A put option in finance allows you to sell a share of stock at a given price in the future. There are different types of put options. A European put option allows you to sell a share of stock at a given price, called the exercise price, at a particular point in time after the purchase of the option. For example, suppose you purchase a six$-$month European put option for a share of stock with an exercise price of $$21.$ If six months later, the stock price per share is $$21$ or more, the option has no value. If in six months the stock price is lower than $$21$ per share, then you can purchase the stock and immediately sell it at the higher exercise price of $$21.$ If the price per share in six months is $$15\mathrm{.}50,$ you can purchase a share of the stock for $$15\mathrm{.}50$ and then use the put option to immediately sell the share for $$21.$ Your profit would be the difference, $$21-$$$15\mathrm{.}50=$$$5\mathrm{.}50$ per share, less the cost of the option. If you paid $$1\mathrm{.}00$ per put option, then your profit would be $$5\mathrm{.}50-$$$1\mathrm{.}00=$$$4\mathrm{.}50$ per share.

$($a$)$ Build a model to calculate the profit of this European put option.

$\backslash$table$[[,$A European Put Option,B$,],[1,],[2,,,],[3,$Parameters,,$],[4,$Cost of Put Option,$$1\mathrm{.}00,],[5,$Exercise Price,$$21\mathrm{.}00,],[6,$Horizon,$6,],[7,,,],[8,$Price at End of Horizon,$$15\mathrm{.}50,],[9,,,],[10,,,],[11,$Model,,$],[12,$Value of the Option,$--$ Select$---,],[13,$Cost of Put Option,$--$ Select$---$ hat$(v),],[14,$Profit per Option,$---$ Select$---$ hat$(v),],[15,,,]]$

$($b$)$ Construct a data table that shows the profit per share for a share price in sin increments of $$1\mathrm{.}00.$

$\backslash$table$[[,$A$,$B$,$C$],[16,$Price at End of Horizon,Profit,$],[17,10,,],[18,11,,],[19,12,,],[20,13,,],[21,14,,],[22,15,,],[23,16,,],[24,17,,],[25,18,,],[26,19,,],[27,20,,],[28,21,,],[29,22,,],[30,23,,],[31,24,,],[32,25,,],[33,26,,],[34,27,,],[35,28,,],[36,29,,],[37,30,,],[38,,,]]$