Enlightenment Exchange: forecasting the future

I was due to be participating in the Edinburgh International Science Festival as part of the  Enlightenment Exchange on forecasting the future, however I am recovering from surgery and so will not make it.  This is an outline of what I planned to talk about.

Needham’s Question is why did the development technology in western European accelerate much faster than in China after 1600.

The issue that Needham wanted to tackle was that China had the physical   and intellectual resources, mathematics, alchemy, astrology and magic, just as Renaissance Europe did, but it did not develop science as Europe did.

One key distinguishing feature between the science that emerged in western Europe in the seventeenth century and other scientific cultures was the use of mathematics.  Aristotle, and his heirs, did not think that physics was reducible to mathematics, similarly the Chinese and Indians did not apply maths to solving scientific problems in the way Newton and Clerk Maxwell did.

One answer to why European science adopted mathematics lies in medieval European finance.  Medieval Western Europe was unique in having to deal simultaneously with a heterogeneity of currency and prohibitions of usury. Muslims had the usury prohibitions but homogeneous currency, the Chinese were lax on usury and generally had homogeneous currencies..

In Western Europe, any local lord would mint their own coin, as soon as they had the power to do so, Italy had 28 currencies in the High Middle Ages while the single authority that ruled the Kingdom of France used  three currencies.  Meanwhile, merchants were prevented from charging for the use of money, usury, but they could ask for compensation for risk, interest.  The skill came in treading a fine line between usury and interest under the scrutiny of the Catholic Church.

As a consequence medieval finance was not simple, on the contrary, having to deal with uncertainty and a complex regulatory system, it was highly sophisticated.  The “modern” financial techniques, such as asset backed securities, collateralisation - “slicing and dicing” - and credit default swaps, were all developed by Europe’s merchants between Charlemagne’s reign and the discovery of North America.

The solution to the problems that Medieval European merchants faced was mathematics, specifically the mathematics Fibonacci described in his 1202 text, the Liber Abaci.  Fibonacci’s mathematics revolutionised European commercial practice.  Prior to the Liber Abaci, merchants would perform a calculation, using an abacus, and then record the result.  The introduction of Hindu/Arabic numbers in the Liber enabled merchants to “show their working” as an algorithm, and these algorithms could be discussed and improved upon.  Essentially after Fibonacci mathematics ceased to be simply a technique of calculation but became a rhetorical device, a language of debate.

The Liber, and the abacco schools that emerged across Europe to train merchants, separate to  the Universities, disseminated and developed practical mathematics and the influence of this training was profound.  The “Merton Calculator” Thomas Bradwardine, who  would have been familiar with medieval mercantile practice and would leave Oxford to to work for the Treasurer of England before becoming the Archbishop of Canterbury  wrote in 1323

 “[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.”

Copernicus, who wrote on money before he wrote on planets, Simon Stevin, the founder of the Dutch Mathematical School that inspired Descartes, Thomas Gresham, Francis Bacon’s influential uncle who established  the first Chair in Mathematics in England and laid the foundations for the Royal Society,  were all trained in the abbaco tradition.

Newton, who spent as much time as Master of the Mint as he did as an academic, stood on the shoulders of merchants.  His derivation of calculus comes from writing a function as a polynomial, mimicking Stevin’s representation of a number as a decimal fraction.

Between 1650 and 1713 extraordinary advances were made in the the mathematical theory of chance in the context of the “fair” pricing of commercial contracts in the context of Christian morality: Faith was associated with statistics, Hope with probability.  The modern conception of probability theory as being based on performing repeatable experiments was simultaneously presented in 1713 by de Moivre and Montmort, this was taken up by Laplace a hundred years later when he argued that nothing was random, it was just humans lacked knowledge to appreciate the links between cause and effect.

“We look upon something as the effect of chance when it exhibits nothing regular,or when we ignore its causes”

To Laplace and scientists that came after him, the problem of uncertainty could be resolved by gathering more data, the mathematics of chance, which developed in the context of ethical finance before calculus was developed in the context of physics, was reduced to the Law of Large Numbers.  Economics developed in the context of deterministic models based on analogies from physics or biology.

The significance of randomness in physics started to emerge in the second half of the nineteenth century in the context of thermodynamics, leading to Einstein’s analysis of Brownian motion, which explained random behaviour of pollen particles in terms of “invisible” atoms, and then in the 1920s with the Copenhagen Interpretation of Quantum Mechanics, prompting Einstein to state that he did not believe God played dice.  Meanwhile, in biology, R A Fisher synthesised Darwinian evolution with Mendelian genetics and in the process revolutionised statistics, the analysis of data.  Meanwhile  economists, such as the American Frank Knight and John Maynard Keynes, started placing uncertainty at the heart of Economics.

At the outbreak of the war in 1939 the vast majority of soldiers and politicians would not have thought mathematicians had much to offer the military effort, an orthodox attitude amongst the military is still that “war is a human activity that cannot be reduced to mathematical formulae”.  However, operational researchers  laid the foundations for Britain's survival in the dark days of 1940-1941. Code-breakers transformed seemingly random streams of letters into messages that enabled the allies to keep one step ahead of the Nazis while Allied scientists had ensured that the scarce resources of men and arms were effectively allocated to achieving their military objectives. By the end of the war General Eisenhower was calling for more scientists to support the military.

