Option Mechanics & Strategies

A trader writes five naked put option contracts, with each contract being on $100$ shares. The option price is $$10,$ the time to maturity is six months, and the strike price is $$64.$ What is the margin requirement if the stock price is $$58?$

A one$-$month European put option on a non$-$dividend$-$paying stock is currently selling for $$2\mathrm{.}50.$ The stock price is $$47,$ the strike price is $$50,$ and the risk$-$free interest rate is $6\%$ per annum. What opportunities are there for an arbitrageur?

What is a lower bound for the price of a six$-$month call option on a non$-$dividend$-$paying stock when the stock price is $$80,$ the strike price is $$75,$ and the risk$-$free interest rate is $10\%$ per annum?