**TOPICS:** Theory, security analysis and valuation

Mitchell Waldrop’s book *Complexity* [1992] describes a revealing anecdote on economics and mathematics from the 1987 Global Economic Workshop. Held at the Santa Fe Institute in New Mexico following the 1980s Latin American debt crisis, the workshop brought together economists, financial theorists, and physicists, mainly from the Los Alamos National Laboratories.

After listening to presentations made by economists—presentations full of complex mathematics—one of the physicists asked Kenneth Arrow why economists use such sophisticated mathematics, given that they have scarce supporting data. According to Waldrop, who attended the meeting, Arrow responded that in fact we use sophisticated mathematics precisely *because* we do not have data: We want to make sure that our reasoning is perfectly correct. This anecdote is revealing of the direction taken by economics and finance theory over the last four decades. (For brevity, we will generically discuss economics, with the understanding that our remarks apply also to finance theory.)

Economics has become a theory that describes an idealized economic and financial reality without any strong connection to data. It is revealing, for example, that financial theorists describe the real behavior of prices as “price anomalies” when prices do not follow the theoretical rules of idealized markets. As we wrote in the Spring 2012 issue of this journal (“What’s Wrong with Today’s Economics?”), “Mainstream economics describes how markets should process information in a rational way. Deviations from the perfect rational behavior are described as defects.”

It is often argued that mainstream economics describes a mathematical idealization of the real world and that the use of idealizations is a legitimate process of science. But in the same article, we raise the objection that “the idealizations used in physics are idealizations of the empirical settings, but not of the basic laws…. Economics, however, studies a complex artefact: the economy or financial markets…. Idealizations change the laws, not only the empirical setting” (Focardi and Fabozzi [2012]). This is something very different.

In view of the above and with the realization that the predictive success of mainstream economics is marginal at best—to the point that a major event such as the financial markets crash of 2008 and the ensuing Great Recession were neither predicted nor explained a posteriori by the equilibrium framework of mainstream economic—one can reasonably ask if the extensive use of mathematics in economics and finance theory is justified. Is the marriage between mathematics and economics on the brink of divorce?

Let’s first observe that the term “mathematics” is too broad to be meaningful in this context. Mathematics is essentially logical deductive reasoning. There are many different types of mathematics: some of them, such as calculus, deal with quantities; others, such as probability and statistics, deal with uncertainty; still others deal with abstract concepts such as shapes and operations. Computer science, for example, is a discipline closely related to mathematics. But the mathematics of computer science is not calculus; it is primarily the mathematics of logical operations.

Economics has chosen calculus as its key mathematical tool and has added the handling of uncertainty (that is, risk) through stochastic calculus. Note that stochastic calculus can be cast in the framework of deterministic calculus by representing the evolution of probability distributions. For example, Itô processes admit a dual representation through Fokker-Planck equations that describe diffusions. Due to this duality, we will refer simply to calculus.

There are many reasons for the choice of calculus as the key mathematical tool in economics: It proved to be extremely successful in physics and engineering, where it acquired its track record. Economists hoped to be able to repeat the extraordinary success obtained by the use of calculus in physics by adopting the same conceptual framework. And the amount of cumulated knowledge in applying calculus to real problems is impressive. But ultimately, calculus has not proved to be well adapted to describing economic and financial phenomena.

There are several explanations for the unsuitability of calculus in economics. Though economic quantities such as prices or economic output can be reasonably well approximated by continuous variables, the interactions between variables and their dynamics is not well described either by differential equations or by stochastic differential equations. The reason is that the basic drivers of economic dynamics are human actions, which are essentially discrete in nature. In order to represent human actions through calculus, economists had to make recourse to the notion of agents who optimize an abstract utility function. But utility is a quantity that has no empirical content.

One might raise the objection that in aggregate, human decisions average out and can be effectively represented through a formalism of a utility-maximizing representative agent. But the theory of complexity, the key topic of the 1987 Global Economic Workshop at The Santa Fe Institute mentioned above, proved this concept wrong. Agents have mutual interactions and do not smoothly aggregate. For example, phenomena of aggregation in complex networks of financial relationships might lead to financial contagion effects that cannot be easily explained within the mathematical framework of calculus. Economic agents make discrete decisions that in aggregate might lead to fundamental instabilities. And when we try to represent agents’ interactions, nonlinearities are always present—leading to intractable chaotic models. The mathematics of *complexity* (not calculus) is called for.

Another important aspect of both economies and financial markets that is not easily represented in the continuous framework of calculus is the extremely high level of uncertainty predominant in some financial phenomena. Over short periods of time, unpredictable fluctuations of variables such as returns might be one or two orders of magnitude larger than their smooth trends. And the building up of instabilities creates uncertainty regarding fundamental trends—that is, it might lead to large market swings and/or economic downturns.

