Magic, maths and money

Why don't more mathematicians see the potential of economics

2012-01-17T01:09:00.000-08:00

The question is, how did economics change its attitude to mathematicsin the forty years between Håvelmo’s The Probability Approach inEconometrics and his Nobel Prize in 1989, when he was pessimistic aboutthe impact the development of econometrics had had on the practice ofeconomics. Coinciding with Håvelmo’s pessimism, many economists werereacting strongly against the ‘mathematisation’ of economics, evidencedby the fact that before 1925, only around 5% of economics researchpapers were based on mathematics, but by 1944, the year of Havelmoand von Neumann-Morgenstern’s contributions, this had quintupled to25%

^{1}

.While the proportion of economics papers being based on maths has notcontinued this trajectory, the inﬂuence of mathematical economics has and theperson most closely associated with this change in economic practice was PaulSamuelson.

Samuelson is widely regarded as the most inﬂuential economist to come out ofthe United States and is possibly the most inﬂuential post-war economist in theworld. He was the ﬁrst U.S. citizen to be awarded the Nobel Prize in Economics in1970 because “more than any other contemporary economist, he has contributedto raising the general analytical and methodological level in economicscience”

^{2}

.He studied at the University of Chicago and then Harvard, were he obtained hisdoctorate in 1941. In 1940 he was appointed to the economics department of M.I.T.,in the ﬁnal years of the war he worked in Wiener’s group looking at gun controlproblems

^{3}

,where he would remain for the rest of his life. Samuelson would comment that “Iwas vaccinated early to understand that economics and physics could share thesame formal mathematical theorems”.

In 1947 Samuelson published Foundations of Economic Analysis, which laidout the mathematics Samuelson felt was needed to understand economics. It issaid that von Neumann was invited to write a review Foundations in1947 declined because “one would think the book about contemporarywith Newton”. Von Neumann, like many mathematicians who looked ateconomics, believed economics needed better maths than it was beingoﬀered

^{4}

.In 1948 Samuelson published the ﬁrst edition of his most famous work,Economics: An Introductory Analysis, one of the most inﬂuential textbooks oneconomics ever published, it has run into nineteen editions and sold over four million copies.

There appears to be a contradiction, Håvelmo seems to think his introductionof mathematics into economics was a failure, while Samuelson’s status seems tosuggest mathematics came to dominate economics. In the face of contradiction,science should look for distinction.

I think the clue is in Samuelson’s attachment to “formal mathematicaltheorems”, and that his conception of mathematics was very diﬀerent from that ofthe earlier generation of mathematicians that included everyone from Newton andPoincaré to von Neumann, Wiener and Kolmogorov.

A potted history of the philosophy of mathematics is that the numerologistPlato came up with the Theory of Forms and then Euclid produced The Elementswhich was supposed to capture the indubitability, the certainty, and immutability,the permanence, of mathematics on the basis that mathematical objects whereReal representations of Forms. This was used by St Augustine of Hippo asevidence for the indubitability and immutability of God, embedding into westernEuropean culture the indubitability and immutability of mathematics. Theidentiﬁcation of non-Euclidean geometries in the nineteenth century destroyed thisediﬁce and the reaction was the attempt to lay the Foundations of Mathematics,not on the basis of geometry but on the logic of the natural numbers.Frege’s logicist attempt collapsed with Russell’s paradox and attentionturned to Hilbert’s formalism to provide a non-Platonic foundation formathematics. The key idea behind Formalism is that, unlike PlatonicRealism, mathematical objects have no meaning outside mathematics, thediscipline is a game played with symbols that have no relevance to humanexperience.

The Platonist, Kurt Gödel, according to von Neumann, has “shown thatHilbert’s program is essentially hopeless” and

The very concept of “absolute” mathematical rigour is not immutable. The variability of the concept of rigour shows that something else besides mathematical abstraction must enter into the makeup of mathematics $^{5}$

Mathematics split into two broad streams. Applied mathematics,practised by the likes of von Neumann and Turing, respondedby focussing on real-world ‘special cases’, such as modelling the brain

^{6}

.Pure mathematics took the opposite approach, emphasising the generalisation ofspecial cases, as practised by Bourbaki and Hilbert’s heirs.

Formalism began to dominate mathematics in the 1940s-1950s. Mathematicswas about ‘rigorous’, whatever that means, deduction from axioms and deﬁnitionsto theorems. Explanatory, natural, language and, possibly worse, pictures, were to beremoved from mathematics. The “new math” program of the 1960s was aconsequence of this Formalist-Bourbaki dominance of mathematics.

It is diﬃcult to give a deﬁnitive explanation for why Formalism becamedominant, but it is often associated with the emergence of logical–positivism, asomewhat incoherent synthesis of Mach’s desire to base science only onphenomena (which rejected the atom), mathematical deduction and Comte’sviews on the unity of the physical and social sciences. Logical-positivismdominated western science after the Second World War, spreading out from itsheart in central European physics, carried by refugees from Nazism.

The consequences of Formalism were felt most keenly in physics. RichardFeynman, the physicists’ favourite physicist, hated its abandonment of relevance.Murray Gell-Mann, another Noble Laureate physicist, commented in 1992 thatthe Formalist-Bourbaki era seemed to be over

abstract mathematics reached out in so many directions and became so seemingly abstruse that it appeared to have left physics far behind, so that among all the new structures being explored by mathematicians, the fraction that would even be of any interest to science would be so small as not to make it worth the time of a scientist to study them.

But all that has changed in the last decade or two. It has turned out that the apparent divergence of pure mathematics from science was partly an illusion produced by obscurantist, ultra-rigorous language used by mathematicians, especially those of a Bourbaki persuasion, and their reluctance to write up non–trivial examples in explicit detail. When demystiﬁed, large chunks of modern mathematics turn out to be connected with physics and other sciences, and these chunks are mostly in or near the most prestigious parts of mathematics, such as diﬀerential topology, where geometry, algebra and analysis come together. Pure mathematics and science are ﬁnally being reunited and mercifully, the Bourbaki plague is dying out. $^{7}$

Economics has always doubted its credentials. Laplace saw the physicalsciences resting on calculus, while the social sciences would rest onprobability

^{8}

,but classical economists, like Walras, Jevons and Menger, wanted their emergingdiscipline economics to have the same status as Newton’s physics, and somimicked physics. Samuelson was looking to do essentially the same thing,economics would be indubitable and immutable if it looked like Formalistmathematics, and in this respect he has been successful, the status of economics has grown faster than the growth of maths in economics. However, while thegeneral status of economics has exploded, its usefulness to most users ofeconomics, such as those in the ﬁnancial markets, has collapsed. Tradingﬂoors are recruiting engineers and physicists, who always looked for therelevance of mathematics, in preference to economists (or post-graduatemathematicians).

My answer to the question “why don’t more economists see the potential ofmathematics” is both simple and complex. Economists have, in the main, beenlooking at a peculiar manifestation of mathematics - Formalist-Bourbakimathematics - a type of mathematics that emerged in the 1920s in response to an intellectualcrisis in the Foundations of Mathematics. Economists have either embraced it, asSamuelson did, or were repulsed by it, as Friedman was.

Why this type ofmathematics, a type of maths that would have been alien to thegreat mathematicians of the twentieth century like Wiener, von Neumann,Kolmogorov and Turing, became dominant and was adopted by economics is more complex and possibly inexplicable. The questionis, can academic mathematics return to its roots in relevance, or will it wither in its ivory towers?

Notes

^{1}

Mirowski (1991, pp 150–151)

^{2}

The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1970.

^{3}

MacKenzie (2008, p 63–64)

^{4}

Mirowski (1992, p 134)

^{5}

Mirowski (1992, p 122, quoting von Neumann)

^{6}

Mirowski (1992, p 122–124)

^{7}

Gell-Mann (1992, p 7)

^{8}

Katz (1993, p 685)

References

Gell-Mann, M. (1992). Nature conformable to herself. Bulletin of the Santa Fe Institute, 7(1):7–8.

Katz, V. J. (1993). A History of Mathematics: an Introduction. Haper Collins.

MacKenzie, D. (2008). An Engine, Not a Camera: How Financial Models Shape Markets. The MIT Press.

Mirowski, P. (1991). The when, the how and the why of mathematical expression in the history of economic analysis. Journal of Economic Perspectives, 5(1):145–157.

Mirowski, P. (1992). What were von Neumannn and Morgenstern trying to accomplish?. In Weintraub, E. R., editor, Toward a History of Game Theory, pages 113–150. Duke University Press.

Why don't more economists see the potential of mathematics

2012-01-06T00:01:00.000-08:00

A research student, working in econometrics has e-mailed me with thecomment

I am a little confused why many economists do not see the potential of mathematics.

The discipline of econometrics was introduced in the 1940’s with the keymonograph being Trygve Håvelmo’s The Probability Approach in Econometrics.Håvelmo’s motivation for writing the paper is eloquently stated in thepreface

The method of econometric research aims, essentially, at a conjunction of economic theory and actual measurements, using the theory and technique of statistical inference as a bridge pier. But the bridge itself was never completely built. So far, the common procedure has been, ﬁrst to construct an economic theory involving exact functional relationships, then to compare this theory with some actual measurements, and, ﬁnally, “to judge” whether the correspondence is “good” or “bad”. Tools of statistical inference have been introduced, in some degree, to support such judgements, e.g., the calculation of a few standard errors and multiple-correlation coeﬃcients. The application of such simple “statistics” has been considered legitimate, while, at the same time, the adoption of deﬁnite probability models has been deemed a crime in economic research, a violation of the very nature of economic data. That is to say, it has been considered legitimate to use some of the tools developed in statistical theory without accepting the very foundation upon which statistical theory is built. For no tool developed in the theory of statistics has any meaning– except, perhaps, for descriptive purposes –without being referred to some stochastic scheme.

The reluctance among economists to accept probability models as a basis for economic research has, it seems, been founded upon a very narrow concept of probability and random variables. Probability schemes, it is held, apply only to such phenomena as lottery drawings, or, at best, to those series of observations where each observation may be considered as an independent drawing from one and the same “population”. From this point of view it has been argued, e.g., that most economic time series do not conform well to any probability model, “because the successive observations are not independent”. But it is not necessary that the observations should be independent and that they should all follow the same one–dimensional probability law. It is suﬃcient to assume that the whole set of, say $n$ , observations may be considered as one observation of $n$ variables (or a “sample point”) following an $n$ -dimensional joint probability law, the “existence” of which may be purely hypothetical. Then, one can test hypotheses regarding this joint probability law, and draw inference as to its possible form, by means of one sample point (in $n$ dimensions). Modern statistical theory has made considerable progress in solving such problems of statistical inference.

In fact, if we consider actual economic research–even that carried on by people who oppose the use of probability schemes–we ﬁnd that it rests, ultimately, upon some, perhaps very vague, notion of probability and random variables. For whenever we apply a theory to facts we do not–and we do not expect to–obtain exact agreement. Certain discrepancies are classiﬁed as “admissible”, others as “practically impossible” under the assumptions of the theory. And the principle of such classiﬁcation is itself a theoretical scheme, namely one in which the vague expressions “practically impossible” or “almost certain” are replaced by “the probability is near to zero”, or “the probability is near to one”.
This is nothing but a convenient way of expressing opinions about real phenomena. But the probability concept has the advantage that it is “analytic”, we can derive new statements from it by the rules of logic.

Håvelmo’s argument can be split into four key points. If economics is to be regardedas ‘scientiﬁc’, it needs to take probability theory seriously. He then notesthat economists have taken a naive approach to probability, and possibly mathematics in general, and introduces the Lagrangian idea of representing

n

points in one dimensional space by one point in

n

-dimensionalspace. Finally he makes Poincaré’s point that probability is a convenientsolution, it makes the scientist’s life easier, and ﬁnally he makes Feller’s point thatit enables the creation of new knowledge, new statements.

Håvelmo then goes on to tackle the issue that goes back as far as Cicero, atleast, “there is no foreknowledge of things that happen by chance” by making thecritical observation, nature looks stable because we look at it in a particularway

“In the natural sciences we have stable laws”, means not much more and not much less than this: The natural sciences have chosen very fruitful ways of looking on physical reality.

Håvelmo is saying that if economists look at the world in a diﬀerent way, if the rightanalytical tools are available to them, they may be able to identify stablelaws.

At about the same time, Oskar Morgenstern was working with John vonNueumann on The Theory of Games and Economic Behavior, a “big bookbecause they wrote it twice, once in symbols for mathematicians and once in prosefor economists”. Morgenstern begins the book by describing the landscape. On thesecond page he, makes the case for using mathematics in economics, just asHåvelmo had, but with a more comprehensive argument. Morgenstern reviewsthe case as to why mathematics is inappropriate to economics, no doubt with vonNeumann at his shoulder,

The arguments often heard that because of the human element, of psychological factors etc., or because there is – allegedly – no measurement of important factors, mathematics will ﬁnd no application [in economics] [von Neumann and Morgenstern 1967 p 3]

However, Morgenstern points out that Aristotle had the same opinion of the use ofmathematics in physics

Almost all these objections have been made, or might have been made, many centuries ago in ﬁelds ﬁelds where mathematics is now the chief instrument of analysis.

While measurement may appear diﬃcult in economics, measurement appeareddiﬃcult before the time of Albert the Great, again before Newton ﬁxed time andspace, when objects were either ‘hot’ or ‘cold’ or before the idea of potentialenergy being released into kinetic energy emerged.

The reason why mathematics has not been more successful in economics must, consequently, be found elsewhere. The lack of real success is largely due to a combination of unfavourable circumstances, some of which can be removed gradually. To begin with economic problems were not formulated clearly and are often stated in such vague terms as to make mathematical treatment a priori appear hopeless because it is quite uncertain what the problems really are. There is no point in using exact methods where there is no clarity in the concepts and the issues to which they are to be applied. Consequently the initial task is to clarify the knowledge of the matter by further careful description. But even in those parts of economics where the descriptive problem has been handled more satisfactorily, mathematical tools have seldom been used appropriately. They were either inadequately handled, as in the attempts to determine a general economic equilibrium …, or they led to mere translations from a literary form of expression into symbols, without any subsequent mathematical analysis. [von Neumann and Morgenstern, 1967, p 4]

Morgenstern makes the critical observation, that the ‘correct’ use of mathematics inscience leads to the creation of new mathematics

The decisive phase of the application of mathematics to physics – Newton’s creation of a rational discipline of mechanics – brought about, and can hardly be separated from, the discovery of [calculus]. (There are several other examples, but none stronger than this.)
The importance of social phenomena, the wealth and multiplicity of their manifestations, and the complexity of their structure, are at least equal to those in physics. It is therefore expected – or feared – that the mathematical discoveries of a stature comparable to that of calculus will be needed in order to produce decisive success in this ﬁeld. [von Neumann and Morgenstern 1967 p 5]

In 1989 Håvelmo was awarded the Nobel Prize in Economics “for hisclariﬁcation of the probability theory foundations of econometrics and his analysesof simultaneous economic structures”. In his speech, the economist Håvelmoreﬂected on the impact of his work,

To some extent my conclusions [are] in a way negative. I [draw] attention to the – in itself sad – result that the new and, as we had thought, more satisfactory methods of measuring interrelations in economic life had caused some concern among those who had tried the new methods in practical work. It was found that the economic theories which we had inherited and believed in, were in fact less stringent than one could have been led to think by previous more rudimentary methods of measurement. To my mind this conclusion is not in itself totally negative. If the improved methods could be believed to show the truth, it is certainly better to know it. Also for practical economic policy it is useful to know this, because it may be possible to take preventive measures to reduce uncertainty. I also mentioned another thing that perhaps could be blamed for results that were not as good as one might have hoped for, namely economic theory in itself. The basis of econometrics, the economic theories that we had been led to believe in by our forefathers, were perhaps not good enough. It is quite obvious that if the theories we build to simulate actual economic life are not suﬃciently realistic, that is, if the data we get to work on in practice are not produced the way that economic theories suggest, then it is rather meaningless to confront actual observations with relations that describe something else. [ Prize Lecture Lecture to the memory of Alfred Nobel ]

Håvelmo’s aim in the 1940s, along with that of John von Neuman, had beento improve economic methodology, the consequence was, in Håvelmo’s case, wasthat it highlighted deﬁciencies in economic theory. The question is, whathappened in economics in the forty years between Håvelmo’s paper on econometrics and his Nobel Prize in1989 to lead to such a negative reflection on the development of economics. I shall come back to this in my next post.

References

J. von Neumann and O. Morgenstern. Theory of Games and Economic Behavior. Wiley, 3rd edition, 1967.

The Girl with the Dragon Tattoo, and why we can't solve the Euro crisis

2011-12-22T01:15:00.000-08:00

I, like a lot of technically minded-people it seems, like a good crime thriller. Having spent a year mesmerised by Sara Lund's sweaters I picked up The Girl with the Dragon Tattoo. It's a good story, but a bit contrived with the plot being resolved by unexplained and unbelievable computer hacking.

So why write about it on a blog about science and finance?

