Question $4$: Commodity Forwards $(2/10)$ The current price of crude oil is $$30\mathrm{.}00$ per barrel. Assume a $2\%$ continuously$-$compounded annual risk$-$free rate, and there is no lease market for crude oil.

$(1)$ Suppose forward price for $6$ months is $$29\mathrm{.}85$ and there is no carry cost. Based on the forward price, what is the convenience yield $c($annual$,$ continuously compounded$)?$

$(2)$ Give an example of convenience yield for holding crude oil.

$(3)$ Suppose forward price for $6$ months is $$32\mathrm{.}31$ and there is no convenience yield. However, there is a lump$-$sum carry cost $$ in $3$ months $($Note this is not the case of continuously compounded carry cost$).$ Based on the forward price, what is the lump$-$sum carry cost $$ in $3$ months?

$(4)$ Suppose the carry cost is the same as $(3)$ and there is no convenience yield, but the forward price for $6$ months is mispriced at $$32\mathrm{.}50.$ How can you arbitrage?