The price of a particular stock today is $$40.$ The price of a $1-$year European put option on this stock with strike price of $$30$ is $$7$ and the price of a $1-$year European call option on this stock with a strike price of $$50$ is $$5.$ Suppose that an investor buys $100$ shares, shorts $100$ call options, and buys $100$ put options.

$($a$)$ Plot the profit$/$loss of this portfolio as a function of the future stock price.

$($b$)$ This strategy is known as a collar. Explain how it combines the effects of the protective put and the covered call.

$($c$)$ Although this is an example of a collar, it is not a zero$-$cost collar. Explain