The plot is set in the context of our hero, a journalist, Blomkvist and his relationship between two businessmen, Wennerström and Vanger. Blomkvist publishes a story claiming Wennerström was involved in a fraud, which he failed to corroborate and the novel begins with his conviction for libel. This makes him vulnerable to an approach from Vanger to investigate the mysterious disappearance of his niece in the 1960s, which is the central mystery in the novel.

What I find interesting is how the different businessmen are presented. Wennerström has become a billionaire from nowhere, building his wealth by participating in such unsavoury activities as options trading and currency speculation. Vanger, on the other hand, is old-money - a traditional industrialist whose family had arrived in Sweden with Napoleon's puppet king, Jean-Baptiste Bernadotte, and was granted land holdings on which they developed paper mills that spawned a firm involved in manufacturing, technology and the media. Within this setting the message is clear: Vanger may be an unpalatable capitalist but Wennerström is beyond the pale.

The sub-text is physiocratic, the Vagner's are legitimate capitalists because they hold land. Wennerström, who has made his own money through speculation, by judgement and foresight, is corrupt. The irony that the hero Blomkvist becomes rich by speculating, that Wennerström is a crook, is passed by. A second irony is that if the events described in the book had actually occurred in England, they might have become of interest to the Leveson Inquiry, looking at actual cases of the relationship between journalists and private detectives involved in phone hacking.

The theme of the corruption of finance is not new, but the novel ignores St Augustine's observation that if a merchant was a cheat, the fault was with the individual, not the profession. What is disturbing is the deep conservatism of the books' sub-text: if some thug in history had not given your ancestors land, you do not deserve to be rich.

I finished reading this book the weekend that France and Germany looked to solve the Euro crisis and impose tighter regulations on financial services. The French President, commenting on the UK's withdrawal from the process, went on to say that "a good part of the world's problems come from the deregulation of financial services".

Both the UK's FSA's, in their recent report on the collapse of RBS, and the US's Financial Crisis Inquiry Commission (FCIC) observe that the regulations existed, but they were not effectively implemented. The UK-US crisis was a failure of government agencies, at the behest of politicians, to enforce rules. The Euro crisis has similar roots in a failure to enforce rules on sovereign debt ratios. The crises seem to have less to do with bankers and more to do with the profligacy of politicians.

Much of the narrative of the ongoing financial crises has been about the imbalance, particularly in the UK, between manufacturing, represented by Vanger, and financial services, the Wennerströms. Implicit is that manufacturing, creating physical objects, is something more real than financial services, moving money around, and moreover, more reputable.

This bias seems to be reflected in UK government science policy. In the recently published Innovation and Research Strategy, the government identifies the following, key, technology based sectors: life sciences, high-value manufacturing, nanotechnology and digital technology. Financial services, while contributing around 10% of UK's GDP, is not mentioned. This approach ignores the advice from the Royal Society in their 2009 report Hidden Wealth.

I would argue that the recent financial crises are about a general lack of understanding of modern finance that enabled institutions to behave irresponsibly. It is hardly surprising that this situation has arisen given the blindness to policy-makers to investing in fundamental research into finance and its underpinning technologies, whether they be the contracts traded, the mathematical models and their computational implementations. This aversion to understanding finance will persist as long as popular culture is dominated by the Vanger=good, Wennerström=bad model.

... but what about the South Sea Bubble...

2011-12-08T01:49:00.000-08:00

David K Waltz has made a comment on my last post, I realised my response was going to be a bit longer than a "comment".

David raises the spectre of the South Sea Bubble and Tuplipmania. I am not an expert on the Tulip Bubble, but I am aware that there is a question as to whether it was a significant as popular imagination suggests (i.e Dumas's The Black Tulip).

The South Sea Bubble (1720) can tell us a lot. In the aftermath the British constitution was completely overhauled - the introduction of the position of Prime Minister (held by Walpole,still the longest serving Prime Minister) was the most obvious impact. In addition, British public finance was put on a firm footing that laid the foundation of successive military victories and the Empire. France's response to the Mississippi Bubble (1719) was less dramatic (they blamed the financier Law) and the consequence was a succession of military defeats.

Also, the popular response included Defoe's Complete English Tradesman where he writes (1726)

A tradesman's books are his repeating clock, which upon all occasions are to tell him how he goes on, and how things stand with him in the world: there he will know when it is time to go on, or when it is time to give over; and upon his regular keeping, and fully acquainting himself with his books, depends at least the comfort of his trade, if not the very trade itself. If they are not duly posted, and if every thing is not carefully entered in them, the debtor's accounts kept even, the cash constantly balanced, and the credits all stated, the tradesman is like a ship at sea, steered without a helm; he is all in confusion, and knows not what he does, or where he is; he may be a rich man, or a bankrupt-for, in a word, he can give no account of himself to himself, much less to any body else.

This contrasts with the Rape of Lady Credit by stock-jobbers that Defoe described in 1709

The first Violence they committed was downright Rape ... these new-fashion'd thieves seiz'd upon her, took her Prisoner, toss'd her in a Blanket, ravish'd her, and in short us'd her barbarously, and had almost murther'd her

Much of the political and public rhetoric in recent years has been focused on criticism of financiers (but, unlike Defoe, not before 2006). However, given the performance of Europe's politicians since 2008 is terrifying (for a European citizen) in that they seem to refuse to take responsibility (which countries were the first to break Euro rule, was it Germany and France?) and prefer to see the crisis as an opportunity to score political points (the Conservative party in the UK).

The British, between around 1688 and 1728, seem to have recognised the democratic nature of markets. Something that Aristotle commented on, and was discussed by scholastics that influenced the development of science through the likes of Bradwardine and Oresme. Napoleon was scornful as Britain as a nation of shopkeepers, but that is precisely where its strength lay. And it is not simply a financial strength, how different is Defoe's description of a Tradesman from Hume's empiricism?

Its not just science that can learn from the markets.

Does finance need a scientific revolution or science need a revolution inspired by finance

2011-12-06T03:26:00.000-08:00

Nature, the British version of Science, rarely makes forays into exploring issues in finance. They invited me to talk at a meeting in September 2009 and I get the sense that they would like to do more but are a bit perplexed by the subtleties of finance. The mind is willing but the body is weak.

In October 2008 Nature published a piece, Economics needs a scientific revolution, by the French econo-pysicist Jean-Philippe Bouchaud, which argued that the fault was with the academic discipline of economics: "Classical economics is built on very strong assumptions that quickly become axioms" where as "Physicists, on the other hand, have learned to be suspicious of axioms. If empirical observation is incompatible with a model, the model must be trashed or amended". More recently they have published a broader article, reflecting the diffusion of the financial crisis into society at large, Science's attitudes mustreflect a world in crisis, by the science writer Colin Macilwain.

Physicists are never slow to criticise the short-comings of other disciplines, but the truth is physics, with its never changing laws, is simple. While physicists are paddling in the play pool, economists are hanging on to a rubber ring in the middle of a North Atlantic gale. It is not surprising they don't look like good swimmers. That said, economists are well aware of the limitations of their discipline. From the mathematician Cournot's criticism of the economists' use of mathematics in the 1830s to the contemporary economist's criticism of the wholesale adoption of the ergodic hypothesis by Paul Davidson, economics could do better.

Since the Second World War, economics has transformed itself from a relatively minor discipline to a dominant field. It is ironic that in the 2008 "Research Assessment Exercise", UK economists judged their work to be "excellent", just as the tsunami of the credit crisis was crashing on the shore. This dominance was built, substantially, on "positive" economics, adopting the ideals of the logical positivists that emerged in central European maths and physics in the aftermaths of the First World War. The authority that these scientist had rested on the fact that they had defeated Germany and Japan.

It was not square-jawed commandos that defeated the Axis, but mathematicians and physicists (like Turing, von Neumann, Kolmogorov, Weiner, Shannon, ... the list is not short) working in Operations Research, the Manhattan Project and code-breaking, that won the war. In particular the code breakers were able to transform streams of random letters into meaningful messages. This analogue seems to have driven economic research through the fifties and sixties: given the right algorithm the randomness of asset prices can be read. Then, just as the idealised economic order of Bretton Woods collapsed under the reality of people and politics, Black-Scholes-Merton and financial mathematics (apparently) emerged to save the world from random chaos.

The financial crisis of 2007-2009 was a demonstration of limitations of the science that emerged out of logical-positivism and in particular the ergodic hypothesis placed at the core of positive economics. It was not a failure of maths and science in general, but a particular type of science that emerged in the 1920s and dominated society between 1940 and the present.

Macilwain is concerned with he fact that many science policy gurus, generally wedded to the tenets of positivism, are "clearly more comfortable discussing the planet's ecological crises than the economic ones currently alarming the general population" and that there is "great danger is that scarce funding will consolidate around single-discipline research". More Big Physics like the LHC, or repeating Tony Blair's "betting the house"on genetics - a bet that has not paid out.

Financial markets are manifestations of random, non-ergodic phenomena, as such they can hardly be amenable to deterministic techniques. Specifically, a single approach to understanding them, whether rooted in analogues from the Olympian disciplines of mathematics, physics or biology, will fail. Scientific revolutions have historically been based on the inter-disciplinary interaction, such as between financiers, lawyers (Bacon, Descartes, Fermat, Huygens) and mathematicians. If society stops to think about the current financial turmoil and what it can tell us, we would surely be at the dawn of another scientific revolution.

De Coding Da Vinci

2011-11-04T02:42:00.000-07:00

It is well known that Leonardo Da Vinci became interested in the "golden ratio" or "divine proportion". It is somewhat less well known is he learnt about the number from Lucca Pacioli, the Francisican friar and grandfather of accounting. What is virtually unknown is that Pacioli probably learnt his mathematics from the financial mathematician and artist, Piero della Francesca (as an artist).

I have written an article, Decoding Da Vinci: Finance, functions and art, on this for plus! an on-line magazine aimed at youngster. The piece explains why the ratio was considered divine because of its form as a continued fraction, and how another financial mathematician laid the foundations for functional analysis by popularising decimal notation.

St Thomas Aquinas comes to the defence of the vampire squid?

2011-10-04T08:36:00.000-07:00

On April 16 2010, the SEC filed fraud charges against Goldman Sachs, the “great vampire squid wrapped around the face of humanity”. The case in court is that the bank claimed the assets in the fund ABACUS 2007-AC1 were selected by an independent advisor, when in fact they were selected by John Paulson to enable his funds to short the assets, and in making the claim, the bank deceived investors. However the case is usually presented as the bank being taken to court for acting immorally structuring an asset designed to fail and selling this on to unsuspecting clients (for example The New York Times’ report and their prediction in December 2009 or in the UK Jeremy Warner of The Telegraph). The vampire squid was sucking the life out of innocent investors.

The SEC could not take Goldman Sachs to court for acting unethically, the courts are about legality not morality. However, it would come as a surprise to many that, according to Catholic doctrine at least, there was probably not much morally wrong with Goldman’s actions.

In the third quarter of the thirteenth century, Thomas Aquinas worked on his Summa Theologica, integrating Aristotelian philosophy with Catholic doctrine and established himself as a “Doctor of the Church”. In the “Second part of the Second part” of the Summa, Aquinas addressed the issue of individual morality, including the concept of justice. One question the theologian considered was “Whether it is lawful to sell something for more than it is worth?” and examined a case presented by Stoic philosophers

A grain merchant from Alexandria arrives at Rhodes, which is gripped by famine. The merchant knows that other merchants are following him with plentiful supplies of grain, though the town’s inhabitants do not know this. How should the merchant price the grain he has? ( De Officiis Book 1, XII)

The Roman jurist Cicero (Tully) was typical in arguing that the merchant should not take advantage of the misfortune of the starving and charge a lower price based on the knowledge of the coming relief.

Aquinas disagrees

in the case cited, the goods are expected to be of less value at a future time, on account of the arrival of other merchants, which was not foreseen by the buyers. Wherefore the seller, since he sells his goods at the price actually offered him, does not seem to act contrary to justice through not stating what is going to happen. If however he were to do so, or if he lowered his price, it would be exceedingly virtuous on his part: although he does not seem to be bound to do this as a debt of justice. (Summa II.II Q77(A3) Reply to Obj. 4)

This is an almost shocking conclusion from a saint. To Aquinas, the merchant, having arrived at Rhodes, may think there are more grain shipments on the way, but does not know. Aquinas argues that

because the just price of things is not fixed with mathematical precision, but depends on a kind of estimate, so that a slight addition or subtraction would not seem to destroy the equality of justice. (Summa II.II Q77(A1) Reply to Obj 1, para 2)

It is not unjust to charge the higher price, since whether of not this price is ‘just’, or not, is uncertain.

Aquinas’ concept of the “equality of justice” comes from Book 5 of Aristotle’s Nicomachean Ethics which had been translated into Latin in 1250. In Ethics, Aristotle considered the morality of economic exchange and argues that market exchange is not performed in order to generate a profit, for gain, but to correct for inequalities and to establish a social equilibrium. Using the example of a builder and a shoemaker, Aristotle argues that

The builder, then, must get from the shoemaker the latter’s work, and must himself give him in return his own. (Ethics p 79)

For justice to exist in the exchange, there needed to be an equality between the shoes the shoemaker produced and the results of the builders’ work. For such an equality to exist, there needed to be a measure of the value of the goods produced by the builder and the shoemaker to enable a just exchange . This measure, the price, was provided by money.

all things that are exchanged must somehow be comparable. It is for this end that money has been introduced, and it becomes in a sense an intermediate; for it measures all things. (Ethics p 79)

The relationship at the heart of the exchange between the shoemaker and the builder that Aristotle went on to establish was explained mathematically, as a geometric relationship. This was significant since Aristotle never used mathematics in addressing questions in physics.

However Aquinas’ use of an analogy with mathematics was not simply following Aristotle, he was re-interpreting in the context of contemporary events. The twelfth century had seen a “renaissance” in western Europe, a population explosion that brought with it Gothic architecture and a commercial revolution.

In the aftermath of the collapse of the Roman Empire in the west, the concept of turpe lucrum, or ‘shameful gain’ emerged and merchants were restricted in what prices they could charge. As Europe emerged out of the ‘castellan’ society, characterised by isolated local lords held together by feudal service relations, into one of integrated trade networks, turpe lucrum was no longer a feasible basis on which to run the economy. It was in this environment that Leonardo of Pisa, Fibonacci, published the Liber Abaci.

One aspect of the complexity of the transactions was to disguise usurious contracts. There is a subtle difference between usury, which is associated with charging for the use of money, and interest which is a compensation for loss. For example a farmer could lend a cow for a year and expect to be re-paid with a cow and a calf, since in the normal course of events a cow would give birth to a calf over the year. Interest could be charged on ‘productive’ assets, with the consequence that all, legitimate, medieval securities would be ‘asset backed’. Gold, being inorganic, was not productive and so charging for its use was unjust.

However, this attitude to money stifled innovation and so more complex structures emerged to disguise the charging of interest on a money loan. For example, the ‘triple contract’ enabled an entrepreneur to raise money to invest in a trading venture. At the heart of the triple contract was a partnership between the entrepreneur and investors, this was the first contract. The second contract would be an insurance contract taken out by the entrepreneur to insure against the loss the investors’ capital. The third contract was another ‘insurance’ contract given to the investors by the entrepreneur, where by the investors surrendered their rights to a share of the profit in exchanged for a fixed payment from the entrepreneur, this payment was guaranteed by the second contract. Not quite a Credit Default Swap but definitely credit insurance.

Eventually, in 1236, the canon (church) jurist, Alanus Anglicus, determined that turpe lucrum did not exist if the future price of the good was uncertain in the mind of the merchant and this ruling was embedded in Catholic doctrine some ten years later. Aquinas, in the Summa was integrating the theories of the newly translated Aristotle with emerging Catholic doctrine and existing commercial practice to argue that a profit was ‘just’ so long as it was uncertain and might lead to a loss, it entailed a ‘risk’.

Islamic societies had equally stringent prohibitions on usury, and so this alone cannot explain why Fibonacci would have such an influence in Europe. The key difference between the environment for medieval European merchants and their colleagues in the Middle East, India or China was the range of currencies being used, a consequence of the castellan society of the ‘Dark Ages’. For example there were 28 different currencies in Italy at one time or another and as Groetzman has observed

Reading Liber Abaci one has the sense that Italian merchants of the 13^th century operated in a world of complete relativism. With no central government, no dominant currency, and even competing faiths and heresies, value is expressed quite abstractly only in a set of relative relations to other items. (NBER Working Paper No. 10352, 2004)

It was in this environment that ‘arbitrage’, the process by which one commodity ‘arbitrates’ between the value of two other commodities, just as money ‘mediates’ between the value of two commodities, emerges and is discussed in the Liber.

Aristotle’s use of mathematics in examining the nature of exchange was first noted by the medieval proto-scientist, and Aquinas’ teacher, Albert the Great. This observation was important philosophically since it separated the measure from the measured, money does not share the same ‘nature’ as shoes. This seems insignificant today, but at the time it was an important conceptual development that spawned a ‘mania’ for measuring and mathematics such as that undertaken by the ‘Merton Calculators’.

