Reviewed by Philip Protter

Mathematical finance (or financial engineering, as it is often known) is a young subject for mathematics, but is highly popular with students. No doubt the allure of being connected to vast sums of money is a part of the attraction. Yet it is a difficult subject, requiring a broad array of knowledge of subjects that are traditionally considered hard to learn. Forty years ago, options and what are now called “financial derivatives” were little known. Options were traded on the Chicago Board Options Exchange (CBOE), primarily for commodities such as pork bellies, orange juice, coffee, and precious metals. Let us take a minute to describe a situation where an option is useful. Imagine a small Indiana farmer raising pigs. The price is high now, but his pigs are only 80% grown. He can market them now and make a handsome profit, or he can wait until they are fully grown and make a larger profit if the price stays up, but end up making significantly less money if the price falls. He could solve the problem by buying a forward contract, locking in a prearranged price and thus a sure profit, but what if the price rises further? Then he will kick himself for having locked in the price. An option, on the other hand, gives him the right, but not the obligation, to sell his pigs at the prearranged price, thus guaranteeing him the nice profit but not excluding a potentially bigger one. The prices of options such as the one just described were set by the market: supply and demand. In the United States there is a fervent belief that the market knows best and, if left alone, will arrive at a fair and just price. There are many unspoken hypotheses involved with this belief, and in the case of commodities, several were violated. It will suffice to point out that small farmers were buying options sold by large banks and companies. In the early 1970s, Black, Scholes, and Merton showed that by using the Itô stochastic calculus and a simple model describing the dynamics of the price of a risky asset, one could arrive at a fair price for an option. They did this using a key idea: if one sells the option for $x , there is a hedging strategy by which one can use that $x to trade in the commodity over time until the option is due and end up with exactly what is owed to the option purchaser at the settlement time. There is no risk at all, except the implicit risk that the model for the dynamic price of the commodity is wrong. Therefore, if the market price is larger than $x , one can charge the market price and match the option and have money left over. If the market price is less than $x , one can buy the options and make money in reverse. It turned out that the market is often wrong, but the breakthrough of Black and Scholes went largely ignored by the financial players. This gradually changed, largely through the efforts of a few visionary people at Wells Fargo Bank, who worked not so much with commodities as with the (then) new concepts of portfolio insurance and index funds (see [2]).