Problem $3.($Collars$,10\text{'})$ The underlying asset for the derivatives below is one ounce of

gold. The expiration date for these derivatives is the end of this year.

$($Hint: please note the difference between payoff and profit diagrams.$)$

$($a$).$ What is the payoff diagram for a long European put with a strike price of $$420?(2\text{'})$

$($b$).$ What is the payoff diagram for a short European call with a strike price of $$440?(2\text{'})$

$($c$).$ What is the payoff diagram of a purchased $420-440$ collar $($European style$),$ i$.$e$.,$ a

portfolio of $($a$)$ and $($b$)?(4\text{'})$

$($d$).$ Briefly discuss when you may want to purchase this collar for risk management. $(2\text{'})$