16 August 2013

Physics, Financial Models, and Predicting the Unpredictable

This essay is principally a review of the book The Physics of Wall Street: A Brief History of Predicting the Unpredictable, 2012, James Owen Weatherall. The first part of the review summarises the key individuals from physics that have made a contribution to financial economics. I then touch briefly on a negative review of Weatherall’s book, noting the complete omission of the important contributions made by non-physicists. The review then turns to a controversial topic – whether models can or should be used when there are extreme and seemingly unpredictable events (“Black Swans”). The thesis of the book is that the assumptions underlying models need to be understood, the models need to be regularly tested, and that better models can be built. Rupture models have been empirically shown to predict when crises will occur, turning the unpredictable Black Swans into predictable “Dragon Kings”). I conclude with some personal comments.

Selected Contributions by Physicists to Financial Economics

In The Physics of Wall Street: A Brief History of Predicting the Unpredictable, 2012, James Owen Weatherall provides a very interesting, if selective, history of the contribution of physicists to quantitative finance. Key contributors covered in the book are:
  • Louis Bachelier – a French physicist who was forced by circumstances to find work at the Paris Bourse and identified that prices are subject to a random walk and are normally distributed. Bachelier used his analysis to calculate the price of options.
  • Maury Osborne – who agreed that markets are random, subject to “Brownian motion”, but it is returns rather than prices that are normally distributed, which means that prices are log-normally distributed. Osborne seems to have considered that there was no way to profit from the stock market and did not extend his work to option pricing until the late 1970s. (p. 45) Osborne also explored the phenomena of mean-reversion or conditional probability in stock prices, where from moment to moment a stock or the market is more likely to reverse itself than continue in a given direction.
  • Paul Samuelson – although Samuelson is a Nobel prize winning economist in his own right,1 Samuelson’s role in this account is his re-discovery of Bachelier’s work.
  • Szolem Mandelbrot – financial markets are complex systems exhibiting wild randomness and fractal patterns, and this wild randomness means that distributions have much fatter tails than the normal distribution, meaning that extreme events occur much more often than these distributions predict. This in turn means that options priced using a based on the normal distribution will not correctly price in extreme events. Mandelbrot argued that rather than being “normal”, distributions are “Lévy-stable” distributions, of which the normal distribution is one extreme and the Cauchy distribution is another extreme.2 Current research indicates that although distributions of financial market returns exhibit fat tails, they are not Lévy-stable (p. 74).
  • John Kelly – who established the “Kelly criterion” for betting, that the proportion of your capital that should be “invested” in a bet is equal to the ratio of your advantage (due to superior information) to the payout.3
  • Edward Thorp – responsible for important advances in information theory. In conjunction with Professor Claude Shannon, the father of information theory, and utilising the Kelly criterion, Thorp devised a scheme for beating casinos at blackjack and roulette, and demonstrated the success of these schemes. Thorp then turned his attention to the stock market and warrants (a form of option). Thorp was able to calculate the “true” price for a warrant using Bachelier’s approach and Osborne’s observation of log-normal prices. As with gambling, Thorp then applied the Kelly criterion to decide how much of his funds to invest. More importantly, Thorp essentially invented delta hedging: the right mix of warrants and stocks can provide a guaranteed profit unless the stock price moves dramatically.
  • Fisher Black – in conjunction with Myron Scholes, the development of the Black Scholes option pricing model (more accurately, derivation of the option pricing model using a new approach). The Black Scholes model relies on the efficient market hypothesis and the assumption that returns are normally distributed. While equivalent to the approach developed by Thorp, the Black-Scholes derivation opened the way for dynamic hedging – the sale of options and purchase of other assets in a way that is, provided certain conditions hold, risk-free. Weatherall argues that this gave the tools that investment banks to manufacture options without (knowingly) taking on huge amounts of risk.
  • Didier Sornette – the application of rupture / catastrophic failure models to predicting market crashes. I discuss this in considerable more detail below.
  • Eric Weinstein and Pia Malaney – the application of gauge theory to the index number problems such as the derivation of a better measure of inflation.