The historian Joel Kaye has argued that the line of thought initiated by Albert and Aquinas developed by Oresme and the Calculators, was based on

the transformation of the conceptual model of the natural world ,…, [which] was strongly influenced by the rapid monetisation of European society taking place [between 1260–1380] (Kaye, p 1)

The scholars took their approach to nature having observed the operation of the markets that had emerged in the century before and in response

[were] more intent on examining how the system of exchange actually functioned than how it ought to function. (Kaye, pp 219-220)

Kaye, and a fellow historian, Alfred Crosby, believe this paradigm shift played a pivotal role in the development of European science. The process of this conceptual change is clear. Innovations in finance led to the development of vernacular mathematics encapsulated in the Liber Abaci and disseminated through abbaco schools. University based scholars then began to try and make sense of what was actually happening and tried to identify the essence of the markets, with Aquinas concluding that a profit was ‘fair’ provided that it was uncertain and came with the risk of a loss. In modern terms, a market was viable if it did not admit arbitrages, so long as there is ‘No free lunch with vanishing risk’.

In much of the public discourse on modern finance, financial institutions are presented as immoral and rapacious beasts, initiating ‘unnatural’ financial instruments in order to take advantage of the innocent. These opinions belie a more complex reality, that finance is integral to society, complex financial products have been with us for at least 900 years, that when people, like Aquinas, employ reason and morality to examine the markets there are some surprising conclusions, and that the study of markets leads to profound insights for science in general. Criticism of finance is as likely to reflect a lack of rationality and morality in society, in general, than a specific lack of ethics and reason, in the markets.

Who was the first Quant?

2011-09-21T01:54:00.000-07:00

Scott Patterson, in his book The Quants, describes Ed Thorp as the 'godfather' of Quants. Without doubt, Thorp heralded the modern age of quantitative finance, but does this mean he was the first Quant?

Many might point to Louis Bachelier as being the original Quant. Bachelier had been born in Le Harve, in Normandy, in 1870. His father was a wine-dealer while his mother came from a banking family. In the six months after graduating from school, both of Bachelier's parents died and he was forced to take over the management of the family firm, Bachelier fils, to provide for his younger siblings. After doing his military service in 1891, Bachelier moved to Paris and became involved in the Paris Stock Exchange. Simultaneously, he enrolled on the mathematics degree at the University of Sorbonne, where Poincare was a professor. He was not a brilliant student but in 1895 he embarked on a doctorate that synthesised the activities being undertaken at the Paris Stock Exchange and the theories of heat and of probability, at the cutting edge of physics at the time. Bachelier then left finance and embarked on an academic career that lasted the following forty years, a career that appears to have been hindered by his early association with the markets.

This, however, is not the career path of a Quant. The likes of Ed Thorp and James Simons had a training in science that they then applied to finance, Bachelier had started working in finance and used this experience as the basis of an academic career: he was a 'reverse-quant'.

The history of reverse-quants is probably more significant than that of Quants. Leonardo Bonacci, better known as Fibonacci, could claim to be the first reverse-quant, taking ideas from contemporary finance to change European culture, by changing not just how merchants undertook their business calculations, but how western science wrote mathematics.

Leonardo's father, Guglielmo Bonacci, was a merchant looking after the Pisan interests in the Algerian port of Bejaia. While we might not imagine that medieval finance was very sophisticated, we would be wrong. The historian Alfred Crosby describes a series of transactions undertaken by an Italian merchant, Datini, which, although they took place two hundred years later, would have been similar to the types of transactions Guglielmo Bonacci would have been involved in. In November 1394 Datini bought wool forward from Mallorca. The wool was sent to Pisa, via Barcelona, in summer 1395, arriving in Italy the following January. Datini sold some wool to a colleague in Florence and processed the the rest into cloth, which he sent to Venice for shipping back to Mallorca for sale in July 1396. However the market in the Balearics was weak and so the cloth was transported to Valencia and North Africa. The last piece of cloth was sold three and a half years after the wool was originally been contracted from the shepherds. Datini would have engaged in forward contracts, loan agreements and transactions in at least five currencies (Arogonese, Pisan, Florentine, Venetian, North African). To make a profit, he needed to be an expert at 'commercial arithmetic', or financial mathematics.

Fibonacci was born in Pisa around 1170 and educated, not only in Bejaia but, as far afield as, Egypt, Syria, Constantinople and Provence. He would write a number of books on mathematics, but his first and most influential was the Liber Abaci ('Book of Calculation'), which appeared in 1202. The Liber was heavily influenced by the Arabic book 'The Comprehensive Book on Calculation by Completion and Balancing' written around 825 CE by al-Khwarizmi, who was himself motivated to write the book because

men constantly require in cases of inheritance, legacies, partition, law-suites and trade [a number]

and his book provided the easiest way of arriving at that number, using al-gabr ('restoration') and al-muqabala ('balancing'). Fibonacci collated these Arabic techniques into a single textbook for merchants, such as Datini, facing the increasingly complex financial instruments and transactions emerging at the time.

The impact of the Liber Abaci was enormous. Fibonacci became an adviser to the most powerful monarch of the time, Frederick II, Holy Roman Emperor and King of Sicily. More significant, Abaco or rekoning schools sprang up throughout Europe teaching apprentice merchants how to perform the various complex calculations needed to conduct their business. Pacioli, who taught Leonardo da Vinci maths, was a well known graduate. Less well known is the fact that Copernicus came from a merchant family and in 1526, seventeen years before his more famous, "epoch-making" 'On the Revolutions of the Heavenly Spheres', he wrote 'On the Minting of Coin' about finance.

The practical usefulness of the reckoning schools was that, by using positional numbers and algebra, merchants could execute complex financial calculations that would typically include an illicit interest charge, hidden from the mathematically unsophisticated, university based, Church scholars. The merchant bankers were using mathematics to keep one step ahead of the regulator and the effectiveness of the non-university mathematics would not have been lost on the sharper scholastics, observing market practice.

The most influential single Abaco graduate has to be the Dutchman, Simon Stevin. Stevin, who was born in 1548 in Bruges, had originally worked as a merchant's clerk in Antwerp then as a tax official back back in Bruges, where he wrote his first book Tafelen van Interest ('Tables of interest') which he published in 1582, before moving to the University of Leiden in 1583. About this time, he was appointed as adviser to Prince Mauritz of Nassau, who was leading the Dutch revolt against the Spanish, and eventually became the Dutch Republic's Finance Minister.

As well as being active in government, Stevin carried out scientific experiments, and it is believed his bookeeping inspired his physics. His most famous experiment showed that heavy and light objects fell to the earth, in the absence of air resistance, at the same speed, an experiment that disproved a belief of Aristotle and is usually attributed to Galileo dropping things from the Tower at Pisa some years later.
One of Stevin's most important posts was as the director of the Dutch Mathematical School, established in 1600 by Mauritz to train military engineers. In this capacity, in 1605, he published a textbook for the School, the 'Mathematical Tradition', which was a comprehensive overview of mathematics and included a whole section on 'Accounting for Princes in the Italian manner'. In a very short period, the Dutch Mathematical School became the centre for merchants' training in north western Europe. This success, in turn, forced the authorities at the University of Leiden, which provided the School with its facilities, to take practical sciences, in particular maths, a bit more seriously. The Dutch Mathematical School would inspire the soldier Descartes to study maths and would train Huygens and a whole generation of European scientists. In addition, it was Stevin's promotion of the use of decimals, to aid accounting, that inspired Newton to think of functions as power-series, giving birth to the discipline of Analysis.

So much for 'reverse-quants'. If a Quant is defined as a someone who moves from an academic career into finance, the most famous Quant is Isaac Newton. Newton essentially finished his work in physics with the publication of Principia in 1687, his last significant work, Optiks, published in English in 1704, was based substantially on research undertaken in the early 1670s. After almost a decade of troubles, Newton moved into finance in April 1696 when he was appointed Warden of the Royal Mint. This was a largely ceremonial post, but Newton took to it so much that he became the Mint's operational manager, its Master, in 1699 (see Isaac Newton: Financial Regulator).

Being Master gave Newton an average income some 16 times what he would have had as an academic. Today a 'starting' salary for a jobbing professor in England is around £60,000, and we could expect the Lucasian Professor at Cambridge to earn somewhat more than this. On this basis, the equivalent salary for Newton as Master of the Mint is in excess of £1 million, which pretty well places him in the bulge bracket. Newton, no longer needing the income from his position in Cambridge, resigned his Professorship at the end of 1701. Newton did become President of the Royal Society in 1703, an institution whose foundations were laid by the banker Thomas Gresham, and Newton followed in the footsteps of his patron, the the English Chancellor of the Exchequer, Charles Montagu. Gresham and Montagu, central to the establishment of the oldest national academy of sciences, are just two more examples of 'reverse-quants'.

Newton might be the most famous Quant, but he was not unique or the first. Huygens identified conditional expectation while investigating the pricing of life annuities, and J. Bernoulli identified the number e when investigating interest payments. Generations before Huygens, Bernoulli and Newton, we have Galileo, who was born in Pisa around 1564. He was appointed to the Professorship of Mathematics the prestigious University of Padua in 1592, where he made his famous astronomical observations. Galileo had been short of money all his adult life, as a mathematician he was poorly paid and was expected to supplement his salary by taking private students, but this interfered with his research. In 1613 the Duke of Tuscany, Cosimo II de Medici, offered Galileo the position of 'First and Extraordinary Mathematician of the University of Pisa and Mathematician to his Serenest Highness (i.e. Cosimio)' with a large salary and no duties, an ideal post for Galileo. It was some time in the next decade or so that Galileo wrote 'Upon the Discoveries of Dice', which he was almost certainly asked to write by Cosimo, who may have been trying to solve a practical problem in gambling. Galileo did not leave Padua to work in finance, but he did become the mathematical adviser to the the head of one of Italy's most important banking families. Ironically, had Galileo stayed at Padua, under the protection of the religiously ambivalent Venetians, he would never had faced the Inquisition, and possibly would not have become as famous as he is today.

European science did not start in the Renaissance, it existed in the High Middle Ages. The 'renaissance' of the 'long twelfth century' resulted in what the historian Joel Kaye describes as

the transformation of the conceptual model of the natural world ,..., [which] was strongly influenced by the rapid monetisation of European society taking place [between 1260-1380].

and played a pivotal role in the development of European science. Thirteenth century scholars

[were] more intent on examining how the system of exchange actually functioned than how it ought to function..

It seems that Fibonacci did not just influence medieval merchants, those scholars keeping an eye on merchant's dubious dealings, also, became obsessed with mathematics. This included the great scholastic scientist, Albert the Great, and the mathematically minded theologian, Thomas Aquinas. But the most famous scholars to turn to mathematics were the 'Merton Calculators'.

The first of the Calculators was Thomas Bradwardine, who entered Merton College in Oxford in 1323. While studying the quadrivium, mathematics, astronomy, music and geometry, Bradwardine observed that

[Mathematics] is the revealer of genuine truth, for it knows every hidden secret and bears the key to every subtlety of letters. Whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start that he will never make his entry through the portals of wisdom.

This was a highly significant point in the history of science since it is the first time that scholars in the Hellenistic tradition, which included Jewish, Christian and Islamic philosophers, innovated in physics by using mathematics. In the footsteps of Bradwardine there was, amongst others, William Heytesbury, who identified the mean speed theorem, and Nicolas Oresme, who introduced the idea of the graph and advised the French king on money supply.

After twelve years at Merton, Bradwardine left the Oxford University to work for the Bishop of Durham, who was the Treasurer and Chancellor of England. So Bradwardine was not only the first Physicist, as we would understand the term today, he was also the first person who left an academic career to take up one in finance. The first Quant was the first Physicist.

Looking at the relationship between scientists and finance reveals some important facts. Firstly, the migration from academic careers in science to finance appear to be embedded, it is not a modern phenomena. However, possibly more significant is the less well-appreciated role of the 'reverse-quants' in the development of science. The influence is captured by events in France in 1304-1305 when economic instability and a market failure led the French King, Philip the Fair, to issue decrees fixing the price of bread. His decrees failed spectacularly, and this was seen by contemporary observers as evidence that 'nature' ruled, and not the authority of the King, and that market prices where an objective, 'scientific' measure. This enabled the likes of Bradwardine to re-assess the role of mathematics in science. Later, people trained in commercial arithmetic - financial mathematics - such as Copernicus and Stevin, were able to challenge the authority of Aristotelian science, and argue that the Earth revolved around the Sun and that heavy and light objects fall at the same speed. Today, financial markets challenge assumptions about determinism and stability of systems, the question is, can science meet those challenges?

Maths and the markets

2011-09-02T02:53:00.000-07:00

Dr Jack Stilgoe, a science policy wonk, has been thinking about Responsible innovation in financial services and asks the question, in relation to the Credit Crisis

Could mathematicians have done more to ensure that their models weren't abused, or is it not really about maths at all?

Jack's deceptively simple question is incredibly intricate. There are many commentators who argue that the complexity of modern markets is such that they are mathematically intractable, and the best approach is analysis through discourse, as was popular in the Dark Ages and between the Black Death and Francis Bacon and Galileo. My (biased) opinion is that these views are held principally by those educated in the ethos that developed before the collapse of Bretton-Woods, when a deterministic economy was managed by wise sages. Unfortunately the world is not deterministic and the sages could not hope to manage the economy by agreeing treaties in luxury hotels.

However, mathematics itself cannot present a unified front. We have Paul Wilmott and Nicolas Nassim Taleb arguing that the mathematical techniques that dominate the markets today, that of Ioannis Karatzas, Steven Shreve, Mark Davis (whom Wilmott has famously libelled in an ad hominen attack) and a Marek Musiela, to name a few, is the wrong sort of mathematics. This is rather like someone claiming a Toyota Prius is not really a car in comparison to a Dodge Pickup, the fact that the Prius is unfamiliar does not mean it is not technically superior.

However, this does not mean that the academic discipline of financial mathematics does not have some issues to address. The publication of the Heath-Jarrow-Morton framework created a demand for stochastic analysis skills in the markets, displacing the skills in the numerical solution of deterministic differential equations familiar to Wilmott, Taleb, physics and engineering. This demand was met by the universities with a plethora of Financial Mathematics Masters degrees. I feel that now the markets have moved on, but whether many of the MScs are keeping up, I am not so sure.

Part of the problem is that many academic mathematicians are more comfortable walking across campus to chat to their colleagues in the economics or finance departments than talk to mathematicians with direct experience of the markets, such as Claude Shannon, Edward Thorp and James Simons. This means that the orthodoxy of Samuelson and his progeny dominates and ideas such as the Kelly Criterion, and those of stochastic control familiar to electrical engineering, have been missing from the rarefied curricula of some financial maths degrees.

But all this is a discussion of plumbing of the markets, a utility, and mathematics is not really a utility. Mathematics is a science.

For Laplace, the roll of a dice is not random, given precise information of the position, orientation and velocity of a dice when it left a cup, the result of the roll was perfectly predictable. At the heart of Laplace's determinism was knowledge, and `probability' was a measure of ignorance, and not of 'chance'. As a product of the Enlightenment, Bernoulli's God is replaced by 'an intellect', Laplace's demon. The positions of Laplace and Bernoulli, however, differ significantly from Cicero who, in De Divinatione, distinguished between the predictable (eclipses), the foreseeable (the weather) and the random (finding of a treasure). But between the Bernoulli's religious and Laplace's atheist conceptions of predestination, there is more than just a change in wording; there is a huge philosophical divide that was one of the key achievements of the Enlightenment.

A persistent problem with determinism is that it, logically, can lead to a collapse in moral responsibility. The syllogistic argument is:
Premise 1.        Actions are either pre-determined or random.
Premise 2         If an action is pre-determined, the entity who performed the action is not morally responsible.
Premise 3.        If an action is random, the entity that performed the action is not morally responsible.
Conclusion.   No one is morally responsible for their actions.

An achievement of the Enlightenment was to realise that moral responsibility should not sit in the conclusion, but as a premise, and the argument became.
Premise 1.        People should be held morally responsible for their actions.
Premise 2.        If someone (i.e. a child) cannot foresee the consequences of their actions they cannot be held morally responsible for their actions.
Conclusion.   Moral responsibility requires that there be foresight.

In order to be 'morally responsible', people needed to have a degree of foresight, which can only be obtained through knowledge, or science. This is the fundamental purpose of science, to enable people to take responsibility for their actions, whether related to the safety of industry or personal diet. This was reflected in Humboldt's view that education should turn 'children into people', but very different from Bacon's opinion that 'knowledge is power'.

Society needs science to interact with the markets because science creates knowledge, knowledge enables foresight and foresight leads to responsibility. If there is no science of finance, there can be no responsibility in the markets (if the Enlightenment was right).

Poincare dismissed the idea of 'science for science's sake', science is not a recreational pursuit. Scientists need to ask the difficult questions at the extremities of knowledge and mathematics role is to tackle the questions that cannot be answered by experimentation. This is why the the $3 billon investment in the Large Hadron Collider, in looking for the Higgs Boson, is seeking to prove a mathematical derivation. Physical sciences are impotent in reaching out to the boundaries of knowledge without mathematics clearing the path.

The financial markets cannot be experimented on. The very fact that they are complex means that the only tool science has in trying to understand them is mathematics. The fact that the, predominantly, deterministic mathematics based on physical phenomena that most people are familiar with (even frequentist or objective probability is rooted in the 'physical' act of counting) is insufficient to understand the markets does not mean that mathematics will not provide the key to understanding the markets. The point is, it will be "mathematics, but not as we know it", it needs to be created.

