Trends In Economics: A Calculus of Risk: Scientific American

Options represent the right (but not the obligation) to buy or sell stock or some other asset at a given price on or before a certain date. Another major class of derivatives, called forwards and futures, obligates the buyer to purchase an asset at a set price and time. Swaps, yet another type of derivative, allow companies to exchange cash flows—floating-interest- rate for fixed-rate payments, for instance.

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Black and Scholes, with Merton’s help, came up with their option-pricing formula by constructing a hypothetical portfolio in which a change of price in a stock was canceled by an offsetting change in the value of options on the stock—a strategy called hedging. Here is a simplified example: A put option would give the owner the right to sell a share of a stock in three months if the stock price is at or below $100. The value of the option might increase by 50 cents when the stock goes down $1 (because the condition under which the option can be used has grown more likely) and decrease by 50 cents when the stock goes up by $1.

To hedge against risks in changes in share price, the investor can buy two options for every share he or she owns; the profit then will counter the loss. Hedging creates a risk-free portfolio, one whose return is the same as that of a treasury bill. As the share price changes over time, the investor must alter the composition of the portfolio—the ratio of the number of shares of stocks to the number of options—to ensure that the holdings remain without risk.

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Investors who buy options are basically purchasing volatility—either to speculate on or to protect against market turbulence. The more ups and downs in the market, the more the option is worth. An investor who speculates with a call—an option to buy a stock—can lose only the cost of purchase, called a premium, if the stock fails to reach the price at which the buyer can exercise the right to purchase it. In contrast, if the stock shoots above the exercise price, the potential for profit is unlimited. Similarly, the investor who hedges with options also anticipates rough times ahead and so may buy protection against a drop in the market.

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