Reviewer Comments and the Contributions of Non-Physicists

Weatherall’s book has received a large number of reviews, both positive and negative. One of the most negative is Aaron Brown, who over the course of a lengthy 18 part series takes exception to many statements made in the book. For example, he notes that Weatherall attributes all advances in modern financial economics to physicists, when pure economists have also made important contributions. Brown comments:4

The problem for The Physics of Wall Street is few physicists were involved in the process at all until the latter years of the 1980s… . Given that it took a few years to gain much influence, it’s fair to say that the role of physicists was largely restricted to the period of expansion [from 1995 to 2010]. This was not a period of major innovation, the basic ideas, tools and institutions were in place by 1995. And even [during this] period, physicists were not over-represented compared to people in other quantitative fields, nor were ideas imported from physics particularly important.

Some of this “glossing over” of the contributions of others is inherent to telling the story that Weatherall is trying to tell, which is really the contribution that physics has made to financial economics. But when acknowledging that techniques and models have been developed for pricing options that better allow for fat-tailed distributions and extreme events, Weatherall could have gone further and explicitly recognised some of the contributors. For example, Robert Merton receives mention as a collaborator in the Black-Scholes model, but he provided an important contribution to adjusting the model for non-normal price behaviour: in 1976 he extended the Black-Scholes model to account for “jumps” in prices.5

Weatherall mentions in passing in the epilogue that Black acknowledged the potential problems with the simplifying assumptions in the Black-Scholes model, in 1988 authoring an article “The Holes in Black-Scholes” in Risk magazine.6 In that article Black discusses a range of issues that will result in the Black-Scholes model mis-pricing options: volatility changes, interest rate changes, borrowing penalties, short-selling penalties, transaction costs, taxes, dividends, and takeovers. Certainly Black was well aware of the short comings of the model, and these would have been well understood within the investment community.

Black Swans and Dragon Kings: Predicting the Unpredictable

Although the investment community clearly had some understanding of the problems of the models, it seems that they were largely unprepared for large-scale correlated events, such as the cascading defaults that led to the sub-prime mortgage crisis in 2008.7 This was an error of assuming that defaults would always remain independent (and hence small scale) and potentially not taking the time to think through the structure of the relevant parts of the financial system.

The crash provided fertile ground for Nassim Taleb who became a celebrity with his book “The Black Swan: The Impact of the Highly Improbable”, 2007. Black Swan events are high-profile, hard-to-predict, and rare events that are beyond the realm of normal expectations in history, science, finance, and technology. By their very nature, Black Swan events have a disproportionately large impact on society and history.8 The essential thesis of Taleb’s book is that Black Swan events occur more frequently than we commonly think, have a very large effect and consequence, and cannot be predicted. In particular, Taleb has championed the view that because Black Swan events cannot be predicted by traditional models (essentially those assuming normally distributed returns) those models should essentially be discarded. For a good review of Taleb see the 2009 review by David Aldous.9

It is very clear that Weatherall has no time for Taleb’s conclusion that models should be discarded:
… the process of building and revising models that I have described here is the basic methodology underlying all of science and engineering. It’s the best tool we have for understanding the world. We use mathematical models cut from the same cloth to build bridges and to design airplane engines, to plan the electric grid and to launch spacecraft. What does it mean to say that the methodology behind these models is flawed – that since it cannot be used to predict everything that could ever happen, it should be abandoned altogether? If Taleb is right about mathematical models, then you should never drive over the George Washington Bridge or the Hoover Dam. After all, at any moment an unprecedented earthquake could occur that the bridge builder’s models didn’t account for, and the bridge could collapse under the weight of the cars. You should never build a skyscraper because it might be hit by a meteor. Don’t fly in an airplane, lest a black swan collide with one of its engines.
J Weatherall, The Physics of Wall Street, p. 215

Indeed, part of the very reason for The Physics of Wall Street is to counter the message of Taleb and others who decry the use of sophisticated mathematical models (for example, Wall Street Journal reporter Scott Patterson in his book The Quants10). Weatherall argues that models should be used with full knowledge of the underlying assumptions and the circumstances in which those assumptions might not hold, and that models should continually be tested to determine whether they are performing as intended.