If society wants to understand the markets, and really wants them to act responsibility, it needs to fund financial mathematics on a par with the investment made into the physically very small or the very distant.

An Engine, Not a Camera: How Financial Models Shape Markets

2011-08-19T01:57:00.000-07:00

This the text of a book review recently published in the Annals of Actuarial Science, Vol. 5, part 2, pp. 297–298

Robert Peston, the BBC’s Business Editor who broke the news of the Credit Crisis to the British public in 2007, has said that in the months leading up to the crisis he had tried to report on Collatorallized Debt Obligations but had not been able to find a banker who could explain them to him (see What is Financial Journalism for? Ethics and Responsibility in a time of Crisis and Change p 20). This point gets to the crux of the Credit Crisis, “no one” saw it coming and the responses of the US and UK governments was inadequate because there had been no public discussion of developments in finance, as there are of developments in, for example, energy, nano or genetic technologies. The result was society was completely unprepared for what would unfold in 2007-2008.

In writing An Engine not a Camera Donald MacKenzie, a sociologist working in ‘Science and Technology Studies’, hopes that there will be “richer conversations” about financial markets, and out of these discussions a better, and stronger, financial system will emerge. Science and Technology Studies examines how social factors affect technological progress and one of its pioneers was Robert K. Merton, father of one Robert C. Merton. While STS is not popular amongst many physical scientists who like to believe their knowledge is based on indisputable fact, there are numerous financial mathematicians who, following Poincaré (see The Value of Science especially Science and Hypothesis) , see facts in the context of theory, which is constructed in a social context.

The central thesis presented by MacKenzie is that modern financial markets ‘perform’ financial theory. That is, finance practioners have been trained by universities to believe a set of cohesive ideas which they then, on graduation, employ in the markets. Since the ideas are the same, whether presented in Europe or in salt-water or fresh-water universities in the US, the heterogeneity of countless individual agents is replaced by a single Homo economicus. The immediate consequence of this is the commoditisation of finance theory; models are ‘shrink wrapped’ and become ‘black boxes’ with the subtleties of their underlying assumptions are irrelevant. The more dramatic result is that, eventually, the ‘performed’ markets become so far removed from the reality of finance that that they collapse in an episode of ‘counter-performativity’. MacKenzie describes in detail two examples of what he sees as counter-perfomativity, Black Monday in 1987 and the failure of Long Term Capital Management eleven years later.

In support of this hypothesis MacKenzie discusses in detail how modern derivative markets emerged and how, alongside the evolving markets, modern finance theory was created. His history of the development of finance theory, between 1950 and the 1990s is possibly unrivalled being based on extensive interviews with pretty much everyone who has made a significant contribution, including Markowitz, Samuelson , Friedman, Merton, Scholes, and Harrison. Underpinning these narratives is a discussion of the relationship between the ‘practice’ of finance and the academic theory, and it is in this context that MacKenzie offers an explanation why Black-Scholes-Merton received the Nobel Prize while Thorp and Kassouf have been generally ignored in the university classroom.

While a reader may have an issue with MacKenzie’s core hypothesis, his account of how finance has developed makes the book is essential reading for anyone who is seriously interested in modern financial theory.

The book’s weakness is that it was written in the aftermath of the failure of LTCM, a time when the importance of the martingale approach to derivative pricing was only beginning to be understood. While the text does discuss the work of Harrison, Kreps and Pliska it does not go on to discuss the importance of the ‘Fundamental Theorem’ (that a market is arbitrage free if a risk neutral pricing measure exists, and complete if the measure is unique). Ironically for sociologists, the Fundamental Theorem is based on Black and Scholes's observation that it should not be possible to make a risk-less profit, which is almost a statement of commercial morality and so is socially constructed.

The strength of the book is that it highlights the dangers of adopting financial models as black-boxes rather than crafting solutions relevant to specific situations. This is highlighted in his analysis of the failure of LTCM, where MacKenzie rejects many of the popular explanations for the fund’s failure, making the observation that criticising hedge funds for taking on risk is rather like criticising aeroplanes for leaving the ground. While the events of 1998 may seem irrelevant in the aftermath of 2007, MacKenzie has undertaken a similarly comprehensive review of the Credit Crisis in the context of performativity and counter-performativity. Again the message he delivers is to employ fundamental skills that can question the received wisdom of the dominant models, a message that finance would do well to heed.

Isaac Newton: Financial Regulator

2011-08-12T08:50:00.000-07:00

Alongside the stock market boom of the 1690s was rampant inflation, caused by the fact that much of the coin circulating in England had been clipped and there was widespread counterfeiting. The inflation came about because, at the time, silver was an absolute measure of price. If English coins lost silver, more were needed to pay the foreigners, and so the importers would ask for more coin from their domestic customers. [Craig, 1946, p 8]

This, physical, process was compounded by the growth in banks’ lending, increasing the supply of money chasing, pretty much, the same volume of goods, further depressing the value of money. The result was a collapse of confidence in the value of the coin that the English government was using to pay its debts. In 1696 the government decided that confidence would be restored only if the Royal Mint replaced all the silver coin in circulation with sound coin.

Almost simultaneously, and apparently not by design, Newton was appointed Warden of the Royal Mint in April 1696. While the publication of Principia had made little impact during the last days of James II Stewart’s reign, when William III took up residence at Hampton Court, Christiaan Huygens, whose elder brother Constantijn Huygens was an adviser to William, introduced Newton to the new court. Another of William’s advisers was the English Puritan philosopher, John Locke, who had been a political exile in the United Provinces since 1683. Locke and Newton became firm friends and Locke took it upon himself to secure a lucrative post for Newton, who seems to have become bored with Cambridge after exposure the excitement around William’s court, filled with soldiers, statesmen, bankers and scientists [Levenson, 2009, p 44–46].

The period 1690–1695 proved difficult for Newton, his work gravitated to theology and he developed an intense relationship with a young Swiss man, Nicolas Fatio de Duiller, which ended in 1693 and shortly after Newton appears to have had a nervous breakdown (He had had one in 1678 also) . In the aftermath, Locke and the President of the Royal Society, who also happened to be the Chancellor of the Exchequer, pulled strings and secured the Wardenship for Newton.

While Newton, along with everyone else, had been asked to give his opinion on what to do about the coinage, it seems the initial appointment was more ceremonial than practical. The Warden was the monarch’s representative at the Mint and had historically been the most important post there until the crown stopped charging seiniorage, a fee for minting coins, in 1666. After that point the Master, the manager of the mint, became its most important, and best paid, employee [Craig, 1946, p 1–4].

The re-coining, by emptying the economy of coins, precipitated what has been described as “the gravest economic crisis of the century” [Murphy, 2009, p 56], a century that included the devastation of the Civil War, the stop on the Exchequer and the boom and bust of the infant stock market. The government were not happy and, in time honoured fashion, pointed the finger at the Mint’s operations rather than blame their own policies.

Newton, as Warden, was called in front of the Parliamentary Committee on Mint Miscarriages in 1696. One of the witnesses for the prosecution, appearing in 1697, was William Challoner who proposed a series of improvements for the Mint, which Newton dismissed. The Committee of politicians preferred the advice of Challoner over that of the physicist and so Newton responded to this turn of events ‘rationally’, by locking Challoner in Newgate prison. Newton, as Warden could do this, but had to release Challoner after seven weeks, at which time Challoner, as might be expected, stirred up a hell of a row.

Around a year after Challoner’s first appearance in front of the Committee, Newton was called before it to justify his his treatment of the man. Newton had spent the intervening year, and £9 10s, investigating Challoner. It turns out that Challoner had started out as a labourer but through counterfeiting, theft, fraud and duplicity, become a wealthy gentleman. His whole ploy, it seems, had been to get appointed to the Mint, to support his criminal activities. Challoner was executed in March 1699 following a prosecution managed by Newton, who was ‘promoted’ to Master of the Mint at the end of the year [Craig, 1946, p 17–19].

Being Master gave Newton an average income some 16 times what he would have had as an academic. Today an ‘average’ salary for a jobbing professor is around £60,000, and we could expect the Lucasian Professor at Cambridge to earn somewhat more than this. On this basis, the equivalent salary for Newton as Master of the Mint is in excess of £1 million [Levenson, 2009, p 239]. Newton no longer needed the security of his position in Cambridge and he resigned his Professorship at the end of 1701.

Newton is often presented as being difficult to get along with, for example a positive assessment published by the Royal Society in 1995 gives the following summary

Isaac Newton was a humourless, solitary, anxious, insecure and private man with obsessional traits. He was poor at human relationship, such as the expression of gratitude [Keynes, 1995]

However this picture is at odds with the picture painted by Sir John Craig, who wrote about Newton’s work at the mint in Newton at the Mint in 1946,

Newton appears to have been a good judge and handler of men, and he had some magnetism which in many engendered an extraordinary regard and respect [Craig, 1946, p 119]

and “He had the tact to allow for political or human aspects”. Moreover,

he was a good bureaucrat, he insisted on the preservation of clear and exact records …[he had] care and restraint in considering new outlays of public money, with indifference to waste or extravagance that had become customary [Craig, 1946, p 120]

Craig's views are supported by the fact that John Maynard Keynes described Newton as “one of the greatest and most efficient of our civil servants,” no small praise from the man who managed Britain’s finances during the First World War.

Newton’s more anti-social behaviour is attributed to insecurity originating in Newton’s difficult childhood, an insecurity that seems to have remained with him throughout his time in science. However, Newton seems to have taken to his role at the Mint like a duck to water, and maybe being able to see the broader relevance of his intelect to the wealth of the nation, gave Newton the confidence to start enjoying life.

Newton did not completely abandon science, he became President of the Royal Society in 1703, holding the post until his death, but his last significant work, Optiks, published in English in 1704, was based substantially on research undertaken in the early 1670s. However, Newton played an active role in finance for the rest of his life. One of his most significant acts was in setting the exchange rate between gold and silver in 1717, an exchange rate that would shift Britain from the silver standard to the gold standard.

English money had always been associated with silver, giving us the term ‘sterling silver’, this was in common with much of Europe, India and China but different from the Middle East, who based money on gold. By 1710 Britain was running short of silver coin because the amount of silver in a coin was making it worthwhile to convert coin into dinner services. Newton’s solution was to reduce the amount of silver in a coin, a pound of silver should be used to make 64.5 shillings instead of 62 shillings. This was not popular with the government, as it was seen as devaluing the coin, and so would result in inflation. Therefore Newton, according to Craig,

translated the conclusion into the mathematically equivalent but impractical proposition that silver coin should retain its weight and be left to rise in value by force of scarcity [Craig, 1946, p 107].

This situation continued until in 1717 the government saw the solution as increasing the value of the Britain’s main gold coin. On Saturday 21 December 1717 the government asked Newton to fix the sterling silver exchange rate with the main gold coins used throughout Europe, in a report ‘blinding with science’ [Craig, 1946, p 108], the value of the British gold guinea increased from being worth only 20s of silver, to being worth 21s. The Law of Unintended Consequences kicked in, Newton had undervalued foreign gold coins. The result was described by Adam Smith in his Lectures some forty-five years later

As silver buys more gold abroad than at home, by sending abroad silver they bring gold in return, which buys more silver here than it does abroad. By this means a kind of trade is made of it, the gold coin increasing and the silver diminishing. Sometime ago a proposal was given in to remedy this, but it was thought so complex a case that they resolved for that time not to meddle with it. [Fay, 1935]

The direct consequence of this was that silver coin became increasingly unpopular, and in 1816, Gold was declared the “sole standard measure of value” [Fay, 1935], putting into the force of law what the markets had started doing a century earlier.

Shortly after defining the Gold Standard, Newton was caught up in the South Sea Bubble of 1720. Newton had been an early investor in the South Sea Company and in April, in the early days of the Bubble, he liquidated, doubling his money with a profit of £7,000 (roughly £700,000 in today’s terms). However, he was drawn into the mania again later that summer, and, according to his half-niece, the famously beautiful Catherine Barton, he lost £20,000 (roughly £2 million in today’s money!) [Kindleberger, 1996, p 28]. Craig notes that Catherine "was a hard woman”, and the ‘loss’ was never realised but rather the potential profit had Newton maintained his original holding and sold at the top of the market [Craig, 1946, p 112]. None the less, Newton was left to observe that

I can calculate the motion of the heavenly bodies, but not the madness of men.

References

J. Craig. Newton at the Mint. Cambridge University Press, 1946.

C. R. Fay. Newton and the Gold Standard. Cambridge Historical Journal, 5(1):109–117,
1935.

M. Keynes. The Personality of Isaac Newton. Notes and Records of the Royal Society of
London, 49(1):1–56, 1995.

C. P. Kindleberger. Manias, panics and crashes: a history of financial crises. Wiley, 1996.

T. Levenson. Newton and the Counterfeiter: The Unknown Detective Career of the World’s Greatest Scientist

A. L. Murphy. The Origins of English Financial Markets. Cambridge University Press,
2009.

History repeating itself?

2011-07-28T07:18:00.000-07:00

The macro-economics of August 2011 looks much the same as August 1971, but have we learnt anything about the markets over the past forty years?

Following the 1929 Crash and the consequent world-wide Depression governments around the world had devalued their currencies in order to make their products more competitive in foreign markets. These ‘beggar-thy-neighbour’ policies created a deflationary spiral that magnified the effects of the Crash. In 1944 the Allied powers met at Bretton–Woods, in New Hampshire, and agreed to fix the gold price of the main currencies, the US$ was fixed to gold at $35/oz, while other currencies were pegged to the dollar with the pound sterling being set at $4.03. Bretton–Woods also established the International Monetary Fund and World Bank.

By the late 1960s the Bretton–Woods system was beginning to creak as the Germans and Japanese exported to the Americans,while the U.S. poured money into th e war in Vietnam. As gold was sucked out of the U.S. the system began to look untenable. In 1971, foreign governments demanded that the U.S. honour its “promise to pay” and convert their dollar notes into gold, in July Switzerland converted $ 50 million into gold. There was an arbitrage, buy gold with dollars and then sell the gold for Deutsche Marks. On August 15, 1971 the U.S. President, Nixon, responded to these activities by abandoning the gold-standard, the “promise to pay”. Bretton Woods collapsed and foreign exchange rates stopped being
certain.

As a consequence of the collapse of the Bretton–Woods system of exchange rates central banks were forced to change the interest rates more frequently. In simplistic terms, the level of interest rates has two effects. If rates are low people will borrow from banks, who will create money for the economy and this may generate inflation which devalues a currency. If interest rates are high, and the currency stable, foreign investors will like to deposit their spare cash in banks paying the high rates of interest, raising demand for the currency. After 1972 interest rate policy became a key lever that governments had to control their economies. In the 27 years between 1945 and autumn 1972, when Bretton–Woods collapsed, the Bank of England changed its lending rate 43 times, in the 27 years after 1972, it changed them 223 times, about every 45 days. Finance had moved from a world of deterministic control to one of stochastic control, and people had to think more carefully about controlling their financial risks.

Business responded to this change in the economic environment by returning to the derivatives markets, using them to provide the tools to hedge the risks, whether as a borrower or a lender, of the fluctuating the interest rates. In the same year that Bretton-Woods collapsed, Nixon appointed William Casey, a spy and tax lawyer, as director of the Securities and Exchange Commission (SEC) and the path for the the derivatives exchanges was opened. A currency future had been created in New York in 1970, but had foundered. However when, on May 16, 1972, the Merc began trading futures on seven currencies the market for FX risk management was there and the Merc was rescued from the doldrums of the 1960s.

While futures or forward contracts, firm agreements to buy or sell an asset at a fixed price in the future, existed in an ethical and legal limbo, option contracts, contracts that gave the holder the right, but not the obligation, to trade were closer to the devil. As late as the 1960s officials of the SEC had compared options to thalidomide and marijuana and claimed that there had never been a case of market manipulation that did not involve options [MacKenzie, 2008, p 149]. Casey, and the SEC, cleared the Chicago Board of Options Exchange and it opened on April 26, 1973. Within days, the Journal of Political Economy, the house journal of the Chicago economists, published a paper, The Pricing of Options and Corporate Liabilities by Myron Scholes and Fischer Black.

The CBOT launched the first interest rate derivative in 1975, where the underlying was linked to mortgages, in 1976 the Merc introduced a future on 30-day U.S. Government Treasury bills, and CBOT launched a future on 30-year U.S. Treasury bonds in 1978. The London International Financial Futures Exchange opened in 1982 and in 1984 the equivalent of a stock–market index for interest rates, the London Interbank Offered Rate, LIBOR, began to be published, and futures, known, confusingly, as Eurodollar futures, began to be traded on the Merc based on this index.

