Thursday 16 February 2012

The mathematical equation that caused the banks to crash (?!?)

Claiming that the Black-Scholes equation had anything to do with the Credit Crisis of 2007-2009 is a bit like claiming Daimler-Maybach were responsible for bombing Hiroshima. Sure the planes used the internal combustion engine, but the causal relationship is being stretched beyond reason. 

The claim that Black-Scholes was involved in the Credit Crisis has been made by Ian Stewart in a piece, apparently promoting his new book, Seventeen Equations that Changed the World, in The Observer.   

Prof Stewart is the most important promoter of mathematics in the United Kingdom, writing his first book in the early 1970s, Concepts of Modern Mathematics, as an exposition of Bourbaki mathematics. In the 1990s, Bourbaki and ‘New Math’ had faded/failed and Prof Stewart turned his attention tho the more pressing question of introducing mathematics into industry, as explained in the Preface to the Dover Publications edition. 

Given this track-record I am at something of a loss to understand what Prof Stewart was trying to achieve in his Observer article, beyond generating publicity for his book. It is unfortunate that he has chosen to approach a critically important issue, the use of mathematics in finance, in such an unthoughtful way as he has. 

His article is little more than a string of factually incorrect statements, my guess is they are culled from the BBC’s Midas Formula which in turn is based on Lowenstein’s When Genius Failed. For example, the piece begins with “It was the holy grail of investors.”. At its heart is the rather depressing statement, from an investors perspective, that profits should equal the riskless rate, that's more a poison chalice than a holy grail - as one student asked in lectures last week “what's the point of that then”.  

What is more, if it was an esoteric secret, why was the paper initially rejected by the Journal of Political Economy on the basis that there was not enough economics in it. Peter Bernstein, in Against the Gods reports that Black felt it was because he was a mathematician and had no qualifications in economics. The paper then went to the Review of Economics and Statistics, again without success. Bernstein reports that the paper was only published by the Journal of Political Economy after the intervention of influential Chicago academics. 

A more thoughtful assessment is that Black-Scholes enabled the CME and CBOT to justify the introduction of financial options trading, and this is touched upon. But there is no accompanying explanation of the collapse of Bretton-Woods, destroying fixed exchange rate and broadly flat bond yields, meaning that volatility suddenly became an issue for the markets. 

The article, then links Black-Scholes with the sub-prime crisis, a gross mis-representation since Black-Scholes played pretty much no (pricing) role in the collapse of LTCM, let alone in the credit crisis. What Stewart fails miserably to understand is that, in typical practice since the 1987 crash, Black-Scholes has not taken volatility as an input and produced prices, but taken prices to imply volatility. The failure is miserable, because this is the genius of the equation, a consistent measurement tool of the markets’ assessment of risk in the future. 

The piece goes on to claim “The idea behind many financial models goes back to Louis Bachelier in 1900”. Well, Prof Stewart should know better. It would be better to write that “The idea behind many mathematical models goes back to the financial mathematician, Fibonacci in 1202” or “The origins of the Black-Scholes formula lie in the Pascal–Fermat solution to the Problem of Points in 1654. The generally accepted origin of mathematical probability”. These are far more interesting points. 

The article continues
The Black-Scholes equation was based on arbitrage pricing theory, in which both drift and volatility are constant. This assumption is common in financial theory, but it is often false for real markets.
This is nonsense, arbitrage pricing is a concept, constant drift and volatility are implementations. Stewart is happy to talk about the economic-physics Black-Scholes equation but seems ignorant of the mathematical Fundamental Theorem of Asset Pricing
A market is free from arbitrage if and only if a martingale measure exists.
A market is complete and free from arbitrage if and only if a unique martingale measure exists.
Why oh why didn’t Prof Stewart talk about this, admittedly not an equation, but profound mathematics that tells us something significant about markets. 

