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 $$28$ per share and $65$ six$-$month European put options with an exercise price of $$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 $0.$

Share Price Benefit of Options

$$20$ $

$$21$ $

$$22$ $

$$23$ $

$$24$ $

$$25$ $

$$26$ $

$$27$ $

$$28$ $

$$29$ $

$($b$)$ Discuss the value of the portfolio with and without the European put options.

The lower the stock price, the beneficial the put options. The options are worth nothing at a stock price of $ or $.$ There is a benefit from the put options to the overall portfolio for stock prices of $ or $.$