It was not only the capital markets that were transformed in the 1970s. Up until 1973 the price of oil was set by the Railroad Commission of Texas, who controlled the oil production in Texas, and hence the oil price in the U.S., which as the world’s main oil consumer, effectively set the world price. Following the collapse of Bretton–Woods, the U.S. dollar’s value fell and as a consequence, the real income non-U.S. oil producers fell, and the Organisation of Oil Producing and Exporting Companies, OPEC, began to price their oil in gold and agree production quotas, setting the global–price. In 1973 the Middle-Eastern members of OPEC imposed an embargo on the west, following the defeat of the Syrian–Egyptian attack on Israel, and cut production, forcing the price of oil up. This, in-turn, prompted the development of alternative oil–provinces, notably the North Sea between the U.K. and Norway, which had been
previously un-economic. When this extra production hit the market, in the 1980s, just as demand fell, as consumers cut back consumption in response to higher-prices, and Iran and Iraq exceeded their quotas to fund their war (1980–1988), prices collapsed along with OPEC’s cohesiveness. In 1985 OPEC’s price-setting mechanism was abandoned, and another key economic input became a stochastic process, and in response, in 1988, the London based International Petroleum Exchange introduced the Brent oil futures contract.

Behind Black Monday, the failures of LTCM and The Equitable and the Credit Crisis of 2007–2008 is the fact that the financial world became stochastic in the aftermath of the collapse of Bretton–Woods,. The derivative markets did not spontaneously appear, they developed in response to the increased uncertainty in key economic drivers.

The world of 1980 was very different to that between 1918 and 1970 when exchange, rates, interest rates and commodity prices were being controlled by the great and the good. One view might be that the abandonment of Bretton–Woods had lead to the chaos of 1970s stagflation, another is that Bretton–Woods shattered under the strains of trying to confine the economy to a specific path. The derivative markets emerged in response to freeing the economy, the deterministic policies created impossible stresses in the global economy, and derivatives enabled risk–management in the resulting uncertainty.

The derivative markets were fundamentally different from the stock-markets, where decisions were made based on an investor’s judgement of the market fundamentals, and the core skills where in understanding economics and a company’s balance sheet, finance. The derivative markets were concerned with comparing the price differences between similar assets across different markets. It was not about the study of objects, but the relations between objects, and the derivative markets needed mathematical skills. In the aftermath of Black Monday significant numbers of applied mathematicians, physicists and engineers began working in the derivative markets, the ‘quants’ had arrived.

The events of August 2011 are not so different to those of August 1971, but then again these events are not so different to those of the 1690s!

Trading in stocks did not take off in England until after the Glorious Revolution of 1688. It is often assumed that this was because, as Geoffrey Poitras puts it, “William III was accompanied by an influx of Dutch persons and practises”. However a market does not create itself and, according to Anne Murphy, a historian who has worked as a currency trader, the root cause of the explosion of stock-market trading, and the accompanying boom, was the Nine Years War [Murphy, 2009, p 10–14], [Poitras, 2000, pp 281–285]. Murphy points to a contemporary account written by John Houghton in 1694

a great many Stocks have arisen since the war with France; for Trade being obstructed at Sea; few that had money were willing it should lie idle [Murphy, 2009, quoting on p 12]

It was not just the shortage of opportunities to participate in regular trade that stimulated the boom. England had grown wealthy under Charles II and alongside the increase in wealth was a growth in the financial services industry, involving life insurance and annuities, general insurance and the trade in the shares of the handful of joint-stock companies that existed at the time, such as the East India, Hudson Bay and Royal Africa [Murphy, 2009, p 12–19]. Evidence of this growth in financial services is provided by the 1673 Act of Common Council that looked to put an end to “usurious contracts, false Chevelance, and other crafty deceits” [Murphy, 2009, p 83].

The additional risks of sea trade resulting from the war with France, and the actions of privateers meant that merchants looked for domestic investment opportunities and the number of joint-stock companies exploded, and with more companies came a more active stock market. John Houghton describes how the market worked

The manner of managing Trade is this; the Monied Man goes amongst the Brokers (which are chiefly upon the Exchange [Alley], and at Jonathan’s Coffee House [the origins of the London Stock Exchange], sometimes at Garraway’s and at some other Coffee Houses) and asks how Stocks go? and upon Information, bids the Broker to buy or sell so many Shares of such and such Stocks if he can, at such and such Prizes [Poitras, 2000, quoting on p 288].

Brokers put buyers and sellers in touch with each other, for a commission but without actually taking possession of any asset. Alongside the monied men and brokers were the stock-jobbers or dealers, the speculators, providing liquidity in the market and trying to turn a profit from their trading. This dual-system, separating brokers from stock-jobbers, existed in London up until the ‘Big Bang’ of 1987.

In 1719 Daniel Defoe published Robinson Crusoe and wrote an article The Anatomy of Exchange Alley in which he described stockjobbing as

a trade founded in fraud, born of deceit, and nourished by trick, cheat, wheedle, forgeries, falsehoods, and all sorts of delusions; coining false news, this way good, this way bad; whispering imaginary terrors, frights hopes, expectations, and then preying upon the weakness of those whose imaginations they have wrought upon [Poitras, 2000, quoted in p 290]

An observation, mentioned by Defoe but more explicitly stated by Thomas Mortimer in 1761, concerned the type of person involved in stockjobbing. Mortimer makes the point that there are three types of stock-jobber, firstly foreigners, secondly gentry, merchants and tradesmen, and finally, and “by far the greatest number”, people

with very little, and often, no property at all in the funds, who job in them on credit, and transact more business in several government securities in one hour, without having a shilling of property in any of them, than the real proprietors of thousand transact in several years. [Poitras, 2000, quoted in p 291]

It was not only stocks that were being traded in the first half of the 1690s. Murphy estimates that around 40% of the trades between 1692 and 1695 were in stock options, that were being traded in order to manage the risks of stock trading. [Murphy, 2009, p 24–30] Evidence of the widespread use of options comes in 1720 when Colley Cibber, who would become Poet Laureate in 1730,, wrote a play, The Refusal (‘The Option’), describing the action in Exchange Alley

There you’ll see a duke dangling after a director; here a peer and ‘prentice haggling for an eighth; there a Jew and a parson making up the differences; there a young woman of quality buying bears of a Quaker; and there an old one selling refusals to a lieutenant of grenadiers [Ackroyd, 2001, p 308]

Clearly, in 1720 the public were familiar with the trading of derivatives, pretty much everyone was involved, and social, religious and political differences were forgotten in the markets.

The stock market boom that started in the late 1680s had gone bust by the middle of the next decade. At the time it was popular to blame stock-jobbers for destabilising the economy by either ramping worthless stock or undermining a going concern (depending on your point of view) [Murphy, 2009, p 33], while, more fundamentally, the stock market was attacked for turning “men away from honest and beneficial trades” [Murphy, 2009, p 68]. More rational explanations were that many of the joint-stock companies were mis-managed and the government’s need for cash sucked funds out of the market, causing prices to collapse [Murphy, 2009, p 35].

Anyone born before 1970, unless they have actually worked in the markets, find it difficult to understand the changes in the financial environment that had occurred after the collapse of Bretton–Woods. Derivatives would not start appearing on undergraduate courses in universities until the mid 1990s, and even then, for many of the lecturers in economics and finance educated in the post-war deterministic economies, they would be unfamiliar beasts. This meant that at the turn of the century there was a dire shortage of people with the skills to understand the complex world of derivatives [Tett, 2009, p 68], which required a unique grasp of financial theory, market practises, applied mathematics, probability and
statistics.

At the same time, some banks chose business managers as their chief executives, such as Fred Goodwin at the Royal Bank of Scotland or Andy Hornby at Halifax–Bank of Scotland, rather than ‘bankers’ bought up on the basic process, and the uncertainties, of converting credit into cash. The consequence of this was that some banks focused on efficient profit generation, the allocation of scarce resources, at the expense of monitoring the risks of their activities, managing an uncertain world. This manifested itself by firms buying in expertise, in the form of ‘black–box’ software systems to value the CDOs combined with external (or internal) consultants for advice. Some banks were not only out–sourcing their call centres, but their brains as well.

In the lead–up to the Crisis of 2007–2008, RBS sponsored sports-stars to the tune of 200 million pounds, in the same period they invested nothing in mathematics. J.P. Morgan was different, they developed, in–house, Value at Risk, CreditMetrics and employed David Li as he thought about using copulas to price CDOs. The quants that that they employed were able to develop these tools because they had a deep understanding of the markets and mathematics, and critically, how and where the mathematical models were weak and needed to be augmented. Not only did J.P. Morgan recognise the need for these skills, a feature shared by all the serious investment banks, but in disseminating their models they advertised
that their expertise was a fundamental component of “first class banking in a first class way”.

The benefit of J.P. Morgan’s approach is described by Gillian Tett in her account of the Credit Crisis, Fool’s Gold. The background is it is 2005–2006 and J.P. Morgan’s shareholders are putting the bank’s managers under intense pressure to mimic the revenues being reported by other investment banks, who were actively investing in CDOs of MBS.

[The J.P. Morgan chief executive] made it clear that he wanted a mortgage production line, so Winters had duly asked his staff to re–examine how to create a profitable business selling mortgage–based CDOs.
When they crunched the numbers, though, they ran into a problem. “There doesn’t seem to be a way to make money on these structures,” Brian Zeitlin, one of the bankers who worked in the CDO division reported. …
Reluctantly, Winters told the J.P. Morgan management should not open the spigots on its pipeline after all. The decision was greatly frustrating, though. The other banks were pushing JPMorgan Chase further and further down the league tables largely due to the bonanza from their mortgage pipelines. So were they just ignoring the risks? Or had they found some alchemy that made the economics of their
machines work? [Tett, 2009, pp 148–151]

The J.P. Morgan quants had taken the prices the traders were observing in the market and reverse engineered them, just like they did with the Black–Scholes pricing formula, extracting the key parameter, ρ. When they told the traders that the basis of the prices in the market was ρ = 0.3, the traders could not believe it. A correlation of ρ = 0.3 implies that there is only a small linkage between defaults, reflecting the fact that if BP went bust it did not mean that Tesco would follow suite. But anyone who had seen an economic downturn would be familiar with whole streets being derelict, the correlation of mortgage defaults was unknown, but not insignificant.

There is another aspect to the approach taken by the banks that weathered the storms of 2007–2008. While there is frequently the claim that the Credit Crisis was a global phenomenon, it was not. Asian banks were unaffected and British and American banks suffered far more than French or German banks. The explanation that banks were not as involved as RBS or Merrill Lynch is not a satisfactory answer (BNP Paribas, SocGen and Deutsche Bank were all heavily involved in credit derivatives. [FCIC, 2011, Fig 20.4, for example]). U.S. bankers have been known to suggest that the non Anglo-Saxon banks played fast and loose with accounting rules, not declaring their losses on credit derivatives. The response to this criticism from French mathematicians is typically Gallic, and Cartesian, “They thought the models were wrong before August 2007, they were certain they were wrong after August 2007, so why should they post losses that they were certain were wrong.”. This point was highlighted by a quote that appeared in The Economist magazine, in January 2008 in relation to a fraud at the French bank, SocGen

In common with other French banks, SocGen was also thought by many to take an overly mathematical approach to risk. “ ‘It may work in practice but does it work in theory?’ is the stereotype of a French bank,” says one industry consultant. (‘No Defense’, The Economist, 31 January 2008.)

The bankers at J.P. Morgan, along with French risk–managers, kept in mind Hume’s observation that “it is never contradictory to deny matter of fact”. Bankers, like all scientists, must use their intellect and constantly ask themselves the questions ‘why’ and ‘how’ to give them the foresight not to act recklessly.

The common thread linking financial crises since Bretton-Woods, Black Monday, the Equitable, the super–portfolio that bought down LTCM, the tech–bubble of 2000 and the Credit Crisis was not the collapse of Bretton–Woods but the adoption of standardised approaches to finance. Had the majority of traders, bankers and regulators thought like mathematicians, or French risk managers, and asked themselves, in a Cartesian manner, “how do I know what I think I know is true?”, then the crises might have been avoided. Banning speculative trading in deriviatives is simplistic, and not the answer.

References

P. Ackroyd. London: The Biography. Vintage, 2001.

FCIC. The Financial Crisis Inquiry Report. Technical report, The National Commission
on the Causes of the Financial and Economic Crisis in the United States, 2011.

D. MacKenzie. An Engine, Not a Camera: How Financial Models Shape Markets. The
MIT Press, 2008.

A. L. Murphy. The Origins of English Financial Markets. Cambridge University Press,
2009.

G. Poitras. The Early History of Financial Economics, 1478–1776. Edward Elgar, 2000.

G. Tett. Fools’ Gold. Little Brown, 2009.

Why magic? ⇔ Why gold?

2011-07-25T07:18:00.000-07:00

Someone has asked me why this blog is Magic, maths and money.

In 1902 two sociologists, Marcel Mauss and Henri Hubert wrote ‘Outline of a general theory of magic’ where they identified that

The magician is a person who, through his gifts, his experience or through revelation, understands nature and natures …Owing to the fact that magicians came to concern themselves with contagion, harmonies, oppositions, they stumbled across the idea of causality, which is no longer mystical even when it involves properties which are no way experimental [Mauss and Hubert, 1902 (2001), pp 94–95]

The origins of magic in primitive societies is the idea that there is some sort of causal link between a ritual and a phenomenon and as such, the individual can control nature, making it less unpredictable.

Recently, the history of science has identified one of the motivations for scientists in Europe during the sixteenth and seventeenth centuries was a desire to find hidden mechanisms of cause and effect, mechanism that they could then employ to control the universe [Dear, 2001, p 52], [Henry, 2008, Chapters 5], [Hall, 1962, p 159]. The belief that these hidden mechanisms actually existed was inspired by what would now be considered as magical beliefs [Dear, 2001, p 25], [Henry, 2008, Chapter 4] and as the historian of science, John Henry, puts it

The history of magic since the the eighteenth century has been the history of what was left to the tradition after major elements of natural magic had been absorbed into natural philosophy. [Henry, 2008, p 57]

By their very nature, these mechanisms were uncommon and had to be discovered and the process of exploration in the world was replicated by experimentation in the laboratory.

In the ‘general theory of magic’ book, Mauss and Hubert distinguish magic and science by observing that magic is based on belief in a set of rituals [Mauss and Hubert, 1902 (2001), p 114]. A person will only consult a magician if they have faith in the actions that the magician will perform. Science is not based on belief in its theorems, the equivalent of magic’s rituals, but on a belief in the process by which science is created. Science is a process that creates rituals, not simply a collection of rituals. This is a subtle point, but the effect is that magic is necessarily static, a contemporary astrologer would have more authority if they claimed to be experts in ancient knowledge. On the other hand, science is necessarily dynamic, we trust modern science’s explanations of the universe more than those of the Babylonians.

While magic and science are distinguished by static or dynamic belief, Mauss and Hubert distinguish magic and religion by hidden and open belief

Where religious rites are performed openly, in full public view, magical rites are carried out in secret. …and even if the magician has to work in public he makes an attempt to dissemble: his gestures become furtive and his words indistinct. [Mauss and Hubert, 1902 (2001), p 29]

The suggestion is, for science to be reputable and maintain a divide with magic, it needs to be carried out, like religion, in the open. As soon as either science or religion takes place out of the public arena, they risk degenerating into magic.

The choice of the word ‘magic’ is to highlight that financial mathematics needs to be a process of developing and refining theories, not simply of adopting sets of rituals (like pricing assets using risk neutral valuation), and that it needs to be conducted in a public forum. The public should be familiar with the “equations that blow up Wall Street” before the explosions.

This might seem a little metaphysical, non-scientific, but then isn’t this why financial mathematics is not just interesting. For example, with concerns of a U.S. default, the price of gold keeps climbing – but why gold?

Around 4,000 years ago, people started making ornaments out of electrum (an alloy of gold and silver), copper and gold [Diamond, 1998, pp 362–363], metals found naturally (i.e. without processing) in nature. Metals have an almost unique, natural, physical property; they reflect light. The only other material that stone-age humans would have come across that reflected light would have been water, so to these people gold would appear to combine the essence of both water and the sun, the basis of life [Betz, 1995], [Landes, 1999, pp 70–73]. Imagine the awe that humans would have felt the first time they spotted a nugget of gold sparkling in a river bed, here was an object that seemed to captured and store life-giving sunlight, the ‘tears of the Sun’ as the Incas said. In the medieval period, European alchemists believed that metals were produced by some mechanism involving rays from different ‘planets’: gold from the Sun, silver from the Moon, mercury from Mercury, copper from Venus, iron from Mars, tin from Jupiter and lead from Saturn.

In ancient Babylon, Egypt and Greece, temples became associated with stores of metals, gold for the Greeks, silver for the Babylonians and copper for the Egyptians. It seems that these metals had developed a religious significance and become important as temple offerings. Consequently followers of the religion would look to acquire the metal, to enable them to make an offering, and so the metal became the commodity in the most universal demand. [Pryor, 1985] Athens treasury was in the Temple of Athena, and Jesus cast the money-lenders, exchanging worldly Roman money for divine shekels, out of the Temple.