Mathematics, in my humble opinion, is not equations, it is concerned with ideas and understanding. Understanding the Fundamental Theorem requires an appreciation that probability is not relative frequency, of epistemology (Black Swans), through the idea of completeness, and ethics, being arbitrage free is about fairness, a martingale measure is about equality. This is all in Aristotle and Aquinas, as described by the mathematician James Franklin, and all in financial mathematics, where contemporary papers include words like ‘belief’ and ‘greed’ in their title. 

Stewart finishes his article by suggesting the solution lies in the mathematics of dynamical systems and complexity and adopting analogues from ecology. The Bank of England has followed this line of thinking building on work of Lord May, the biologist and past President of the Royal Society. The problem is, Lord May’s analysis is based on the model of ‘contagion’, the banking system is an ecology through which default spreads, in the same way that mad-cow disease spreads through farms. The response is to build firewalls, quarantine zones, around banks. The Bank of England seems reluctant to develop this line of thinking, possibly because they now realise that the model is just plain wrong. A more appropriate model for banking is the internet, the Credit Crisis was a result of linkages in a network collapsing not of some invisible infectious agent spreading throughout the network. The preferred model now seems to be electricity grids (the article was apparently prompted by a discussion with Andy Haldane). So the Bank is still using a physical analogue, but a better analogue. 

Stewart would see the solution in dynamical systems and complexity, because that is an area of mathematics that the British are strong in. But, dynamical systems are typically ergodic (in the 2003 Review, “dynamical systems and complexity” was labelled “dynamical systems and ergodic theory”) , and, unfortunately, the economy is not ergodic

Mathematicians must start taking the economy seriously, and not try and shoe-horn economic problems into a framework of the mathematics they understand. Ian Stewart is extraordinarily influential in defining what mathematics is and how it can be applied. Academics should not publicly discuss topics they are not expert on, that is the realm of journalism (or blogs). In this article, Prof Stewart has misrepresented mathematics, damaging its reputation. For this he should be ashamed.

Thursday 9 February 2012

The intellectual void at the heart of the Occupy movement

The Bayesian algorithm at Amazon recommended David Graeber’s book, Debt: The First 5,000 Years with the product description
Economic history states that money replaced a bartering system, yet there isn’t any evidence to support this axiom. Anthropologist Graeber presents a stunning reversal of this conventional wisdom. For more than 5,000 years humans have used elaborate credit systems to buy and sell goods. Since the beginning of the agrarian empires, humans have been divided into debtors and creditors. Through time, virtual credit money was replaced by gold and the system as a whole went into decline. This fascinating history is told for the first time.
I tell my students, almost exclusively looking towards careers in banking or insurance, that they should take some interest in the “nature of money”, given this is the central topic of their current studies and future careers. Practising what I preach I thought I would get Graeber’s book.

The claim that “This fascinating history is told for the first time.” is publicist’s hyperbole. Within economics there has always been a debate as to the nature of money. Nicole Oresme wrote a Treatise on the Origin, Nature, Law, and Alterations of Money in the mid fourteenth century and then Copernicus wrote on On the Minting of Coin long before he wrote on the planets. In the first decades of the nineteenth century, the Bullionist Debates dominated British economics while the 1900 book The Wonderful Wizard of Oz is sometimes seen an allegory in favour of the State Theory of Money, that money is created by governments, as opposed to the Commodity Theory, that money is the most convenient commodity to facilitate exchange.

In the twentieth century it was anthropologists, like Malinowski and Mauss, who challenged the standard economic argument that money emerged out of barter.  More orthodox was  John Maynard Keynes’s A Treatise on Money , motivated by the conundrum, in the historical record, that credit existed before money. In 1985 the anthropologist Caroline Humphrey summarised the situation saying that
Barter is at once a cornerstone of modern economic theory and an ancient subject of debate about political justice, from Plato and Aristotle onwards. In both discourses, which are distinct though related, barter provides the imagined preconditions for the emergence of money …[however] No example of a barter economy, pure and simple, has ever been described, let alone the emergence from it of money; all available ethnography suggests that there never has been such a thing.1
A modern narrative of this story, though not novel, would be interesting, and Graeber’s book is full scholarship, summarising the work of other social scientists who have addressed the basic question “what is money”. It presents a strong case that the standard economic argument, that first there was barter and then money emerged as a commodity to facilitate exchange, is myth unfounded in fact. This is important since so much of contemporary monetary policy has been based on the assumption of money as a commodity.