The part that I found most fascinating in The Physics of Wall Street was the discussion of Sornette’s work, from the application of rupture models to the Kevlar pressure vessels used in Ariane space rockets, to earthquakes, and then to financial markets. In contrast to Taleb’s “Black Swans”, Sornette calls these events “Dragon Kings”, and has proven that at least some of these events can be predicted. Sornette had sufficient faith in the predictions generated by his analysis that he bought cheap out-of-the-money put options and profited from the October 1997 market crash. Importantly, Sornette turns Taleb’s thesis on its head and, while acknowledging that there are extreme events that are completely outside what can be captured within financial models, other models can be built to identify when crashes are likely to occur. These models essentially identify when conditions within the financial system are aligning so that a perturbation will precipitate a crisis. Knowing this, regulators, policy makers, and even financial market participants may be able to intervene or at least find ways to mitigate the effects of pending crises.

As an interesting aside, when I visited Taleb’s website, he now also refers to a "Mandelbrotian Grey Swan", which he defines as:11
Black Swans that we can somewhat take into account – earthquakes, blockbuster books, stock market crashes – but for which it is not possible to completely figure out their properties and produce precise calculations.

One is tempted to conclude that the work of Sornette and others has had such a marked impact that Taleb has had to invent this new classification so that the essence of his thesis remains unchanged.

For those that want to read more of Sornette’s work, the following 98 page paper gives comprehensive coverage:
Didier Sornette, “Critical Market Crashes”, Physics Reports 378(2003):1-98.

A shorter, more accessible paper is
Didier Sornette, “Dragon Kings, Black Swans and the Prediction of Crises”, International Journal of Terraspace Science and Engineering 2(1): 1-18

A more recent paper summarising the work on explaining and predicting Dragon Kings is
Didier Sornette and Guy Ouillon, “Dragon Kings: Mechanisms, statistical methods and empirical evidence”, European Physics Journal Special Topics, 205(2012):1-26

Didier also gave a TED talk on the same topic here or here with some supporting commentary.


I found that Weatherall’s book resonated with me. It was not intended to be a history of the development of financial modelling; it was intended to be a selected history of how physicists (and mathematicians with some physics background) have contributed to financial economics. I came to economics not with a PhD in physics, but with a strong enough background in mathematics and physics that I was granted direct entry into a second year physics paper requiring advanced mathematical skills. My Master’s thesis was in an area of economics other than financial markets, but did include substantial mathematical modelling, and the matrix algebra was solved using a computer programme I wrote myself.

I also have an interest in economic history, so the mix of physics, mathematical modelling, and economic history was a fascinating mix of topics for me. That it might be selective is the nature of any historical work; the real contribution of this book is to advance the position that advanced mathematical modelling has more to offer to finance and economics, and that ideas from other mathematically-based disciplines such as physics may be able to provide new ways of understanding economic problems.

[1] For more on Paul Samuelson see Wikipedia article http://en.wikipedia.org/wiki/Paul_Samuelson
[2] For more on Lévy-stable distributions see the article "Stable Distribution", Wikipedia http://en.wikipedia.org/wiki/Stable_distribution
[3] See "Kelly criterion", Wikipedia, http://en.wikipedia.org/wiki/Kelly_criterion
[4] http://www.minyanville.com/business-news/editors-pick/articles/The-Physics-of-Wall-Street-physics/12/17/2012/id/46719?page=full#ixzz2azZI5RoO
[5] Robert C. Merton “Option Pricing when Underlying Stock Returns are Discontinuous”, Journal of Financial Economics 3(1976):125-144. Available online at http://www.people.hbs.edu/rmerton/optionpricingwhenunderlingstock.pdf
[6] Fisher Black, “The Holes in Black-Scholes”, Risk magazine, available online at http://www.risk.net/digital_assets/5955/The_holes_in_Black-Scholes.pdf
[7] See, for example, http://en.wikipedia.org/wiki/Subprime_mortgage_crisis
[8] "Black Swan Theory", Wikipedia, http://en.wikipedia.org/wiki/Black_swan_theory
[9] David Aldous, http://www.stat.berkeley.edu/~aldous/157/Books/taleb.html
[10] Scott Patterson, The Quants, is given a non-critical review in the Wall Street Journal at http://online.wsj.com/article/SB10001424052748704509704575019032416477138.html
[11] http://www.fooledbyrandomness.com/glossary.pdf