Max Houck holds $500$ shares of Boulder Gas and Light. He bought the stock several years ago at $$52\mathrm{.}89,$ and the shares are now trading at $$77\mathrm{.}00.$ Max is concerned that the market is beginning to soften. He doesn't want to sell the stock, but he would like to be able to protect the profit he's made. He decides to hedge his position by buying $5$ puts on Boulder G&L$.$ The three$-$month puts carry a strike price of $$77\mathrm{.}00$ and are currently trading at $$3\mathrm{.}45.$

a$.$ How much profit or loss will Max make on this deal if the price of Boulder G&L does indeed drop, to $$62\mathrm{.}50$ a share, by the expiration date on the puts?

b$.$ How would he do if the stock kept going up in price and reached $$84\mathrm{.}00$ a share by the expiration date?

c$.$ What do you see as the major advantages of using puts as hedge vehicles?

d$.$ Would Max have been better off using in$-$the$-$money puts$-$that is$,$ puts with an $$88\mathrm{.}50$ strike price that are trading at $$10\mathrm{.}49?$ How about using out$-$of$-$the$-$money puts$-$say, those with a $$73\mathrm{.}50$ strike price, trading at $$0\mathrm{.}90?$ Explain.