## The Most Successful Theory

What is it that physicists do on Wall Street? Mostly, they build models to determine the value of securities. Buried in investment banks, at hedge funds, or at financial software companies such as Bloomberg or SunGard, they tinker with old models and develop new ones. And by far the most famous and ubiquitous model in the entire financial world is the Black-Scholes options pricing model. Steve Ross, a famous financial economist, options theorist, and now a chaired professor at MIT, wrote in the Palgrave Dictionary of Economics that "...options pricing theory is the most successful theory not only in finance, but in all of economics"

The Black-Scholes model allows us to determine the fair value of a stock option. Stocks are commonplace securities, bought and sold daily, but a call option on a stock is much more arcane. If you own a one-year call option on IBM, for example, you have the right to buy one share of IBM one year from today at a predetermined price: say, $100. The value of the option on that future date when it expires will depend on the prevailing value of a share of IBM. If, for example, a share sells for $105 on that day, the option will be worth exactly $5; if a share sells for less than $100, the option will be worth nothing. In a sense, the option is a bet that the stock price will rise.

An option is a special case of a more general derivative security, a contract whose value is derived from the value of some other simpler underlying security on which it "rests." A derivative security's payoff at expiration is specified in a contract via a mathematical formula that relates the payoff to the future value of the underlying security. The formula can be simple, as is the case with the stock option just described, whose payoff is the amount by which the final stock price exceeds the value of $100, or it can be extremely complicated, with a payoff that depends on the prices of several underlying securities through detailed mathematical expressions. During the past twenty years derivative securities have become widely used in the trading of currencies, commodities, bonds, stocks, mortagages, credit, and power.

Derivatives are more intricate than unvarnished stocks or bonds. Then why do they exist? Because derivatives allow clients such as investment banks, money managers, corporations, investors, and speculators to tailor and fine-tune the risk they want to assume or avoid. An investor who simply buys a share of IBM takes on all the risk of owning it; its value waxes and wanes in direct proportion to IBM's share price. In contrast, an IBM call option provides potentially unlimited gain (as the share price rises above $100) but only limited loss (you lose nothing but the cost of the option as the stock price drops below $100). This asymmetry between upside gain and downside loss is the defining characteristic of derivatives.

You can buy or sell options retail on specialized options exchanges, or you can trade them with wholesalers, that is, the dealers. Options dealers "make markets" in options; they accomodate clients by buying options from those who want to sell them and selling options to those who want to acquire them. How, then, do dealers handle the risk they are forced to assume?

Dealers are analogous to insurance companies, who are also in the business of managing risk. Just as Allstate must allow for the possibility that your house will burn down after they sell you an insurance contract, so an options dealer must take a chance of a rise in IBM's stock price when he or she sells you a call option on IBM. Neither Allstate nor the options dealer wants to go broke if the insured-against scenario comes to pass. Because neither Allstate nor the dealer can foretell the future, they both charge a premium for taking on the risks that their clients want to avoid.

Allstate's risk strategy is to charge each client a premium such that the total sum they receive exceeds the estimated claims they will be obliged to pay for future conflagrations. An option dealer's risk strategy is different. In an ideal world, he or she would simply offset the risk that IBM's price will rise by buying an IBM option similar to the one he or she sold, from someone else and at a cheaper price, thereby making a profit. Unfortunately, this is rarely possible. So instead, the dealer manufactures a similar option. This is where the Black-Scholes model enters the picture.

The Black-Scholes model tells us, almost miraculously, how to manufacture an option out of the underlying stock and provides an estimate of how much it costs us to do so. According to Black and Scholes, making options is a lot like making fruit salad, and stock is a little like fruit.

Suppose you want to sell a simple fruit salad of apples and oranges. What should you charge for a one-pound can? Rationally, you should look at the market price of the raw fruit and the cost of canning and distribution, and then figure out the total cost of manufacturing the hybrid mixture from its simpler ingredients.

In 1973, Black and Scholes showed that you can manufacture an IBM option by mixing together some shares of IBM stock and cash, much as you can create the fruit salad by mixing together apples and oranges. Of course, options synthesis is somewhat more complex than making fruit salad, otherwise someone would have discovered it earlier. Whereas a fruit salad's proportions stay fixed over time (50 percent oranges and 50 percent apples, for example), an option's proportions must continually change. Options require constant adjustments to the amount of stock and cash in the mixture as the stock price changes. In fruit salad terms, you might start with 50 percent apples and 50 percent oranges, and then, as apples increase in price, move to 40 percent apples and 60 percent oranges; a similar decrease in the price of apples might dictate a move to 70 percent apples and 30 percent oranges. In a sense, you are always trying to keep the price of the mixture constant as the ingredients' prices change and time passes.The exact recipe you need to follow is generated by the Black-Scholes equation. Its solution, the Black-Scholes formula, tells you the cost of following the recipe. Before Black and Scholes, no one even guessed that you could manufacture an option out of simpler ingredients, and so there was no way to figure out its fair price.

This discovery revolutionized modern finance. With their insight, Black and Scholes made formerly gourmet options into standard fare. Dealers could now manufacture and sell options on all sorts of underlying securities, creating the precise riskiness clients wanted without taking on the risk themselves. It was as though, in a thirsty world filled with hydrogen and oxygen, someone had finally figured out how to synthesize H2O.

Dealers use the Black-Scholes model to manufacture (or synthesize, or financially engineer) the options they sell to their clients. They construct the option from shares of raw stock they buy in the market.

Conversely, they can deconstruct an option someone sells to them by converting it back into shares of raw stock that they then sell to the market. In this way, dealers mitigate their risk. (Since the Black-Scholes model is only a model, and since no model in finance is 100 percent correct, it is impossible for them to entirely cancel their risk.) Dealers charge a fee (the option premium) for this construction and de construction, just as chefs at fancy restaurants charge you not only for the raw ingredients but also for the recipes and skills they use, or as couturiers bill you for the materials and talents they employ in creating haute couture dresses.

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