A trader writes $5$ naked put option contracts with each contract being on $100$ shares. The option price is $$10,$ the time to maturity is $6$ months, and the strike price is $$64.$

$($a$)$ What is the margin requirement if the stock price is $$58?$

$($b$)$ How would the answer to $($a$)$ change if the rules for index options applied?

$($c$)$ How would the answer to $($a$)$ change if the stock price were $$70?$

$($d$)$ How would the answer to $($a$)$ change if the trader is buying instead of selling the options?