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 financial theories (or, more accurately, theories are plagued by their inability to systematically account for the actual behavior of markets). For example, the negative return on a single day during the crash of 1987 was so many historical standard deviations away from the mean that it should never have occurred in our lifetime if returns were normally distributed. More recently, JPMorgan called the events of the “London Whale” an eight-standard-deviation event (JPMorgan Chase & Co. 2013). Stock evolution, to take just one of many examples, isn't Brownian.1 So, while financial engineers are rich in mathematical techniques, we don't have the right laws of science to exploit – not now, and maybe not ever.
Because we don't have the right laws, the axiomatic approach to finance is problematic. Axiomatization is appropriate in a field like geometry, where one can postulate any set of axioms not internally inconsistent, or even in Newtonian mechanics, where there are scientific laws that hold with such great precision that they can be effectively regarded as axioms. But in finance, as all practitioners know, our “axioms” are not nearly as good. As Paul Wilmott wrote, “every financial axiom.. ever seen is demonstrably wrong. The real question is how wrong.” (Wilmott 1998). Teaching by axiomatization is therefore even less appropriate in finance than it is in real science. If finance is about anything, it is about the messy world we inhabit. It's best to learn axioms only after you've acquired intuition.
Mathematics is important, and the more mathematics you know the better off you're going to be. But don't fall too