This level of uncertainty, combined with the small size of available samples, makes estimation very challenging. Essentially, it is very difficult, if not outright impossible, to choose one’s models. Does it really make sense to use the complicated machinery of stochastic calculus when uncertainty is so large? For example, many economists argued that the 2008 financial markets crash was simply a possible random outcome of current theories. If correct, this argument casts doubts on the usefulness of current models.

These considerations do not imply that we have to give up mathematics in economic reasoning and adopt a framework of qualitative reasoning typical of the social sciences. The key point is this: In economics, we have to adopt a mathematical and logical framework suitable for the data and the phenomena that we have and that needs to be analyzed.

Many other disciplines went through the same process of choosing a conceptual framework. Biology and linguistics are typical examples. It would make little sense to study DNA sequencing with the tools of calculus. We can even say that many disciplines have developed their own mathematics: linguistics, for example, has developed a mathematical theory of languages.

Economics and finance theory need the same type of creative thinking. Arguably, there are areas in which the framework of calculus and stochastic calculus is applicable. One such area is the analysis of financial derivatives. These instruments are artificial assets whose value is prescribed by essentially the same models that analyze them. In some sense, the world of financial derivatives is a self-referential world—where all participants adopt similar models to represent the underlying instruments and to price them. However, when we need to analyze the real financial world, the usual tools of calculus and stochastic calculus are much less useful. The study of truly complex phenomena in the last 40 years has shown that the paradigm of economic equilibrium with a representative utility-maximizing agent is a poor idealization for real-world economies.

What is called for is the representation of the interactions between real individual agents and the representation of situations of instability. We need to take into account the interactions between the financial and the real aspects of the economy. Somehow, we need to represent cultural phenomena that have a critical influence on patterns of spending and consumption. And we need to analyze the formation and workings of the power structures present in real economies.

What type of mathematics is called for? What do we need to teach at universities to educate students interested in becoming professionals in the finance industry? This is an open question that the economics and finance departments at universities must have a conversation about. At this stage, we can say that economics is a science in the making: Essentially, we need to rebuild economics as an empirical science. Some results have been obtained. Network theory has made significant progress in representing interactions among economic agents. Chaotic models and their relationships to statistics are now better understood. A new kind of statistics may be needed to work with the level of uncertainty that characterizes economics and finance.

Currently, we make a distinction between robust statistics for the bulk of the data and extreme value theory (EVT) to model the tails. We have learned how to make rough predictions of possibly very large outcomes never experienced in the past. But we do not have tools to deal with very high levels of uncertainty. For example, what is the meaning of statistics when estimation is essentially nonsignificant?

Most estimates made in economics would become nonsignificant if we were to take into account all required parameters. Currently, to solve this problem, we constrain models to become sufficiently simple to allow their estimation. But we might need some conceptual breakthrough to deal with highly uncertain phenomena.

Typically, university curricula in economics and finance are divided today: There are those programs with mathematics and those programs without mathematics. Those with mathematics teach sophisticated calculus and stochastic calculus; those without mathematics still feel that it will be necessary and try to teach diluted and simplified versions of calculus and stochastic calculus. This situation is arguably unsatisfactory. Students of highly mathematical curricula end up feeling they are in a kind of ivory tower and do not develop the hard data discipline of the empirical sciences. In contrast, students of non-mathematical curricula end up feeling that logic and mathematics are optional and will not be used in real life.

Both positions are unreasonable. In the practice of investment management, highly sophisticated calculus is used primarily in the financial derivatives business. Today, students who want to become “quants” need to become familiar with calculus and stochastic calculus. But these students should keep in mind that the evolution of modern economies will likely require new, possibly different, mathematical concepts. They should keep a very open mind to new ideas.

But the opposite position that mathematics is a useless option is also very dangerous. Investment management requires rigorous logical thinking. The often-cited opposition of intuition and mathematics is profoundly wrong and might have pernicious consequences on the investment process.

Humans have intuition, in the sense that they develop beliefs that they would be unable to justify through a mathematical process. Although intuition is a fundamental component of human thinking, it cannot be the only source of knowledge; it is actually frequently wrong. Intuition must be combined with data and with the ability to reason about data. Students must learn how to reason logically—and how to test (the products of) their intuition.

The challenge to universities and business schools is to teach students to reason rigorously without constraining the teaching in a straightjacket of calculus. Teaching nonrigorous mathematics encourages sloppy reasoning; reasoning must be logical and rigorous, even if we have to be content with an approximate view of economic reality.

The marriage between economics and mathematics can (and must) be preserved, but much has to change. We advocate a serious reflection on the use of mathematics in university curricula, and in the industry. On one hand, it should now be clear that mathematics other than calculus will likely be needed in economics and in the management of investments. But it should also be clear that students (and practitioners) must develop the habit of employing rigorous logical thinking, regardless of the level of approximation used in modeling reality.

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