The earliest tokens used as ‘money’ were not specific weights of a certain metal but roughly cut pieces of metal with an official stamp on them [Ingham, 2004, p 98] – monopoly money as it were. The emergence of money, in the sense of coins, in Greece coincides with the emergence of mercenary troops, the term ‘soldier’ is derived from the word for a Roman gold coin, solidus. A simple economic model developed, states paid soldiers in gold, who then spent it in the community. The government then recovered the gold by taxing the merchants and innkeepers that the soldiers had paid for food and lodgings. This model would survive and drive colonialism until the modern age. A power, such as Alexander’s Greece, Imperial Rome, Napoleonic France or Industrial Britain, would take control of a region through force of arms. They would then demand tax from the conquered nation, which would have to be paid in currency specified by the coloniser. The conquered nation could only obtain the currency by exchanging their produce for the specified currency. For example, in the 1920s the British taxed Kenya at a rate of about 75% of wages, while the Belgians did not tax the Congo – they relied on forced, rather than ‘free’ wage labour [Ingham, 2004, p 76].

In the 1920’s, John Maynard Keynes became obsessed with the origins of money [Ingham, 2004, p 209, Note 16]. In neo-classical economic analysis, money is a consequence of exchange, and from money comes banking and then the ‘credit relations’ between borrowers and lenders, pretty much as Aristotle had conceived things. However, at the start of the twentieth century, anthropologists studying primitive societies observed that cultures that had not developed money, tokens to facilitate exchange, none the less, lent and borrowed.

One of the most studied examples of these sorts of systems was that of indigenous people around Vancouver in North Western Canada. A young man would lend five blankets to an older, richer person, for a year and they would be repaid with ten blankets. A similar situation existed in the Southwestern Pacific were strings of shells were lent by a young man, sometimes to an unwilling borrower, at very high rates of interest. The value of the shells was was purely ceremonial, they bought entry to the fraternities of adult men [Homer and Sylla, 1996, pp 22–23]. In fact many cultures had systems where by a gift had to be reciprocated by a greater gift in return [Mauss, 1924 (2001)] and it could be disastrous to be given a present, since it would require the return of more han what was given. What these very primitive societies had done was create credit-debit relationships, reciprocity, which played a critical role in gluing society together by creating bonds between the rich and poor, the old and young. [Sahlins, 1972 (2003], [Mauss, 1924 (2001)]

The anthropologist Marcel Mauss summed up the research in a paper on 1923, ‘An essay on the gift: the form and reason of exchange in archaic societies’

The evolution in economic law has not been from barter to sale, and from cash sale to credit sale. On the one hand, barter has arisen through a system of presents given and reciprocated according a time limit. …On the other hand, buying and selling arose in the same waywith the latter according to a fixed time limit, or by cash, as well as by lending. For we have no evidence that any of the legal systems that have evolved beyond the phase we are describing [primitive society] (in particular, Babylonian law) remained ignorant of the credit process that is known in every archaic society that still survives today. [Mauss, 1924 (2001), pp 46–47]

Keynes knew that he Babylonians had established credit relations. There was a Babylonian prototype of the Bill of Exchange from a time when the Babylonians did not use coins and in the 1930 Treatise on Money Keynes, like Mauss, argued that money did not, in fact, originate in the market place, rather it was created by the state or community ‘the age of money had succeeded to the age of barter as soon as men had adopted a money of account’ [Keynes, 1971, p 4]. Here, ‘money of account’ is the generally accepted medium by which debts are settled, for example the Babylonians settled debts, typically tax debts or loans from temples, in either grain or an amount of silver, and this medium was defined by the state and based on trust in credit relations, the state theory of money or money as a ‘social relation’.

References

O. Betz. Considerations on the real and the symbolic value of gold. In G. Morteani and J. P. Northover, editors, Europe: Mines, Metallurgy and Manufacture, chapter 2, pages 19–28. B. B. Price, 1995.

P. Dear. Revolutionizing the Sciences. Palgrave, 2001.

J. M. Diamond. Guns, Germs and Steel: A short history of everybody for the last 13,000 years. Vintage, 1998.

A.R. Hall. The Scientific Revolution 1500-1800. Longmans, 1962.

J. Henry. The Scientific Revolution and the Origins of Modern Science. Palgrave, 2008.

S. Homer and R. Sylla. A History of Interest Rates. Rutgers University Press, 3rd edition, 1996.

G. Ingham. The Nature of Money. Polity Press, 2004.

J.M. Keynes. The collected writings of John Maynard Keynes. Vol. 5 : Treatise on money 1: The pure theory of money. Macmillian, 1971.

D. S. Landes. The Wealth and Poverty of Nations. Abacus, 1999.

M. Mauss. The Gift: Form and Reason for Exchange in Archaic Societies. (Routledge), 1924 (2001).

M. Mauss and H. Hubert. A General Theory of Magic. (Routledge), 1902 (2001).

F. L. Pryor. The origins of money. Journal of Money, Credit and Banking, 9(3):391–409, 1985.

M. Sahlins. Stone Age Economics. (Routledge), 1972 (2003)

Structured finance in the twelfth century

2011-07-19T06:44:00.000-07:00

From about the twelfth century the Catholic Church began to examine closely the concept of usury, elevating it, without scriptural justification, to a sin equivalent to one of the seven deadly sins [MacCulloch, 2009, p 369].

Usury is sometimes equated with the charging of interest, but by the thirteenth century it was recognised that the two ideas were different. Usury derives from the Latin usura, meaning ‘use’, and referred to the charging of a fee for the use of money. Interest comes from the Latin intereo, meaning ‘to be lost’, and originated, in the Roman legal codes as the compensation someone was paid if they suffered a loss as a result of a contract being broken[Homer and Sylla, 1996, p 73]. So a lender could charge interest to compensate for a loss, but they could not make a gain by lending [Kaye, 1998, p 83].

It is easier to understand this with a simple example. A farmer lends a cow to their cousin for a year. In the normal course of events, the cow would give birth to a calf and the cousin would gain the benefit of the cow’s milk. At the end of the loan, the farmer could expect the cow and the calf to be returned. The interest rate is 100%, but it is an interest since the farmer, if they had not lent the cow to their cousin, would have expected to end the year with a cow and a calf. Similarly, if the farmer lent out grain, they could expect to get the loan plus a premium on the basis that their cousin planted the grain, he would reap a harvest far greater than the sum lent.

Another aspect, which was important in the Jewish tradition, but is, in fact, deeply rooted in all societies [Homer and Sylla, 1996, Chapter 1], [Mauss, 1924 (2001], was that loans should be made to support members of the community who were in distress, and not for the benefit of the lender [Homer and Sylla, 1996, p 71]. This approach was taken to maintain the cohesiveness of the community, a distinguishing feature of Judaism, and so the Talmud prohibited the charging of interest within the Jewish community, but allowed a Jew to charge interest to an outsider [Poitras, 2000, p 77].

By the start of the twelfth century different streams of thought, Greek, Roman, Biblical and feudal, came together at a time of great economic growth and the question of where the dividing line between usury and interest lay became one of the most important questions of the age. At the extreme where the ‘manifest usurers’ who were quite content with flouting Christian doctrine. The most well known example of these usurers were the Jews, who were immune from the Church’s sanction of excommunication and were able to take usury outside of their community.

Pawnbrokers were another class offer manifest usurers. Pawnshops were widely tolerated, and frequently defended by local magnates in retrain for a license fee [Poitras, 2000, p 43], [Homer and Sylla, 1996, p 72]. The last group of recognised usurers were the Lombards. Originally from modern day Hungary, they dominated Italy during the Dark Ages and centred around the port of Amalfi, the Lombards produced communities of “remarkably creative and energetic” merchants [Swan, 1999, p 100], who, for a reason not well understood, “showed a strange insensitivity to ecclesiastical and social censure” [Poitras, 2000, p 43, quoting de Roover, 1948] and quite happily engaged in usury. Over time, the term ‘lombard’ was used to describe any Christian usurer and today many European cities have a Lombard Street at the heart of the financial district.

These manifest usurers were beyond the influence of the scholastics, who focused on establishing the legitimacy of various types of related contracts; poena, census, prestiti, societas and, most widely used, the Bill of Exchange.

In Roman mythology, Poena was the goddess of punishment who accompanied Invidia (the Latin version of Nemesis), who dealt out retribution for excessive pride, undeserved happiness or good fortune, and the absence of moderation. The word poena became synonymous with a penalty for late payment under the terms of a contract. In the medieval world, this Roman principle evolved into the practice of entering into ‘legitimate’ loan contracts that included the implicit understanding that the borrower would delay payment, by and agreed period, incurring the poena, which could be justified to the clerics as a licit interest payment [Poitras, 2000, p 87].

A census originated in the feudal societies as an “obligation to pay an annual return from fruitful property”[Homer and Sylla, 1996, p 75, quoting Noonan], [Poitras, 2000, p 91]. What this means is that the buyer of the census would pay a landowner, for example, for the future production from the land, such as wheat or wine, over a period of time. As economic life in western Europe became based on money transactions rather than barter transactions, censii lost the link to specific produce, cartloads of wheat or barrels of wine. The buyer of the census would accept regular cash payment instead of the actual produce, and this was legitimate in the eyes of the canon lawyers as long as the lump-sum paid buy the buyer ‘equated’ with the value of the ‘fruitful property’ being produced by the seller.

Anyone who could became involved in censii. A labourer might sell a census based on the future revenue from their labour, states sold them based on the future revenue from taxes and monopolies, and the Church invested bequests by buying censii [Homer and Sylla, 1996, pp 75–76], [Poitras, 2000, pp 31–33]. Censii issued by governments, usually linked to specific tax revenues, became known as rentes [Poitras, 2006, p 82]. Censii could be ‘temporary’, lasting a few years, or ‘permanent’, until one of the parties died. In today’s terms, temporary censii resemble modern mortgages, permanent censii resemble the ‘annuities’ pensioners live off today. They could be ‘redeemable’, by one or both parties, meaning that the contract could be cancelled.

The complexity of these innovative contracts meant that they did not have a clear price. For example, a census to supply three cows, every November, for five years had a price that was fairly explicit. But what about a permanent cash census based on someone’s labour? How much was the labour worth each year, and how long would the seller live?

Prestiti were a development from the rentes created by states. Around the twelfth century the Italian city-states of Venice, Genoa and Florence began to forcefully sell temporary rentes to their rich citizens. By the mid-thirteenth century the different issues of rentes were consolidated into a mons (mountain) and everyone who had been made to buy a rente was given a share, proportionate to their contribution, in the mons.

Venice created its mons, the monte vecchio, in 1262 and the shares, known as prestiti, entitled the holder to be paid 5%, a year, of the sum they lent, which was written on the prestiti and known as the ‘face value’. While there was no obligation for the states to pay the coupon, the annual payment, there was an expectation that they would if it could be afforded and the mountain itself was paid back as and when funds allowed [Poitras, 2000, pp35–36].

Quickly a market for Prestiti emerged, where holders who needed ready cash would trade them with people who had a surplus of cash and wanted to save. During times of peace and prosperity they had a high price, but during war and uncertainty, they traded at a low price.

For example, Venetian prestiti traded for their face value around 1340 when the Republic paid off a lot of the mons, but in 1465, during a disastrous war with the Ottoman Turks, they fell to 22% of face. The Florentine prestiti actually had a built in facility where a holder could go to the state and sell them for 28% of their face value, however their market price was never so low as to make this profitable.

The legitimacy of the prestati was debated by the canon lawyers. On the one hand the coupons, the regular cash payments can be seen as compensation for the forced nature of the original loan. The lender had no choice and so does suffer a loss. However, if a prestiti with a face of 100 ducats was sold for 22 ducats, the buyer would be receiving interest at a rate of 5∕22 = 23%; in what way had this buyer of the prestiti been forced to enter into the contract? An interest payment of 23% in these circumstances seemed to be “asking for more than what was given”.

Prestiti are important in that are one of the earliest representations of an actively traded financial instrument. The prestiti does not represent bushels of wheat or barrels of oil, it is a contract where by a state promises to pay a specified amount of money. Whether or not the state does pay out on the contract, is unknown and uncertain, hence the value of the contract is also unknown and uncertain.

The Franciscans, in the fifteenth century, developed the state run mons into the Montes Pietatius (‘mount of piety’). This mons was created by bequests from the wealthy and then the friars would make low interest loans to those not able to borrow money elsewhere. The Montes Pietatius can be seen as a medieval version of micro-finace schemes, now spreading throughout the developing world.

A Societas was a partnership contract. Originally each partner would put a sum of money into an enterprise and at the end of the activity, the enterprise was capitalised, all its assets sold off and converted into cash, which was divided between the partners in the same proportion as their original investments. For example a group Venetian merchants might form a societas to trade with Caffa, the the trading post on the northern shore of the Black Sea. Pooling their money, they would buy a shipment of manufactured goods and hire a ship to transport the goods, where they, or their agents, would trade the manufactured goods for furs and amber, which they would ship back to Venice and sold. The proceeds from these sales would be shared amongst the partnership.

A societas was obviously legitimate, and this type of contract still represents the cornerstone of Islamic finance. The agreement was usually created by a close knit group¹ meaning that an outsider, a foreigner or a poor entrepreneur, without the right connections would find it difficult to participate in a societas. To get around this barrier, the ‘triple contract’ was created.

At the heart of the triple contract was a societas between the entrepreneur and an investor, this was the first contract. The second contract would be an insurance contract taken out by the entrepreneur to insure against the loss the investor’s capital. The third contract was another ‘insurance’ contract given to the investor by the entrepreneur, where by the investor surrendered their rights to a share of the profit in exchanged for a fixed payment from the entrepreneur, this payment was guaranteed by the second contract. In effect, the triple contract was a ‘usurious’ loan to the entrepreneur, wrapped up in a complex financial structure to hide its nature from the prying eyes of the Church. [Poitras, 2000, p38] What is most striking is the similarity between a ‘triple contract’ and the Credit Default Swap.

All in all, at the start of the thirteenth century western Europe was going through a financial revolution (See also [Usher, 1934]). The creation and management of the poena, censii, prestiti, societas and Bill of Exchange required complex negotiation and calculation. The merchant would be looking to ensure they did not lose money on a transaction, for the sake of their family, while the Church tried to ensure the contracts were not usurious, for the sake of their souls.

¹If the partners did not know each other, it was an anonymous partnership, and today limited companies are known as S.A., or “anonymous societies” across continental Europe.

References

S. Homer and R. Sylla. A History of Interest Rates. Rutgers University Press, 3rd edition, 1996.

J. Kaye. Economy and Nature in the Fourteenth Century. Cambridge University Press, 1998.

D. MacCulloch. A History of Christianity. Allen Lane, 2009.

M. Mauss. The Gift: Form and Reason for Exchange in Archaic Societies. (Routledge), 1924 (2001).

G. Poitras. The Early History of Financial Economics, 1478–1776. Edward Elgar, 2000.

G. Poitras. Life annuity valuation. In G. Poitras, editor, Pioneers of Financial Economics: contributions prior to Irving Fisher, pages 79–99. Edward Elgar, 2006.

E. J. Swan. Building the Global Market: A 4000 year history of derivatives. Kluwer Law, 1999.

A. P. Usher. The origins of banking: The primitive bank of deposit, 1200–1600. The Economic History Review, 4(4):399–428, 1934. URL http://www.jstor.org/stable/2245435.

What is financial mathematics

2011-07-13T05:24:00.000-07:00

My article "What is Financial Mathematics" was included in the anthology "The Best Writing on Mathematics: 2010".

Following this I have received a number of enquiries about books to read.

Currently, my favourite introductory textbook is Mark Joshi's The Concepts and Practice of Mathematical Finance. I think it has a good balance of correct theory and practical application.

The canonical text is Steven Shreve's Stochastic Calculus for Finance I and II.

I tell all my students to read Donald MacKenzie's analysis of the use of mathematics in derivative markets, such as An Engine, Not a Camera: How Financial Models Shape Markets and his recent review of the Credit Crisis, The Credit Crisis as a Problem in the Sociology of Knowledge.

Martin Creed

2011-07-13T04:22:00.000-07:00

Simon Singh passed my details on to the Fruitmarket Gallery when they contacted him about talking about the work of the artist Martin Creed. Initially the gallery were interested in a mathematician talking about the concept of zero.
However, the exhibition was full of works like Work No. 928 or Work No. 958 and it struck me that the concepts of the discrete vs continuous and limits were much more interesting, and relevant and bought in zero's alter ego, infinity.

The talk, ‘Much Ado about Nothing’ took place on 13 October 2010.

Back from a blog sabbatical

2011-07-13T03:55:00.000-07:00

This blog has been dormant for over a year, a year where I have done more research into the links between science and finance, and I now plan to start posting more regularly.

In April 2010 the Edinburgh International Science Festival hosted a panel discussion that I organised involving Prof Donald Mackenzie, Dr Gillian Tett and, Ms Terri Duhon. This is the text of an article published by The Scotsman (3/4/2010) in the lead up to that event, and the joint BMC/BAMC held in Edinburgh at the same time..

On 2 November 2008 the former French Prime Minister, Michel Rocard, was reported in Le Monde as saying that “mathematicians are guilty (unwittingly) of crimes against humanity” in reference to the financial crises that engulfed the world that year. The following March, our own Financial Services Authority, in their review of the financial crisis, listed one of the causes as “a misplaced reliance in sophisticated mathematics”, and pointed their finger at a technique known as Value at Risk (VaR) (see the FT's response at Maths and markets and my own comment at Maths and the Markets )

A series of events taking place in Edinburgh through April, as part of the Edinburgh International Science Festival and Maths2010, the largest meeting of mathematicians in the UK for five years, will address the question of the role of mathematics in finance, and correct any misconceptions that French premiers and UK regulators might have had – since the FSA have revised their assessment on consideration of the facts.

