Question $5$: Forwards and Futures $(\frac{2}{10})$ Suppose the risk free interest rate is $10\%($annual$,$ continuously compounded$).$ That is$,$ the daily interest rate is approximately $1+r^{\text{d}\text{a}\text{y}}=e^{10\%\frac{1}{365}}.$ There are two contracts on one HKUST share: a forward contract that allows you to buy one HKUST share on Dec $3$ and a future contract $($no margin requirement$)$

$2$

that allows you to buy one HKUST share on Dec $3.$ The spot price of the share on Dec $3$ is $S_3.$

Suppose you are taking positions on these two contracts on Dec $2.$ The forward price on Dec $2$ is denoted as $F{O}_2$ and the future price is $F{U}_2.$

$(1)$ If you long one forward contract, what is your initial cost on Dec $2$ and what is your payoff on Dec $3?$

$(2)$ If you short one future contract, what is your initial cost on Dec $2$ and what is your payoff on Dec $3?$

$(3)$ If you long one forward contract and short one future contract, what is your initial cost on Dec $2$ and what is your payoff on Dec $3?$ What does no arbitrage condition tell you about the relationship between $F{O}_2$ and $F{U}_2?$

Now suppose you are taking positions on these two contracts $($recall the maturity dates are Dec $3)$ on Dec $1$ and selling these positions on Dec $2.$ The forward price on Dec $1$ is denoted as $F{O}_1$ and the future price is $F{U}_1.$

$(4)$ If you long $1$ forward contract on Dec $1,$ what is the value of the forward contract $($that you entered on Dec $1)$ on Dec $2?$ Note this is the amount of money you get if you sell $($close out$)$ the forward contract on Dec $2.($Hint: the value of a forward contract depends on its original forward price $(F{O}_1$ in this case$),$ and the forward price now $(F{O}_2$ in this case$).)$

$(5)($Optional$)$ If you short $\frac{1}{1+r^{\text{d}\text{a}\text{y}}}$ future contracts on Dec $1,$ what is your profit $($or loss$)$ on Dec $2?$ Note this is the amount of money you get if you close out the future positions on Dec $2$ because closing a future contract earns$/$costs nothing and simply gives back the profit you have earned so far. $($Hint: the profit $/$ loss of a future contract depends on its original future price $(F{U}_1$ in this case$),$ and the future price now in this case$).)$

$(6)($Optional$)$ Consider the portfolio that longs $1$ forward contract and shorts $\frac{1}{1+r^{\text{d}\text{a}\text{y}}}$ future contracts on Dec $1$ and closes out both positions on Dec $2.$ what is your initial cost on Dec $1$ and what is your payoff on Dec $2?$ What does no arbitrage say about the relationship between $F{O}_1$ and $F{U}_1?$

$(7)($Optional$)$ From above, what can we deduce about the prices of forwards and futures when interest rates are constant?