Pricing the Future focuses on the search for a solution to the pricing of options. An option is a kind of contract that gives you the right — but does not oblige you — to buy or sell something (usually goods or a security) on a fixed date for a fixed price from a broker. Since the future price is unknown, the broker assumes a risk, but you have no risk since you can decline to exercise the option. How much should you pay the broker to bear the risk?
Options have been traded since at least the sixteenth century. Until late in the twentieth century, nobody had any real notion of how to price an option. Mostly, option prices were set based on gut feeling. The work of Fisher Black and Myron Scholes (published in 1973), together with parallel development by Robert Merton, produced the option formula called the Black-Scholes equation. Scholes and Merton received the Nobel Prize in economics in 1997. (Fisher Black had died in 1995.) Subsequently, as associates at Long-Term Capital Management, Scholes and Merton became part of a gigantic flop that nearly brought down the banking system in the U.S. It would be unfair to blame the Black-Scholes equation for that debacle, though perhaps the confidence it inspired reinforced the far more questionable reliance on the efficient market model.
This book tells the story of the historical and intellectual developments that led to the options pricing formula. Many fascinating personalities were involved with the development; the list includes Robert Brown (of Brownian motion fame), Henry Lefevre, Louis Bachelier, Einstein, Norbert Wiener, Kolmogorov, Kiyoshi Ito, and the economist Paul Samuelson.
The author has written his book for the general public. It has very little explicit mathematics, though there is an appendix with a kind of rough, non-rigorous derivation of the Black-Scholes equation. The author has a background in mathematics and physics, with a doctorate in finance and mathematical economics, so he is an ideal position to present the many and varied aspects of the story. Generally he deals well with mathematical explanations suitable for a general reader. But every now and again there is an odd slip. When he is describing various forms of the Chapman-Kolmogorov equation and the role they play in modeling price movements, he writes, “But did such equations actually exist?” The equations certainly exist. It’s unclear if he’s asking whether solutions to the equations exist or something else, but it certainly is confusing. In another place he writes about Wiener and his creation of the field of cybernetics; he says that cybernetics is “a statistical approach to the theory of communication”, but this misses the mark. It’s about control and feedback mechanisms. There are perhaps a dozen more such odd statements throughout the book, but they are mostly harmless. The author writes so well that it seems petty to complain about small potatoes.
A curious aspect of the option pricing story is that around 1968 Fisher Black, a Harvard Ph.D. in applied mathematics, had the basic partial differential equation in hand, but could not solve it. He evidently had little background in differential equations and didn’t recognize what was essentially the heat equation. So the incomplete work sat in his file cabinet for two or three years. Then Black and Scholes got together and eventually guessed a solution.
The interested reader is urged to read the short essay The Financial Modelers' Manifesto by Emanuel Derman and Paul Wilmott. The Modelers’ Hippocratic Oath is included. One of its tenets is:
I will remember that I didn’t make the world, and it doesn’t satisfy my equations.
Derman and Wilmott argue that the Black-Scholes model, often unjustly reviled as the cause of so much mayhem over the past few years, is clear and robust. It is this story of what they call a “model of models” that Szpiro tells so well in his book.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.