The fact that Graeber castigates modern orthodox economics is important since it sets the benchmark against which he should be measured.

In addition, there is  a note referring to the novelist Margaret Atwood
[Atwood] then proceeds to explore the nature of our sense of economic morality …Despite the brilliance of many of its arguments, the result is a rather sad testimony to how difficult it is for the scions of the North Atlantic professional classes not to see their own characteristic ways of imagining the world as simple human nature.2
As a (middle class) novelist, Atwood is falling into the trap of subjective analysis, something the scientist, even political scientists and certainly anthropologists, should avoid.

The problem is, that when Graeber begins to consider exchange in the context of financial markets, he relies onsimilar economic and subjective assumptions.
In the case of …commercial exchange, when both parties in the transaction are only interested in the value of the goods transacted, they may well – as economists insist they should – try to seek the maximum material advantage3
Given that up to this point the book has given a series of examples of when observed behaviour does not mimic economic theory, it is striking that this one point is made based on that same economic theory without further comment. Graeber goes on to say
What marks commercial exchange is that it’s “impersonal”: who it is that is selling something to us, or buying something from us, should in principle be entirely irrelevant.4
The fact is, that when sociologists and anthropologists observe the actions of financiers, they realise that the economic theory is not, in fact, put into practice. While the market is capable of automating trade, so that decisions are based only on value, the markets emerged, evolved and exist on the basis of trust between participants, an emotions centred on personal relationships.

The sociologist Donald MacKenzie highlights this when discussing how Leo Melamed, the chairman of the Chicago Mercantile Exchange negotiated with the academic, Milton Friedman, about sponsoring a paper advocating the introduction of currency futures in the early 1970s. The speculators at the Merc, and the CBOT, were acting together in, what they believed, was the common good, to create the market, and not in their personal interest, maximising their wealth by focussing on competitive trading. This leads to the “delightful paradox”, that it seemed
the very markets in which Homo economicus, the rational egoist, appears to thrive cannot be created (if they require the solution of collective action problems, as in Chicago) by Homo economicus.5
The belief that markets are about maximising wealth, rather than creating networks, comes out of political philosophy, emerging in early Victorian Britain with the liberal philosopher John Stuart Mill arguing that economics
is concerned with [man] solely as a being who desires to possess wealth, and who is capable of judging the comparative efficacy of means for obtaining that end.6
Around the same time, the poet-, Alfred, Lord Tennyson, wrote about nature “red in tooth and claw”. In 1859 Darwin published the Origin of the Species which explained evolution in terms of natural selection. In the popular perception, nature became seen as being driven by a bitter struggle for survival, un-regulated by the ethics of divine architect. It was the leading economist at Cambridge University of the time, Alfred Marshall, who would synthesise Mill’s approach to economics with Darwinian metaphors7 in the late nineteenth century. In the twentieth, Friedman defined ‘positive economics’ as being disconnected from morality and his views have been summarised as
Any deviation from that single–minded pursuit of profit–maximisation by the admission of some other social responsibility is “fundamentally subversive”, “pure and unadulterated socialism”, something which could “thoroughly undermine the very foundations of our free society.” Businessmen subjected to “a social responsibility other than making maximum profits for stockholders” cannot know what interests to serve.8
While this is the view from broader society, it is not natural in finance. This point is captured in a key case in the English courts in 1950, Buttle v Saunders. Saunders managed a trust for Buttle, and had agreed to sell a piece of land owned by the trust for £6,142 to a Mrs Simpson. Before the transaction had become legally binding, another person offered the trust £6,400 for the land. However, Saunders believed “my word is my bond” and declined the higher offer in favour of the original agreement with Mrs Simpson. The beneficiary of the trust, Buttle, took Saunders to court, and the court ruled in favour of Buttle
The only consideration which was present to [the trustees] minds was that they had gone so far in the negotiations with Mrs Simpson that they could not properly, from the point of view of commercial morality, resile from those negotiations.
‘Commercial morality’ was not a valid consideration and the sale to Mrs Simpson was declared null and English home-buyers could never again be certain that a purchase would be completed, ‘gazumping’ had arrived, not at the instigation of the financial adviser but on the insistence of a judge.

