

a$.$ What is the equilibrium forward price of silver if the continuously compounded lease rate is $5\%?$

b$.$ If the $1-$year forward price of silver is $$25,$ is there an arbitrage opportunity? If so$,$ what is the minimum profit per troy ounce? Explain the transactions you could use to make money with zero initial investment and no risk.

c$.$b$.$ If the $1-$year forward price of silver is $$22,$ is there an arbitrage opportunity? what is the lease rate under the transactions? Explain the transactions you could use to make money with zero initial investment and no risk.