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 $$26.$ If six months later, the stock price per share is $$26$ or more, the option has no value. If in six months the stock price is lower than $$26$ per share, then you can purchase the stock and immediately sell it at the higher exercise price of $$26.$ If the price per share in six months is $$22\mathrm{.}50,$ you can purchase a share of the stock for $$22\mathrm{.}50$ and then use the put option to immediately sell the share for $$26.$ Your profit would be the difference, $$26-$$$22\mathrm{.}50=$$$3\mathrm{.}50$ per share, less the cost of the option. If you paid $$1\mathrm{.}00$ per put option, then your profit would be $$3\mathrm{.}50-$$$1\mathrm{.}00=$$$2\mathrm{.}50$ per share. The point of purchasing a European option is to limit the risk of a decrease in the per$-$share price of the stock. Suppose you purchased $200$ shares of the stock at $$28$ per share and $60$ six$-$month European put options with an exercise price of $$26.$ Each put option costs $$1.$

$($a$)$ Using data tables, construct a model that shows the value of the portfolio with options and without options for a share price in six months between $$20$ and $$29$ per share in increments of $$1\mathrm{.}00.$ What is the benefit of the put options on the portfolio value for the different share prices? For subtractive or negative numbers use a minus sign even if there is a $+$ sign before the blank $($Example: $-300).$ If you answer is zero, enter $"0".$

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