The strike price of a futures option is $550$ cents, the risk$-$free rate of interest is $3\%,$ the volatility of the futures price is $20\%,$ and the time to maturity of the option is $9$ months. The futures price is $500$ cents.

$($a$)$ What is the price of the option if it is a European call?

$($b$)$ What is the price of the option if it is a European put?

$($c$)$ Verify that put$$call parity holds.

$($d$)$ What is the futures price for a futures style option if it is a call?

$($e$)$ What is the futures price for a futures style option if it is a put?