On April 8 at 11:30, Professor Paul Embrechts, an internationally recognised expert on financial risk management will give a public lecture “Did mathematics really blow up Wall Street”. Paul’s talk will focus on what went wrong; why it went wrong and how, by finance and mathematics working together future crises can be avoided.

The following week, at 8pm on 14 April, Gillian Tett, the Markets Editor at the Financial Times will chair a discussion between Professor Donald Mackenzie, a sociologist at the University of Edinburgh, Terri Duhon, who was involved in the innovation of the financial products traded by the banks and Dr Tim Johnson, a mathematician at Heriot-Watt.

Gillian Tett has emerged as one of the leading authorities of on the credit-crisis, based on her understanding of the activities the banks were involved in, which she describes in her book Fool’s Gold. Donald Mackenzie has been observing the financial markets from his perspective of a sociologist of science and technology for over a decade. He is regarded, both within the industry and by research mathematicians, as having some of the clearest understanding of how and why the mathematics technology failed. Terri Duhon worked for J.P. Morgan when the American investment bank developed the VaR technology and pioneered the use of Collatrorallised Debt Obligations. However, while J.P. Morgan innovated it did not engage in the reckless activities that bought Lehmans down. The key points of Terri’s experience being that not all banks got it wrong and financial innovation is not intrinsically dangerous. Finally Tim Johnson can give insights on the, surprisingly, close relationship between maths and finance and how science can learn as much from finance as finance can learn from mathematics.

Despite the very different backgrounds and perspectives of the panellists, it is likely that there will be some agreement in their conclusions, and like Paul Embrechts’s views, the solution is for mathematicians, social scientists and financiers to work more closely, and also for the public to become more involved in finance, just as society is involved in medical or energy technology.

Gillian Tett mentioned the meeting in here FT column

Greedy, or desperate, bankers

2010-04-19T00:24:00.000-07:00

I am completely bemused.

This morning I hear Robert Peston (the BBC's Business Correspondent) arguing the case of the airline industry, that maybe the scientific advice is too conservative and maybe airplanes could fly through the ash cloaud.

The point is that airplanes have been shown to fall out o f the sky if they fly through ash. Clearly not all planes do this but some will. An airline, facing bankruptcy will be risk seeking and so is going to be comfortable about losing a plane or two, may happen, may not but being grounded will result in failure.

What the airlines will do is force their customers to make a choice - to fly or not to fly. Customers with tickets will be faced with an impossible decision. Do I risk death or risk losing the airfare? This is an unfair choice to put on the public.

Above all, I cannot believe that Peston has not learned the lesson of the credit crisis. Businesses will take unacceptable risks to remain in business, and typically it is the public who end up bearing those risk. Banks and regulators ignored the advice of scientists, if we allow the airlines to ignore the advice, we deserve extinction.

The whole of western science is based on financial maths

2009-12-07T02:07:00.000-08:00

A historian Joel Kaye has written widely on the development of science in medieval Europe. In Economy and Nature in the Fourteenth Century he presents a thesis that

In broad terms, the conceptual landscape that emerged in the fourteenth century resulted from a striking shift in the models derived to represent order and activity in the natural world: from a static world of numbered points and perfections to a dynamic world of ever-changing values conceived as continua in expansion and contraction; from a mathematics of arithmetical addition to a mathematics of geometrical multiplication, newly accepting of the approximate and the probable; from a world of fixed and absolute values to a shifting, relational world in which values were understood to be determined relative to changing perspectives and conditions; and from a philosophy focused on essences and perfections to one dominated by questions of quantification and measurement in respect to motion and change. Each of these new directions proved to be of great importance to the future of scientific thought.

Mathematics, Economics and Decision Making

2009-12-07T01:55:00.000-08:00

The 2009 Alan Tayler Lecture was delivered by Prof Lord Desai of the London School of Economics and Political Science at the University of Oxford on 30 November.

I did not attend the lecture, but I understand that Lord Desai called for economists to use more sophisticated modeling techniques - i.e. not to rely on single-factor linear models when trying to understand the economy. Makes sense to me.

I shall try and get a copy of the talk.

I also recently attended a talk by Prof Lord Skidelsky who was talking about Keynes, and publicising his book Keynes: The Return of the Master. A key point made by Skidelsky was Keynes belief in "irreducible uncertainty". From the perspective of maths, there is nothing controversial in Keynes is position, the Fundamental Theorem of Asset Pricing, in noting that there are infintely many martingale measure in an incomplet market, makes the same point. As a Russian mathematician has pointed out to me, when economists talk about "rational expectations", ask them under which measure they are calculating the expectation.

Culture clash?

2009-08-30T23:55:00.000-07:00

The FSSC report Graduate Skills and Recruitment in The City has the following quotes (p 45) from a graduate recruitment manager at an investment bank

“Sales managers prefer graduates who are academic, with some interest in finance but more social skills with extra-curricular activities e.g. being the Head of the rugby/cricket club. They’re less fussy on the degree subject, but quite fussy in terms of the university they have attended – Oxford, Cambridge, LSE, Imperial.

The [derivatives] trading/research side is much more focussed on the technical. Many of our senior managers are from French universities, with experience of engineering schools and écoles polytechniques in France. They look for expertise in engineering, maths, qualitative [quantitative?] finance. They like Oxford, Cambridge, Warwick”.

This quote resonated by an interview with a former Lehman's executive published in the Sunday Times on 23 August, Lehman Brothers failures exposed:

"Instead of reviewing the bank’s risk management systems, the executive directors would have ndless meetings discussing the corporate dress code as Fuld was a stickler for appearances. A angerous lack of awareness about the technical financial products the bank was playing with didn’t elp matters."

Most people employed in finance are involved in sales and marketing activity, and so the soft skills needed for these roles are frequently highlighted by employers to the likes of the FSSC and Universities. However, the core of the banks' business is is highly quantitative, to quote from p 14 of the FSSC report:

“…[banks] need high level maths skills because that’s how the bank makes money – vanilla roducts have very little margin.”

The financial innovation (or more accurately, the re-introduction of derivative products into financial markets) since the 1970s has changed the nature of banking, has the culture of banking kept up?

Perhaps a solution is to separate financial institutions into "traditional banks", undertaking lending and M&A activities relying on sales and marketing skills and "speculative banks" specialising in trading, derivatives and risk management. However, this is not the only model. Many manufacturing firms, particularly pharmaceutical and oil companies, are based on core expertise in science and engineering but where the majority of employees are involved in sales and marketing. So it appears the cultures can co-exist.

It's academic

2009-08-26T01:43:00.000-07:00

On of my favourite quotes from the credit crisis comes from the 2/02/08 edition of The Economist, the article "No Defense" on fraud issues at Societe General, states

"In common with other French banks, SocGen was also thought by many to take an overly mathematical approach to risk. “‘It may work in practice but does it work in theory?' is the stereotype of a French bank,” says one industry consultant."

In response to "Of couples and copulas", Prof Paul Embrects sent the following letter to the FT

Dear Sir
The article "Of couples and copulas", published on 24 April 2009,
suggests that David Li's formula is to blame for the current financial
crisis. For me, this is akin to blaming Einstein's E=mc² formula for
the destruction wreaked by the atomic bomb.

Feeling like a risk manager whose protestations of imminent danger
were ignored, I wish to make clear that many well-respected
academics have pointed out the limitations of the mathematical tools
used in the finance industry, including Li's formula. However, these
warnings were either ignored or dismissed with a desultory
response: "It's academic".

We hope that we are listened to in the future, rather than being
made a convenient scapegoat.

The Royal Society calls for more funding of financial maths

2009-08-26T00:39:00.000-07:00

The Royal Society published a report on Science, Technology, Engineering and Maths impact on the service sector. They devote a whole chapter (3) to financial innovation, see Hidden Wealth. The report was covered in the FT.

I was contact by the Royal Society in mid-July, this is (their) summary of what I said:

1. The financial crisis was not a homogeneous events; some institutions did (much) better than thers.

2. The root of the problem is in banks mis-pricing assets. When pricing assets the banks were relying on mathematical models. Some banks had "engineered" their own models, others "bought in" models (either by buying "off the shelf" or by hiring individuals from the innovating banks). Banks who treated models as "black boxes" have done far worse than those that had a reputation or developing their own.

3. Many quants (the majority ?) have a background in physics and engineering (there is a factoid that the majority of engineering graduates from top UK universities go into finance rather than engineering - you could check this. Similarly The City is the largest employer of physics PhDs). They understand deterministic systems but only have a rudimentary understanding of modern probability theory (the majority of maths graduates are in a very similar position, I spoke to a maths teacher who had a degree from Glasgow who told me he had not done probability since he was 16).

4. Financial economics developed in the late twentieth century using relatively straightforward maths. The models it produced are "too simple" (see pp 4-12 in "An Engine, Not a Camera"), but are "elegant".

The physicists / engineers were given a simple framework in which deterministic approaches appeared to give definite results. This approach is related to the "Crash of 87" and "When genius failed" in '98.

5. Throughout the industry a belief emerged that maths, "rocket science" removed risk. In the 1990s, mathematicians began to investigate financial models and re-evaluate them. A more rigorous approach to financial economics revealed the naivety of the assumptions that led to the simple models.

6. Because the financial mathematics community is small and peripheral in British science, it lacks authority , and the theory (from the mathematicians) became disconnected from the practice in industry.

7. In Europe, because they approach probability as a branch of analysis rather than from the perspective of statistics, there seems to have been a better appreciation that simple models that fitted data were inadequate. This is a subtle point. The issue is whether the "quants"
really understand stochastic systems. Does the UK education system (schools to universities) produce the volume of people with (basic to advanced) skills in probability and statistics. There is a view that France & Germany are better at this, as demonstrated by the large number of continental scientists and engineers employed in London banks.

8. The credit crisis provides a tangible example and introduction to a wider problem about our poor understandings of complex and stochastic systems. The concern is that many of the global challenges we face involve these sorts of systems. Is our science up to the task?

9. There is a basic competency issue but also a need to develop new mathematics able to describe what Lord May describes as "ephemeral" systems, but in other areas relating to fundamental maths.

Driving by using the rear view mirror

2009-07-06T01:43:00.000-07:00

Last October the EPSRC, my funders, asked me to contact the Science Media Centre about media coverage of the credit crisis. This was because there was a risk, which materialized, that mathematics would come into criticism in light of the credit crisis. A the time the SMC took the view, on the basis I was the only person making the point, that the credit crisis had nothing to do with science. Following my "Was the Credit Crisis a Science Story" piece and an intervention by Lord Drayson’s office, the Science Media Centre arranged for a session on the credit crisis to be included at the World Conference of Science Journalists, which I attended.

At the conference I explained the links between modern financial markets and science, gave a “scientific” perspective of what happened and why, discussed the role of mathematics in relation to finance and in relation to science in general, emphasizing the fact that maths is looking to describe complex and stochastic systems. An issue critical to the future development of all of science.

Three other people were on the panel. During the Q&A an Italian print journalist (I believe from the Italian equivalent of New Scientist) reported that his magazine had devoted a whole issue to the crisis and it had generated the most positive feedback the magazine had had all year. At the end of the session, the Chair asked me what the next big story in finance would be, I said the big story is that successful finance is based on good science, the Chair informed me this was not a story that could be sold to her editor.

Following the session a stream of European science journalists came up to me and told me that in their countries, science journalists had been covering the story and generally there had been a very positive response from the public.

In particular I had a lengthy discussion with two German journalists. One reported that the main German TV science magazine show had, also, devoted a whole program to the crisis, and again it had been very well received. What was disturbing was that he was scathing about the level of understanding of the crisis his British counterparts had displayed in the session. The fact that one of them did not recognize the importance of basing finance on good science “completely misses the point”.

We talked about why British science journalists take this view, and the conclusion was it came down to the British tradition of empiricism; science is about gathering data to build models that explain the world. There is a blind spot to Cartesian rationalism, that science also involves abstract reasoning. In relation to the credit crisis this explains the British (and American) dismissal of French Risk managers with the saying that worry “That’s all very well in practice but what about the theory”. The point is, Anglo Saxon risk management techniques were all very well in practice, up until the point they went wrong. French mathematicians (Artzner) had a theoretical basis for dismissing Value at Risk long before British financiers found out that in practice, it was flawed.

The empiricist-rationalist divide also explains why Continental Europeans have a more catholic view of science. If science is about rational thought it encapsulates mathematics, sociology, anthropology and even history. If it is about the interpretation of numerical data, it does not. Ironically, it looks as if to understand one aspect of the credit crisis, the fact that bankers made decisions based on what they saw in their rear view mirrors, seems to be a sociological question, not a question that can be addressed by the physical sciences.

P.S. Gillian Tett (17/7/09) has discussed the sociological interpretation of "herd behaviour".

Maths and the Markets (II)

2009-06-10T06:46:00.000-07:00

Following my last post I drafted a letter on behalf of the leading researchers in financial maths in the UK. After some edits, Sir David Wallace, Chair of the Council of Mathematical Sciences, sent this on our behalf to Lord Turner.

The FSA have responded positively to this, as discussed by Clive Cookson in the FT today.

Its a good result for both the FSA, who need the support of mathematicians that is out of their price range to employ and tghe research community will get some sort of support.

There is an angle that is not so positive. The government is keen that academics do more public engagement, and I think this is a good thing. If financial mathematicians do something that is not pointless, they should be able to inform the FSA on good regulatory practice. My concern is that the institutions between the government and academics are not efficient at enabling academics to engage. Typical examples are the RAE/ REF process, which focus academics on learning more and more about less and less until they know everything about nothing. Similarly the Research Councils are keen to facilitate public engagement, but at the end of the day most grant applications are assessed by a panel who are only (in maths at least) in world clas smathematics research - and pure mathematicians don't like being associated with anything as messy as the real world.

These are general comments, there is a specific example. The Technology Strategy Board recently conducted a call to establish a Knowledge Transfer Network in Financial Services. Some of the signatories to Lord Turner's letter submitted a proposal, which was unsuccessful. The question is, did the successful group make any submission to the FSA regarding the Turner Review? My point is that sometime government funding of research activity results in activites that "tick the boxes" but that is about all. These comments are informed by a private conversations with people from a number of consotiums who submitted proposals to the TSB.

Is there a better way? When I explore issues like the shortcomings of the RAE and research councils, members of the government counter that "that is what the (elite) universities want". It was Universities UK that lobbied government to keep the RAE as was/is. The high and mighty like the research councils panels, because it enables them to control science funding. However, people tell me that the US system, managed by the National Science Foundation, is more researcher focussed and less buraucracy focussed.

Maths and the Markets

2009-03-26T09:49:00.000-07:00

The Turner Review of Financial Services Regulation identifies one of the causes of the financial crisis as a "misplaced reliance on sophisticated maths". The FT was quick to criticise this naive and ill-informed view.

The banks that failed seem to have abused maths, financial firms that succeed use maths.

Take for example BNP Paribas, Bank of the Year in 2008. The lead quant is Prof Marek Musiela, a well regarded research mathematician who would be snapped up by the maths department of any prestigious university. In employing Prof Musiela, BNP is setting the culture of the company.

Much of financial innovation is coming out of hedge funds, and while these "shadow banks" come in for some criticism, a number of hedge funds were created by the frustration innovative scientists felt working in the blue chips banks (which are failing).

Last year's top earner was Jim Simons, who runs Renaissance Technologies. Jim is another research mathematician, having won the American Mathematical Society’s Oswald Veblin
Prize in Geometry. A third of Renaissance's 200 staff in its main offices have PhDs and one senior manager is quoted as saying “I’ve always said Renaissance’s secret is that it didn’t hire MBAs,” - i.e. it does not hire "finance" experts, it hires scientists.

In the UK, while the MAN Group has posted a halving of profits (on a falling market), their share price rose, the reason being they did better than expected. The MAN Group takes maths so seriously they fund Oxford University mathematicians, drawing world-leading researchers to Oxford and into dialogue with the hedge fund. Not only does this give MAN and edge, but it gives the UK as a whole an edge by being a "brain sink" rather than a "brain drain".

Have these leading firms really been getting their use of maths and science wrong? Anecdote is not evidence, but while RBS was investing millions in "sports ambassadors", BNP, Renaissance Technologies and the MAN Group were all investing in mathematics and science. The impact of these two strategies appears obvious, and investors should bear this in mind in the future.

A Brief History of Financial Engineering

2009-02-25T07:14:00.000-08:00

c 550 BCE Thales, the father of western science makes money by using his scientific knowledge to become rich through olive speculation (Aristotle, Politics Book 1 part 11).

1202 Leonardo the Pisan (Fibonacci) introduces Hindu-Arabic numbers to Europe to aid in commercial calculations.

c 1250 Pope Innocent IV, in a commentary on cannon law, justifies the charging of risk premium for assets (Murray Rothbard, Economic Thought Before Adam Smith , Edward Elgar,1996)

c 1260 St Thomas Aquinas endorses insider trading (making profits based on information not know to the buyer) (Summa Theologica, Second part of the second part, 77, 3)

c 1564 Girolamo Cardano identifies the concept of mathematical probability.

