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 $$24.$ If six months later, the stock price per share is $$24$ or more, the option has no value. If in six months the stock price is lower than $$24$ per share, then you can purchase the stock and immediately sell it at the higher exercise price of $$24.$ 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 $$24.$ Your profit would be the difference,

$$24$ $$22\mathrm{.}50=$ $$1\mathrm{.}50$

per share, less the cost of the option. If you paid $$1\mathrm{.}00$ per put option, then your profit would be

$$1\mathrm{.}50$ $$1\mathrm{.}00=$ $$0\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 $230$ shares of the stock at $$29$ per share and $90$ six$-$month European put options with an exercise price of $$24.$ 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 $$15$ and $$35$ per share in increments of $$1\mathrm{.}00.$

$\backslash$table$[[,$A$,$B$,$C$,$D$],[1,,$European Put Option,$],[2,$Share Price,Value with Options,Value without Options,$],[3,$$$15\mathrm{.}00,\backslash$bar $($sl$),$$$,],[4,$$$16\mathrm{.}00,$s$||,$$$1,],[5,$$$17\mathrm{.}00,$s$,$$$,],[6,$$$18\mathrm{.}00,$sprod,$$,],[7,$$$19\mathrm{.}00,\backslash$bar $($s$!),$$$[,],[8,$$$20\mathrm{.}00,$s$||,$$$1,],[9,$$$21\mathrm{.}00,$s$||,$$$,],[10,$$$22\mathrm{.}00,$s$,$$$,],[11,$$$23\mathrm{.}00,$s$||,$$$,],[12,$$$24\mathrm{.}00,\backslash$bar $($s$!),\backslash$bar $($$$1),],[13,$$$25\mathrm{.}00,$s$||,$$$||,],[14,$$$26\mathrm{.}00,$$$||,$$$,],[15,$$$27\mathrm{.}00,$sprod,$$1,],[16,$$$28\mathrm{.}00,$s$||,$$$,],[17,$$$29\mathrm{.}00,$s$,$$$,],[18,$$$30\mathrm{.}00,$s$||,$$$,],[19,$$$31\mathrm{.}00,$s$,$$$,],[20,$$$32\mathrm{.}00,$s$]],$$$[,],[21,$$$33\mathrm{.}00,$s$,$$$[,],[22,$$$34\mathrm{.}00,$s$,$$$,],[23,$$$35\mathrm{.}00,$s$||,$$$,],[24,,,,],[25,,,,]]$