The scientist Graeber has fallen into the trap that he criticises the novelist Atwood of having succumbed to; he is imagining financiers rather than studying them. This is significant because it means his whole thesis is based on an assumption of what finance is about, this assumption is based on what academic economists imagine what it is about, rather than what the actual behaviour of bankers tells us.

Googling Graeber I discover that he is a central figure in the “Occupy” movement and is regarded as providing an intellectual justification for the physical manifestation of the movement.  However, I view this justification as being, itself, based on a false axiom/assumption, and that is the assumption about the nature of finance.

While we can acknowledge and accept the point the FCIC makes in concluding that
there was a systemic breakdown in accountability and ethics. The integrity of our financial markets and the public’s trust in those markets are essential to the economic well–being of our nation. The soundness and the sustained prosperity of the financial system and our economy rely on the notions of fair dealing, responsibility, and transparency.9
The critical question is whether the lack of ethics the FCIC report is endogenous to the markets, and possibly incompatible with them, as an Occupy protester may argue, or whether the natural ethics of the market have been expunged by a series of commentators external to the markets, from Mill to Friedman. The point is, coming to this question with anarcho-communist preconceptions is not going to help a serious analysis.

That said, Graeber does acknowledge that finance has not always been as it is now. He remarks that the Medieval understanding of finace was concerned with maintaining social relations. This leads me to suggest that there is a blind-spot for many of the commentators on finance, and that is in regard to the problem of randomness.

Mill, Marx, Darwin and Dickens were all contemporaries living at a time when science was attempting to relegate randomness, chance, to history. In the 1830s English law criminalised most forms of gambling, the consequence was the difficulty Melamed had in creating a foreign exchange futures contract in 1970. Marxism proved popular not with the industrial proletariat, but in agrarian economies of Russia, China and Cuba, societies exposed to random climatic changes. Richard Dawkins is often heard stating that “evolution is not a random process”. Novels of the nineteenth century repeatedly use the device of a character being ruined by reckless speculation or feckless gambling.

The economic theory that developed at this time followed the dominant cultural direction. Laplace had advised that mechanics should be confined to the physical science, while the social sciences should employ probability to manage uncertainty10, however, economists ignored this advice and built their science on deterministic mechanics11.

The point is, earlier generations had accepted chance as an integral part of life, in particular economic life. Scholastic analysis of finance, which Graeber commends, was based on a distinction between what would happen with certainty and what was subject to chance.  The pioneers of mathematical probability, Pascal, Fermat, Huygens and J. Bernoulli all came to the topic trying to understand finance. 

Further back in time, gambling is an almost universal feature of primitive society. For the Greeks, the brothers Zeus, Poseidon and Hades cast lots to divide up the universe, Zeus winning the sky, Poseidon the sea and Hades the underworld. Hindus believe the world was a game of dice played between Shiva and his wife, while at the heart of the epic tale of the Mahabharata is an, unfair, dice game between the Kauravas and the Pandavas.12

Contemporary anthropologists recognise that gambling plays a fundamental role in contemporary neolithic communities. Consider a case observed in an Australian aboriginal group, the Momega in a remote area of Arnhem Land around 1980. The community had access to social security payments and there was often a surplus left over after essentials had been bought. As the anthropologist, Jon Altman, studying the group observed
this surplus was not equally bestowed. …This variability in bestowal was extremely arbitrary and it resulted in inter–household variability in access to cash.13
This variability can seen as subjective discrimination of the community by the Australian government. Gambling, according to Altman, “acted effectively to both redistribute cash …[it] provided a means for people with no cash income to gain cash”14 and from a small stake a larger cash reserve could be generated. The random distribution created by gambling, while not uniform, some would lose a lot, some win a lot, was none the less objective and most people ended up with a fair share of the cash this was important in a non-hierarchical community because it meant that the arbitrary bestowal of money was not corrected by another subjective distribution, such as redistribution by a chief.