1610 Galileo becomes the first “quant”. Having used a telescope to observe Jupiter, Galileo publishes his results, leaves Padua, and becomes “First and Extraordinary Mathematician of the University of Pisa and Mathematician to his Serenest Highness Cosimo II de Medici”. At Cosimo’s request, he investigates a gambling problem and publishes Sopra le Scoperte dei Dadi (Upon the Discoveries of Dice).

1654 The first derivative pricing formula is developed by Pascal and Fermat in answering the Problem of Points. The solution to the problem of points is essentially the same as the Cox-Ross-Rubenstein model. A special case of the discrete time CRR model converges to the continuous time Black-Scholes option pricing model.

1696 Newton becomes a quant, moving from Cambridge to the Royal Mint. In 1717 Newton moves England from the silver standard to the gold standard, fixing the price of silver to gold.

1738 Daniel Bernoulli publishes “An Exposition on a New Theory of the Measurement of Risk” in Russia. The manuscript appears to have been lost, reprinted in Germany in the late 19th century and then in English in 1954, 10 years after von Neuman and Morgenstern had introduced Expected Utility in the Theory of Games and Economic Behavour.

1860-1926 Maxwell develops the kinetic theory of gases, investigated by Einstein who discusses Brownian motion, which is defined mathematically by Weiner in 1926. In the late 1960s Robert Merton develops the theory of continuous time finance based on the Weiner process.

1933 Kolmogorov identifies probability with measure, enabling financial mathematicians to use the ideas of conditional probability and equivalent measures, which are the fundamental tools of derivative pricing.

What mathematics can offer finance

2009-02-25T07:11:00.000-08:00

I emphasise to my undergraduate students that Kolmogorov’s 1933 identification of probability with measure theory is, in my opinion, the most important scientific development of the twentieth century. To paraphrase Bertrand Russell, probability is important because all science is inductive, and prior to Kolmogorov there were significant problems, philosophical, theoretical and practical, with identifying probability with the frequency of events.

The context is as follows. Students learn that if a market is arbitrage free then asset prices today are simply expectations of future asset prices calculated using, special, risk neutral probabilities, rather than any “natural” probabilities. The idea of a probability as an abstract measure is introduced along with the idea of equivalent measures, so the students understand that there is nothing special about, so called, natural probabilities. They are then shown that if a market is complete (idealised) there is a unique risk neutral probability, while if the market is incomplete (for example, it includes transaction costs) then there are an infinite number of possible choices of risk neutral probabilities. Finally, in a complete market, all risk can be removed in pricing derivatives; the same is not true for incomplete markets, which are the reality.

The next question is, how do we choose what risk neutral probabilities we should use to price assets in a market if there are infinitely many choices? A number of solutions have been presented, the most widely accepted is to use utility-based methods, an approach initiated by Prof Mark Davis in his paper Option Pricing in Incomplete Markets (1998). Essentially, given an agent’s utility function, we can identify the correct risk neutral probabilities to use in asset pricing.

Why is this significant in economics? Von Neumann-Morgenstern Expected Utility Theory, published in 1944, states that given a set of natural probabilities associated with lotteries (generating a probability distribution function) an agent’s utility function can be identified (preferences over distribution functions precede preferences over outcomes). However, von Neumann-Morgenstern Expected Utility assumes probabilities are objective and exogenously defined by nature (the Kolmogorov formulation of probability was not accepted in the US until around 1948). Derivative pricing tells us to ignore these probabilities and price assets using equivalent risk neutral probabilities. However, in realistic markets, there are infinite choices of the correct probabilities to use and we should use utility functions to identify the right one. Pricing assets starts with utility functions. The implication is, we need to understand utility to understand asset pricing; finance needs to be interdisciplinary.

Within the banks, with probability 1, there are scientists who are comfortable with probability as measure. I would be willing to wager that there are few people managing banks who are familiar with the idea that financial mathematics tells us that there is no unique probability measure under which to calculate expectations, rational or not.

Another topic I raise with students is why Pythagoras was responsible for the Black Monday in 1987. The context here is that the ’87 crash was associated with portfolio insurance, based on hedging investments in portfolios by taking out positin in index futures. I have to teach the students (because it is part of the Actuarial syllabus) a the optimal hedge ratio. This is derived on the assumption that investors choose between expected returns and variance of returns. This idea was introduced by Markowitz (mean variance portfolio selection) and developed by Sharpe, Litner (the capital asset pricing model) and yielded a handful of Noble Prizes. Why did Markowitz suggest this approach? There is no evidence to suggest that investors actually do choose between expected returns and variance of returns, Markowitz suggested it because, thanks to Pythagoras’s theorem, comparing expectations and variances is mathematically tractable.

Markowitz is a nice example of mathematics driving financial theory, rather than, as happens in the natural sciences, the observations driving the development of mathematics. Newton developed calculus to build his models. I tell my students that we are in an exciting time in financial maths and compare the state of the science to that of aviation around 1904. We are using glue and tying things together with string, but we are on the right lines. Much of finance theory was developed in the 30 or so years following the Second World War based on assumptions of probability that date from the seventeenth and eighteenth centuries. Using measure theoretic probability, scientists are able to look at financial problems without the constraints of nineteenth century mathematics and as a result, are able to build more accurate, though still simple, economic models.

The following is a personal selection of papers that I feel are significant contributions to finance theory from mathematics

Fernholtz, R., Karatzas, I, Stochastic Portfolio Theory: an Overview

This addresses issues relating to Markowitz style portfolio selection

Musiela, M., Zariphopulou, T., Portfolio choice under space-time monotone performance criteria

This addresses a significant issue in classical finance. At time t=0 the agent sets their objectives at some time t=T>0. In the interval ]0,t[ the agent locks themselves in a room, closes their eyes and sticks their fingers in their ears. The reality is that in ]0,T[ the investor is affected by the economy. Energy managers (a PhD in maths and a PhD in theoretical physics) at Scottish Power recently raised this issue with me, they need to manage portfolios dynamically, not statically.

Hugonnier, J., Kramkov, D. Schachermayer, W., On Utility-Based Pricing of Contingent Claims in Incomplete Markets, Mathematical Finance, Vol. 15, No. 2, pp. 203-212, April 2005

A refinement of Prof Davis's work.

H. Jin and X. Zhou, Behavioural portfolio selection in continuous time Mathematical Finance, Vol. 18 (2008), pp. 385-426.

Provides a mathematical basis for behavioural finance, as introduced by Kahneman and Tversky.

Don't blame the quants

2009-02-13T15:44:00.001-08:00

Thalia Zariphopoulou has a couple of articles on her website about the financial crisis. A piece by Steven Shreve that appeared in Forbes Magazine and an article by Philip Protter.

Getting serious about banking

2009-02-11T05:17:00.000-08:00

We do not think banking is a serious occupation. If we take a job seriously, like the law, medicine, accountancy, engineering, teaching or plumbing, we expect the workers to have passed exams and to keep up with developments in the profession throughout their careers. The Financial Services Authority, rightly, does not allow anyone to sell an endowment mortgage to a private individual unless they have passed a series of exams. If someone fails their FSA exams, not to worry, they could get a job in an investment bank and sell Mortgage Backed Securities to other investment banks. It is up to the banks to make the judgement as to whether or not their employees are competent. The consequence of the chairmen and chief executives of RBS and HBOS not being qualified, is that they were not competent to judge if their employees were capable.

Hector Sants, the Chief Executive of the FSA, said in October that he had over thirty years experience in banking. That is nice to know, but if I owned a jet aeroplane in 1950 I might prefer a mechanic who has three years experience of jet engines to one with thirty years in aviation. This is the root of the problem; finance is developing too fast for people with thirty years experience. As a financial mathematician, researching and teaching how you make good, scientific, decisions in the uncertain world of finance, I am not overawed by the complexity of the financial products the banks are trading. I am stunned by the simplicity of the mathematics that many bankers were using to manage these products. Nuclear power plants are complex, and so we expect the engineers who operate them to use cutting edge technology. The human body is complex, we expect doctors to use up to date research in treating us. In investment banking, Collateralised Debt Obligations are complex but the reaction of the "quants" designing them was to dumb down the maths so that the managers, who did not have the professional qualifications of an engineer or a doctor, could keep up.

What happened in the lead up to the credit crisis is not difficult to explain. When I was in my late teens, l embarked on a short-lived experiment in producing ginger beer. Having brewed the beer in a large bucket, I bottled it in twenty-four bottles and stored them in my wardrobe. If I had presented a newly sealed bottle to an independent expert and asked them to asses the probability of the bottle exploding, they may have given the chance as one in twenty four, and so I could expect one of my twenty four bottles to explode. If I was concerned with losing many bottles, I could do some basic maths and conclude that there was a 99.99% chance that I would not lose more than five bottles.

A few weeks later one bottle exploded, followed by the remaining 23 over the course of a few days. When I had done my sums, I had treated each bottle as being independent of the others. Of course, since each bottle contained the same product, if one exploded, they probably all would. This was an object lesson in what are called dependent risks, essentially there was a one in twenty four chance of losing everything rather than a near certainty of only losing a little.

Investment banks had bought beer, sub-prime mortgages, and sold crates of bottles to investors on a sort of "money back guarantee" deal. Like some of Sir Alan's naive apprentices, they focused on turnover, since salesmen are paid on commission, rather than worrying about profitability.

Bankers, not just in the UK but all around the world, blame "nature" for mis-behaving when all the bottles exploded. This is a smokescreen to protect their reputation. Since 2000, there have been a series of concerns raised by mathematicians as to whether the equations bankers were using to manage the risk of their investments were adequate. In particular, that the key equation they use would underestimate the number of bottles that would blow up, and another, that valued their finished products would over re-act to a bottle being lost and the value of all the beer, not just the ginger beer, they were selling would collapse. The problem is, not only did bank management not understand this, but also, it is exactly what happened.

Was the credit crunch a science story?

2009-02-05T13:59:00.000-08:00

A recent report from the LSE has raised some important questions concerning financial journalism. Specifically, in relation to the credit crisis it reports that Robert Peston, the BBC’s Business Editor, had found it difficult to explain some of the complex derivatives being constructed by banks since “even the bankers creating the CDOs were unable to explain them in terms that make sense to non-specialists”. The role of these complex derivatives is central to understanding the credit crisis, and key to understanding the derivatives is mathematics. This raises the question, could science journalists have contributed to reporting the credit crisis.

The storm clouds of the credit crisis broke following BNP Paribas’s announcement that it was unable to value assets on 9 August 2007. The following day, Goldman Sachs reported that one of its funds had lost 30% of its value in the week, equiring a, now paltry, $3 bn bailout. In explanation the Chief Financial Officer of Goldman Sachs said

“We were seeing things that were 25-standard deviation moves, several days in a row. There have been issues in some of the other quantitative spaces. But nothing like what we saw last week.”
Financial Times, August 13 2007, Goldman pays the price of being big

Essentially, his point was that the economic environment had been so perverse that the losses, incurred by the most respected investment bank in the world, were inevitable. The Financial Times2, along with much of the established financial press, took this as a reasoned explanation but, almost immediately, commentators in the blogosphere revealed it to be what it was, rubbish. Goldman’s models were wrong, not nature.

It is unlikely that a science journalist would have accepted the pseudo-scientific explanation Goldman’s offered. Would they have done a better job understanding the crisis as a whole?

Since the 1960s investment banks have been using increasingly complex mathematics, literally rocket science, to price and manage the risks of the assets they trade. The current credit crisis is unique in that for the first time, quants, physicists and engineers, using mathematical formulae rather than economists using their knowledge of markets, have been managing the assets at the heart of the crisis. Since experienced managers in banks have been at a loss to grasp the mathematics, it is hard to see how a financial journalist should be expected to do so. However, a journalist who is accustomed to investigating safety critical systems in technology is well capable of coming to terms with risk management systems in an investment bank. While a journalist who is able to explain the latest research coming from a physicist in a research lab, would be able to describe the pricing algorithm developed by a physicist in a bank.

If the types of journalists investigating science turn their attention to banks, would the risk of future crises be reduced? There is a strong argument that it would. A key role that science-reporting plays is in ensuring that technology is not misused and that it is properly regulated. Medicine is better off for the reporting of Thalidomide and the environment benefits by journalists scrutinising the energy industry.

In the 1990s, central bankers became concerned that the rules for defining how much capital a bank held in reserve were quickly becoming redundant, given the explosive growth of sophisticated derivative products. In 1996, the Bank for International Settlements, the gnomes of Basel who regulate the banks in relation to the credit crisis, began a process of re-writing banking regulation, called Basel II. Realising that they would never be able to keep up with the activities of the banks, the regulators specified a framework for regulation rather than a set of hard and fast rules. The framework, which is only now beginning to be implemented, is based on three pillars; the minimum standards for calculating how much money a bank needs to keep in reserve; the supervisory review process in the bank overseeing those calculations and market discipline ensuring the reviews and calculations are adequate.

In 2000, a mathematician working at the US Federal Reserve, checked the formulae used in Basel II to calculate capital reserves, but raised concerns about whether it would be adequate if finacial products became too sophisticated (in the Discussion). The Basel committee had selected a formula that did not model the connections in the economy in a sophisticated manner and so it was inadequate if loans were inter-connected. The root cause of the credit-crisis has been that loans were connected, by the housing market, banks ran out of capital, precipitating the liquidity crisis that led to the collapse of the retail banks and the credit crunch.

The Basel II accord still uses the simplified formula, because the accord is clear; under the supervisory review process, a bank needs to match the calculation of capital reserves to the omplexity of their operations. The accord expects that banks will adequately execute the supervisory review process because of market scrutiny. Consequently, informed journalism plays a fundamental role in the new framework for banking regulation. Science journalists should take up this role given their knowledge and skills.

The Public Relations machines of banks are happy that discussion around regulation focuses on issues such as bonuses and banning complex derivatives, because the banks know that governments will not set such regulations. Turning the public’s attention to the science underpinning the bank’s business will not only put the focus on where the problem is, but is already part of the regulatory framework. In addition, putting banking managers under pressure to explain the details of the technology they are using will force better communication, debate nd understanding within the banks bridging the gap in banks between the quants and conventional financiers.

Finally, if the failure, because of poor use of science, of a section of the energy or pharmaceutical businesses had lead to a trillion-dollar rescue package, it is inconceivable that the managers of he industry would still be in place. Why have so few bankers been fired by their shareholders? Is it because the bankers have successfully blinded the shareholders with pseudo-science?

Do the maths

2009-02-05T13:51:00.000-08:00

In a week that has seen the disappearance from the financial landscape of such venerable firms as Lehman Brothers, Merrill Lynch and the Bank of Scotland it is inevitable that people question the competence of bankers. Along with many financial mathematicians, in both the financial industry and academia, I take the view that if you give people who have only ever driven bullock carts, Ferraris, they are likely to end up in a ditch. The people running the banking industry have not had a firm grasp of the technology underpinning their business.

Financial mathematics supports the practical disciple of constructing financial products in the same way that mechanics and thermodynamics supports motor engineering. Derivatives, the products that financial mathematicians construct, transfer risk in financial markets just as engines transfer energy in the physical world. Fixed rate mortgages are a popular product of financial mathematics, the borrower passes the uncertainty in the future interest rate to the lender, who takes it on for a fee. Just as in any industry, most people working in the financial markets are involved with the sales and servicing of products and only a few people actually design, and so have a deep understanding of, the products themselves. Jeremy Clarkson does not need to understand automotive engineering to be considered an expert on cars.

It is not surprising that many bankers did not understand the science they work with, because financial mathematics is literally rocket science; it comes directly from the mathematics developed in the fifties and sixties to control rockets. The international investment banks are well aware of this and they are becoming less interested in recruiting graduates with a degree in economics from the LSE, who understand theory, and are more interested in engineers, physicists or mathematicians, who can do the complex calculations underpinning derivatives. This is nothing new, for around a decade the most popular destination of physicists on completing their PhDs has been The City and more graduate engineers in the UK now head for careers in finance than in engineering.

Although young British scientists can contemplate fat salaries with investment banks, they are at a disadvantage. If you visit the sharp end of a City firm, you will notice that there are a large number of continental graduates working there. This is no accident; continentals have a much better education in probability than UK graduates. The reason for this is cultural. The UK was at the heart of the development of statistics in the nineteenth century but at that time most mathematicians thought statistics heresy, because it is inductive rather than deductive, and today most UK universities still have separate maths and statistics departments. Seventy-five years ago this summer, a young Russian mathematician Andrey Kolmogorov defined probability in terms of rigorous mathematics placing probability firmly in the mathematics syllabus taught at school and through university in the rest of Europe.

There is a conceptual gulf between British and French graduates. When using probability to think about the future, English speakers will talk about an expectation, the French talk about an esperance, which is associated with 'hope'. The British and American bankers expect something, the French hope for it. The bankers creating asset-backed securities believed that they had removed risk because they lacked a real understanding of probability. If we want to maintain our position in the lead of finance, we need to train more people in probability so that they are better able to deal with an uncertain future and look after our money.