same expiration over a range of strikes would be a flat line. Figure 1.1 shows what three-month equity index implied volatilities looked like before the Black Monday stock market crash of 1987.
Figure 1.1 Representative S&P 500 Implied Volatilities prior to 1987
Prior to the crash, therefore, the BSM model seemed to describe the option market rather well, at least with respect to variation in strikes. Figure 1.2 shows typical three-month implied volatilities after the crash of 1987. Even though all the options used to generate the smile were written on the same underlier, each option had a different implied volatility. This is inconsistent with the BSM model, which assumes that implied volatility is a forecast of actual volatility, for which there can be only one value. You can think of options as metaphorical photographs of the stock's future volatility, taken from different angles or elevations. While photographs of a building taken from different points might look different, the actual size of a building remains the same. In a similar way, if the BSM model were truly reliable, the implied volatility of the stock would be the same, no matter which option you chose to view it with. The option price is derived from the stock price, but the stock's volatility should not depend on the option.
Figure 1.2 Representative S&P 500 Implied Volatilities after 1987
Though the smile appeared most dramatically in equity index option markets after the 1987 crash, there had always been a slight smile in currency option markets, a smile in the literal sense that the implied volatilities as a function of strike resembled one:
Since the crash of 1987, the volatility smile has spread to most other option markets (currencies, fixed income, commodities, etc.), but in each market it has taken its own characteristic form and shape. Traders and quants in every product area have had to model the smile in their own market. At many firms, not only does each front-office trading desk have its own particular smile models, but the firm-wide risk management group is likely to have its own models as well. The modeling of the volatility smile is likely one of the largest sources of model risk within finance.
No-Nonsense Financial Modeling
During the past 20 years there has been a tendency for quantitative finance and asset pricing to become increasingly formal and axiomatic. Many textbooks postulate mathematical axioms for finance and then derive the consequences. In this book, though, we're studying financial engineering, not mathematical finance. The ideas and the models are at least as important as the mathematics. The more math you know, the better, but math is the syntax, not the semantics. Paul Dirac, the discoverer of the Dirac equation who first predicted the existence of antiparticles, had a good point when he said:
I am not interested in proofs, but only in what nature does.
About Theorems and Laws
Mathematics requires axioms and postulates, from which mathematicians then derive the logical consequences. In geometry, for example, Euclid's axioms are meant to describe self-evident relationships of parts of things to the whole, and his postulates further describe supposedly self-evident properties of points and lines. One Euclidean axiom is that things that are equal to the same thing are equal to each other. One Euclidean postulate, for example, is that it is always possible to draw a straight line between any two points.
Euclid's points and lines are abstracted from those of nature. When you get familiar enough with the abstractions, they seem almost tangible. Even more esoteric abstractions – infinite-dimensional Hilbert spaces that form the mathematical basis of quantum mechanics, for example – seem real and visualizable to mathematicians. Nevertheless, the theorems of mathematics are relations between abstractions, not between the realities that inspired them.
Science, in contradistinction to mathematics, formulates laws. Laws are about observable behavior. They describe the way the universe works. Newton's laws allow us to guide rockets to the moon. Maxwell's equations enable the construction of radios and TV sets. The laws of thermodynamics make possible the construction of combustion engines that convert heat into mechanical energy.
Finance is concerned with the relations between the values of securities and their risk, and with the behavior of those values. It aspires to be a practical field, like physics or chemistry or electrical engineering. As John Maynard Keynes once remarked about economics, “If economists could manage to get themselves thought of as humble, competent people on a level with dentists, that would be splendid.” Dentists rely on science, engineering, empirical knowledge, and heuristics, and there are no theorems in dentistry. Similarly, one would hope that finance would be concerned with laws rather than theorems, with behavior rather than assumptions. One doesn't seriously describe the behavior of a market with theorems.
How then should we think about the foundations of finance and financial engineering?
On Financial Engineering
Engineering is concerned with building machines or devices. A device is a little part of the universe, more or less isolated, that, starting from the constructed initial conditions, obeys the laws of its field and, while doing so, performs something we regard as useful.
Let's start by thinking about more familiar types of engineering. Mechanical engineering is concerned with building devices based on the principles of mechanics (i.e., Newton's laws), suitably combined with empirical rules about more complex forces that are too difficult to derive from first principles (friction, for example). Electrical engineering is the study of how to create useful electrical devices based on Maxwell's equations and quantum mechanics. Bioengineering is the art of building prosthetics and biologically active devices based on the principles of biochemistry, physiology, and molecular biology.
Science – mechanics, electrodynamics, molecular biology, and so on – seeks to discover the fundamental principles that describe the world, and is usually reductive. Engineering is about using those principles, constructively, to create functional devices.
What about financial engineering? In a logically consistent world, financial engineering, layered above a solid base of financial science, would be the study of how to create useful financial devices (convertible bonds, warrants, volatility swaps, etc.) that perform in desired ways. This brings us to financial science, the putative study of the fundamental laws of financial objects, be they stocks, interest rates, or whatever else your theory uses as constituents. Here, unfortunately, be dragons.
Financial engineering rests upon the mathematical fields of calculus, probability theory, stochastic processes, simulation, and Brownian motion. These fields can capture some of the essential features of the uncertainty we deal with in markets, but they don't accurately describe the characteristic behavior of financial objects. Markets are plagued with anomalies that violate standard
