The Money Formula. Wilmott Paul

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СКАЧАТЬ is that every agent is actuated only by self-interest.”18 People also had a fixed set of preferences. So if they liked cereal for breakfast, they didn't suddenly swap over to eating toast. And people always acted in a completely rational fashion.

      Thus was born the notion of homo economicus, or rational economic man. While these assumptions had obvious flaws – surely we do change our minds? – they did allow economists to construct elegant mathematical models of the economy.

Finding Equilibrium

      An obvious difference between economics and physics was that physical quantities could be measured in well-defined units, while “utility” was rather vague and no one knew what its units were (“utils” was suggested). However, Jevons argued that in reality we can never directly measure a force like gravity, only its effects. Even if utility could not be directly measured, or even defined, we could infer it from market prices. Today, it is fair to say that quants do not suffer from lack of data – we have more information about markets than we have about other things that we wish to predict, such as the weather.

      Another problem was that, while the atoms of physics are believed to have the same properties everywhere in the universe, people – who are the atoms of the economy – show a high degree of variability. According to Jevons, though, what counted was the behavior of “the single average individual, the unit of which population is made up.”19 This meant that the agents in the economy – i.e., individuals and firms – could be treated as if they were all the same. The idea was inspired by the “social physics” of the 19th-century Belgian scientist Adolphe Quetelet, who wrote of l'homme moyen, or the average man.20 He claimed that “the greater the number of people observed, the more do peculiarities, whether physical or moral, become effaced, and allow the general facts to predominate, by which society exists and is preserved.”21

      Here we have a social science version of the probabilist's Law of Large Numbers. This mathematical law states that the average of a large number of trials will converge to the expected value of a single trial.A die has 21 spots and six sides, giving an expected throw of 21/6 = 3.5. As you roll the dice more frequently, the average will converge to this expected value. The idea that the expected behavior of humans, as with dice, is all that matters in the long run could be an explanation behind Isaac Asimov's fictional character Hari Seldon in the Foundation series of novels. Professor Seldon is one of the creators of “psychohistory,” a science that makes predictions about the future based on the statistics of large groups of people. When asked “Can you prove that this mathematics is valid?” he replies, “Only to another mathematician.” Nobel Laureate Paul Krugman says that he became interested in economics thanks to Hari Seldon and his ability to predict mankind's actions.22 Of course, it's all poppycock, but entertaining reading nonetheless.

      Whatever the equations governing man's economic behavior, the neoclassical economists faced a rather daunting computational problem. One way to make it tractable was to assume that prices were at equilibrium. Jevons compared the price mechanism to the motion of a pendulum, which came to rest at the ideal balance between supply and demand. Even if one could not compute the daily permutations of the markets, it should be possible to compute the average equilibrium position to which the invisible hand was pushing them. Furthermore, it made sense that markets should be at or near equilibrium; since if prices were too low or too high, then this would imply that market participants were not making rational decisions. The assumption of equilibrium was therefore also tied up with the idea of rationality.

Intrinsic Value

      As economics developed in the 20th century, concepts such as rationality and equilibrium remained at the heart of the theory. In the 1960s, economists Kenneth Arrow and Gérard Debreu created a model of an idealized market economy, and famously showed that it would reach a kind of optimal equilibrium (a result that did not displease their sponsors at the US Department of Defense, at a time when the country was embroiled in an ideological conflict with its communist foes23). But to prove its results, its authors had to assume that market participants act rationally to maximize their utility, not just now but also in the future. Since the future is unknown, this means they have to know what is the best course of action for every possible future state of the world – something which implied infinite computational capacity. The Arrow–Debreu model of the economy served as the theoretical foundation for general equilibrium models, versions of which are used today to determine the effects of policy changes on the economy.

      Unfortunately, these models – despite being “aesthetically beautiful” to theoreticians24 – turned out to be little better at predicting the economy than random guessing (which is why they are not used by quants). Psychohistory they weren't. However, the University of Chicago's Eugene Fama came forward with a convenient excuse for why economists were doing such a poor job of predicting the future, at least for markets. His efficient market hypothesis portrayed the market as a swarm of “rational profit maximizers” who drive the price of any security to its “intrinsic value.” It was therefore impossible to beat or out-predict the market, because any information would already be priced in. The invisible hand of the market was the epitome of rationality. This leads to the weird situation where individuals are assumed to be able to make perfect predictions (Arrow–Debreu), but this in turn means that no one can predict the markets (Fama).

      This would normally be the point at which most investors turned their backs on too much theorizing – as ex-Fidelity fund manager Peter Lynch told Fortune magazine, “Efficient markets? That's a bunch of junk, crazy stuff” – but it is precisely the elegance of this “result” that excites the academic economists.25 As discussed further below, the efficient market idea formed the backbone of academic models used in risk analysis, and much of quantitative finance in general. As with Adam Smith and the neoclassical economists, the central idea was of the market at equilibrium, with the invisible hand constantly restoring it to what Smith called a “tone of tranquillity and composure.”

      Quants in general have a somewhat conflicted attitude toward the efficient market hypothesis. If it were really true, then they would be out of a job. On the contrary, many quants came out of the Chicago School of Economics, or were otherwise influenced by Fama and his academic accolades, so at least pay lip service to the idea.26 From a quant survey we performed at wilmott.com, some 43 % of respondents described it as true. One way to square the circle is for quants to see themselves as enforcers of efficiency, whose job it is to drive prices to their correct level – even if that means driving them off a cliff. (We'll give our own verdict on the theory in the next chapter, but basically, Lynch is right.)

      The assumptions of neoclassical economists therefore had a dual nature. On the one hand, they were designed to make the economy mathematically tractable. It is obviously easier to model people who are selfish, have fixed preferences, and are completely rational than it is to model people who are influenced by the opinions of others, change their minds for no reason, and make puzzling and bizarre life choices. On the other hand, they shaped the way that we see and model the economy – as a beautifully rational, stable, and efficient system – which as we'll see, shaped the economy itself.

      Of course, no one – even business school lecturers – thinks that people are perfectly rational, or that markets are perfectly stable or uniform, or that models are perfect. Much work has been done exploring deviations from these assumptions. As we will see, though, the models used in finance continue to treat the world as a very rational, stable, and symmetric place – and this has as much to do with aesthetics, mathematical ego, and the desire to impress and intimidate as it does with making money. In the next chapter, we look at how these elegant but unrealistic assumptions and formulas were made to seem compatible with markets СКАЧАТЬ