Question $5$: Commodity Futures $(\frac{2}{10})$ Table $1$ shows the hypothetical daily closing prices for one unit of corn, and the daily prices for the corn futures contract with maturity date Jan $4.$ The futures contract size is $100$ units of corn.

The initial margin required for the futures contract is $$10,000$ per contract. The maintenance margin is $$7,000$ per contract. The risk free rate is $2\%($annual$,$ continuously compounded$).$ Assume this is the rate you earn on your margin account.

$2$

$(1)$ Suppose you take a long position in $10$ futures contracts on Jan $1.$ What is your profit$/$loss on Jan $2$ and Jan $3?$ What is the balance in your margin account at the end of Jan $2$ and Jan $3?$ Do you face any margin calls?

$(2)$ Following $(1),$ at the end of Jan $4,$ you will pay $390$ per unit to purchase the corn, and your margin account is returned to you. What is your total profit$/$loss from Jan $1$ to Jan $4,$ both including interest and excluding interest? $($Hint: when excluding interest, total payoff should be the same as the formula $N_0(S_T-F_{0,T}).)$

$(3)$ Repeat part $(1)$ for a short position of $5$ futures contracts.

$(4)$ Suppose there is a lease market for corn. Using the Jan $1$ corn price and futures price, what is the implied equilibrium lease rate $($annual$,$ continuously compounded$)$ on corn?