Consider the situation of pricing a stock option as in Example 4.1.14.We want to prove that a price other than $20.19 for the option to buy one share in one year for $200 would be unfair in some way.
a. Suppose that an investor (who has several shares of the stock already) makes the following transactions. She buys three more shares of the stock at $200 per share and sells four options for $20.19 each. The investor must borrow the extra $519.24 necessary to make these transactions at 4% for the year. At the end of the year, our investor might have to sell four shares for $200 each to the person who bought the options. In any event, she sells enough stock to pay back the amount borrowed plus the 4 percent interest. Prove that the investor has the same net worth (within rounding error) at the end of the year as she would have had without making these transactions, no matter what happens to the stock price. (A combination of stocks and options that produces no change in net worth is called a risk-free portfolio.)
b. Consider the same transactions as in part (a), but this time suppose that the option price is $x where x < 20.19. Prove that our investor loses |4.16x − 84| dollars of net worth no matter what happens to the stock price.
c. Consider the same transactions as in part (a), but this time suppose that the option price is $x where x > 20.19. Prove that our investor gains 4.16x – 84 dollars of net worth no matter what happens to the stock price. The situations in parts (b) and (c) are called arbitrage opportunities. Such opportunities rarely exist for any length of time in financial markets. Imagine what would happen if the three shares and four options were changed to three million shares and four million options.