When I started working on Wall Street, I simply assumed it made good sense to apply the techniques of physics and applied mathematics to financial modeling. Differential calculus, partial differential equations, Fourier series, Monte Carlo calculation, stochastic calculus-all these tools for describing continuous motion also seemed unquestionably useful for describing the movements of markets and stocks, yield curves and volatilities. In particle physics, the field from which I came, these methods and the axioms they were applied to, had triumphed. There, people with a penchant for overly cute names dreamed of GUTs (Grand Unified Theories) and even TOEs (Theories of Everything).
It was Einstein who brought to fruition this mental approach to discovering the laws of the universe, transforming into a methodology the almost unbelievable insights that lay behind Newton’s mechanics and Maxwell’s electromagnetics. Einstein’s method wasn’t based on observation or empiricism; he simply tried to perceive and then enunciate the very principles that dictated the way things should work. His theories were almost meta-theories, rules about the allowed forms that, once adopted, would almost strip the theorist of any subsequent choice. In a speech on the principles of research, given in honor of Max Planck, the discoverer of the quantum, in 1918, Einstein said, “There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them.” In this way, he discovered special and general relativity, as well as many aspects of the quantum nature of matter. (If you want to get an entertaining feel for the persistent struggle for vision behind Einstein’s work, take a look at Dennis Overbye’s recent biography, Einstein in Love.)
Looking at the motion of yield curves in the mid-1980s, I at first saw no reason why financial theorists shouldn’t shoot for their theory of everything too. Why shouldn’t there be one model that described all interest rate motions, producing one set of rational market prices for all interest-rate-sensitive instruments? If you’d asked me in 1986 what yield-curve theories would look like in 2000, I would have imagined everyone using one theory for valuing all instruments from bond options through caps and swaptions to mortgages. Despite my naive idealism, I nevertheless knew that you shouldn’t expect the sort of accuracy in finance that you obtain in rocketry or hydrodynamics, let alone in the atomic physics of electromagnetic radiation.
I wasn’t alone in expecting the financial world to be deeply amenable to theory. Recently, I met again, for the first time in 13 years, someone who was an analyst with an undergraduate degree at Goldman Sachs in 1985, and is now a professor of finance. He told me that he chose to study for a Ph.D. in finance because he expected it to become the physics of the late 20th century.
Fifteen years on, I say without regret that things aren’t the way I had expected. There is no unified theory. Interest rate trading desks are pragmatic, and typically use a variety of models-one for bond options, perhaps another for caps, one for swaptions, often a totally different one for the Bermudan variety. Of course, the desks try to calibrate all their models consistently, but there’s no really grand attempt at comprehensiveness.
Some of this willingness to compromise stems from the need for computational speed, but even with infinitely fast computers I doubt that there would be an ultimate model on the horizon. So-called market models derive simple valuation formulas for complex derivatives by the trickery of conveniently choosing as a currency unit whatever traded instrument makes the payoff for each complex derivative look simple. This makes any single complex product easy to value, but veils the complexity of the relationship between different products. Using all these models at once is a bit like living in Manhattan and telling someone about a forthcoming trip: “It’s a $40 cab ride to Kennedy, then a six-hour flight to L.A., and finally a hundred-mile drive to Palm Springs.” You’ve used three different distance metrics-dollars, hours and miles-but it’s not hard to relate them to each other if you know geography. The geography of currency volatilities, however, is less simple.
So why is it that the techniques of physics and applied mathematics hardly ever give you more than the most approximate version of the truth about financial values?
In the end, there may be no absolute truth about financial values. In my experience, most finance practitioners raised in the tradition of the physical sciences don’t expect too much from financial theory. It’s not that physics is “better”; rather, finance seems to be harder. Paul Wilmott, in his textbook Derivatives, writes that “every financial axiom I’ve ever seen is demonstrably wrong…The real question is how wrong is the theory, and how useful is it regardless of its validity. Everything you read in any theoretical finance book, including this one, you must take with a generous pinch of salt.” I couldn’t agree more. In fact, the very title of Wilmott’s latest book, Wilmott on Derivatives, aptly illustrates his point. The “Wilmott” in Wilmott on Derivatives lends a touch of authority to the exposition of the subject, but in so doing it implies a deficiency of authority in the subject matter itself. Imagine a 1918 textbook called Einstein on Gravitation. Unlike finance, the theory of gravitation gets it weight from the ineluctability of its internal arguments and its accounting for previously inexplicable subtleties, and needs no Einstein to lend it gravitas.
So why do the methods of hard science work less well in finance?
One possible answer is that, in physics, once you know the dynamics, the parameter values are universal. No one disagrees significantly about the value of the gravitational constant or the mass of the earth you must use to calculate a satellite trajectory. Ignoring quantum mechanics, your accuracy is limited in practice by how well you know these values and, of course, how well you’ve taken account of all other known influences. But in finance, to calculate the fair value of a stock, the parameters you need are the expected dividends; to calculate the value of an option, you need the expected volatility. Science uses theory to move from known to unknown. Finance uses theory to move from one expectation to another.
When you propose a model or theory, you’re saying “Let’s pretend …” and then you see what happens as you work out the consequences. But what are you pretending about? When you propose a model of the physical world, you’re pretending you can guess the structure God created. It sounds a plausible thing to do; every physicist believes he has a small chance of guessing right, else he wouldn’t be in the field. But when you propose a financial model, you’re pretending you can guess another person’s mind. When you try out a simple yield-curve model, you’re saying, “Let’s pretend that people care only about future short rates, and that they think they’re distributed lognormally.” As you say that to yourself, if you’re honest, your heart sinks. You know immediately that there is no chance you are truly right. To pretend you can figure out God’s intention doesn’t sound preposterous. To pretend you can figure out man’s, does. Perhaps it’s because God doesn’t pretend, so when you tackle nature, you’re the only one doing the pretending. But when you tackle people, you’re pretending you can figure out another pretender.
I find myself relying on a critical difference between people and Nature as an explanation of the inadequacies of financial theory. But aren’t people part of Nature too? Schrodinger, the unconventional father of the wave equation in quantum mechanics, wrote a short summary of his personal views on determinism and free will in the epilogue to What is Life? his influential lectures on the physico-chemical basis of living matter. “My body functions as a pure mechanism according the Laws of Nature,” he wrote. “Yet I know, by incontrovertible direct experience, that I am directing its motions, of which I foresee the effects, that may be fateful and all-important, in which case I feel and take full responsibility for them.”
The only way he could reconcile these two apparently contradictory experiences-his deep belief in the susceptibility of Nature to human theorizing and his equally firm sense of the individual autonomy that must lie beneath any attempt to theorize-was to infer that “every conscious mind that has ever said or felt ‘I’ … [is] the person, if any, who controls the ‘motion of the atoms’ according to the Laws of Nature.”
Schrodinger was following a long line of earlier German philosophers who thought that all the various worldly voices referring to themselves in conversation as ‘I’ were not really referring to independent I’s, but to the same universal I-God or Nature. It’s a comforting notion. But it still doesn’t explain why, if all the I’s add up to God, it’s so much harder to predict the world of I’s than the world of God.