Another anthropologist, William Mitchell considered the role that gambling plays in disrupting hierarchical social structures, such as the Indian caste system, by studying the Wape in New Guinea around the same time
An important task of Dumont’s classic study of Indian caste was to demonstrate how inequality is maintained. My task is the obverse, that is, to reveal how the Wape defeat the formidable principle of hierarchy to maintain male equality. How do the Wape, who, as individuals, desire wealth and who, since the 1930s, have been directly tied to a world capitalist market system, prevent wealth from being successfully manipulated by a few men to raise themselves above others? The paradoxical answer is deceptively simple: through gambling.15
This is an explanation for the pervasive nature of gambling in neolithic communities, appearing in the Vedic scriptures, potlach ceremonies of North America, and in aboriginal Australia and New Guinea and the Hazda16; it is an objective, fair, mechanism for the redistribution of wealth.

We can criticise Graeber’s methodology and his reliance on unsupportable assumptions, but the real failure of the Occupy movement is their inability to appreciate the positive aspects of gambling and speculation, central to the markets.Perhaps they should read more anthropology and less Dickens.


1 Humphrey (1985, p 48)
2 Graeber (2011, p 404, n5)
3 Graeber (2011, p 103)
4 Graeber (2011, p 103)
5 MacKenzie (2008, p 151)
6 Persky (1995, quoting Mill, p 223)
7 Backhouse (1985, 10.1), Thomas (1991)
8 Watchman (2001, p 27)
9 FCIC (2011, p xxii)
10 Katz (1993, p 685)
12 Sahlins (2003, p 27), Brenner and Brenner (1990, p 1–5)
13 Altman (1985, p 56)
14 Altman (1985, pp 60-61)
16 Sahlins (2003, p 27)


    Altman, J. (1985). Gambling as a mode of redistributing and accumulating cash among aborigines: a case study from Arnhem Land. In Caldwell, G., Dickerson, M., Haig, B., and Sylvan, L., editors, Gambling in Australia, pages 50–67. Croom Helm.
    Backhouse, R. (1985). A History of Modern Economic Analysis. Blackwell.
    Brenner, R. and Brenner, G. A. (1990). Gambling and Speculation: A theory, a history and a future of some human decisions. Cambridge University Press.
    FCIC (2011). The Financial Crisis Inquiry Report. Technical report, The National Commission on the Causes of the Financial and Economic Crisis in the United States.
    Graeber, D. (2011). Debt: The first 5,000 years. Melville House.
    Humphrey, C. (1985). Barter and economic disintegration. Man, 20(1).
    Katz, V. J. (1993). A History of Mathematics: an Introduction. Harper Collins.
    MacKenzie, D. (2008). An Engine, Not a Camera: How Financial Models Shape Markets. The MIT Press.
    Ménard, C. (1987). Why was there no Probabilistic Revolution in economic thought? In Kruger, L., Gigerenzer, G., and Morgan, M. S., editors, The Probabilistic Revolution: Volume 2: Ideas in the Sciences. MIT Press.
    Mitchell, W. E. (1988). The defeat of hierarchy: Gambling as exchange in a Sepik society. American Ethnologist, 15(4).
    Persky, J. (1995). Retrospectives: The ethology of Homo economicus. The Journal of Economic Perspectives, 9(2):221–231.
    Sahlins, M. (1972 (2003)). Stone Age Economics. (Routledge).
    Thomas, B. (1991). Alfred Marshall on economic biology. Journal of Financial Intermediation, 3(1):1–14.
    Watchman, P. (2001). A legal framework for the integration of environmental, social and governance issues into institutional investment. Technical report, UNEP Finance Initiative/Freshfields Bruckhaus Deringer. .