During the war many economists had worked alongside mathematicians solving military problems, such as Paul Samuelson.  In the post war decades mainstream economics transformed itself from a discursive discipline into a mathematical science.  However the biggest change in the field came in 1972 with the Nixon Shock and the collapse of the Bretton-Woods system of exchange rates.  Overnight finance transformed from a broadly deterministic system, fixed exchange rates, static interest rates and constant commodity prices, into a stochastic system.  As exchange rates fluctuated, interest rates started changing monthly and the commodity prices became a random processes.

The markets evolved quickly in response to the changed environment.  Financial instruments that had not been a feature of financial economics since 1914 re-appeared, and over the next decades techniques that would have been familiar to Renaissance bankers like the Fuggers, such as securitisation, emerged to cope with the fundamental uncertainty of the markets.

Mathematics enables science without experiments, the Large Hadron Collider was built on the basis of a mathematical theory, and so is essential in the markets that are too dynamic for experimentation.  The question is, was the post-Laplacian science, with an implicit rejection of randomness, up to the task of  dealing with the fundamentally uncertain markets?

The UK’s Financial Services Authority in their 2009 review of the Financial Crisis of 2007-2009 identified a “mis-placed reliance on sophisticated mathematics” as a root cause of the Crisis.  Last June, the Director of Financial Stability at the Bank of England, identified basic flaws in mathematical understanding as a “top 5 but not top 3” cause of the crisis, more significant was the fact that Banks were not maintaining adequate records of their loan portfolios.  The Financial Crises since 2007 have been less to do with fancy finance backed up by even fancier maths, it was more to do with banks, and their regulators, not keeping an eye on their core business of lending prudentially.  It is worth noting that Fred Goodwin at RBS and Andy Horner at HBOS both came from outside banking.

But the facts are dull, it is more interesting, and convenient for many, to talk about the slicing and dicing of loan portfolios as the cause of the Crisis.  In the aftermath of the Crisis, the technology magazine Wired identified the “Formula that Destroyed Wall Street”, the Gaussian Copula.  The formula is a function that captures the distribution of loan defaults in an abstract  portfolio of infinitely many infinitesimally small loans based on a single parameter,rho,  representing the dependence of loans in the portfolio failing.  If rho=0 then the loans were completely independent, one default would not affect any other loan,  if rho=1.0 they were completely dependent, one default would lead to all the loans to default.  This formula was applied to Mortgage Backed Securities across investment banks with rho set at 0.3.  This choice of rho was based on historical data based on the defaults of corporate bonds, and it resulted in low chances of significant portfolio defaults.  (For a full analysis see MacKenzie and MacKenzie and Spears)

What is most significant about the Gaussian Copula is that it was popularised in investment banking by J.P. Morgan, a very old fashioned bank, who relied on the moral character of their employees and employed a lot of good mathematicians.  J.P. Morgan did not create, distribute or invest in many Mortgage Backed Securities.  The reason is described in Gillian Tett’s book on the crisis, Fool’s Gold.  The mathematicians at Morgan’s reverse engineered prices of traded MBS and deduced that they were generally based on rho=0.3, the bankers then asked themselves was this reasonable, and decided that since corporations where different to mortgage borrowers, it was not.  rho should be higher indicating a greater probability that if one homeowner defaulted, another would.  On this basis, of a rho=0.5,  there was no chance of making a profit on MBS.  J.P. Morgan were right, other banks, who relied on rho=0.3, were wrong.

Were the failed banks bad scientists as well as bad bankers?  While the banks that chose rho=0.3 were wrong, their behaviour was not too different from that encountered in more traditional branches of science and engineering.  Official assessments of the probability of a serious failure of a nuclear power plant do not match the empirical evidence that the have been 5 major failures of commercial, not just experimental, nuclear power plants, most recently at Fukushima.  Similarly, in the 1980s, the official NASA assessment of  the probability of a Shuttle failure was 1 in 10,000, while engineers on the shopfloor reckoned it to be 1 in 200.  In fact, reading Richard Feynman’s Appendix to the Rogers’ Commission Report on the Challenger Failure is interesting in context of the banking failures.

A common feature of the under-reporting of the chances of failure in nuclear engineering, NASA and banking has been a belief in “perfect” systems so that data that can be extrapolated from the known into the unknown.  Today, as “big data” becomes the current buzz-word, with Wired declaring in 2008 the “end of theory”, the power of data enables the scientific method, that searches for coherence and causality, to be replaced by the identification of correlations.

To understand why this is relevant, observe that the Financial Crisis on 2007-2009 was not a global event, whatever Gordon Brown says.  While British and American banks failed, German, and particularly French, banks did not.  At the time there was anger in the Anglo-Saxon banking community, who knew their Continental colleagues were involved in exactly the same practices but did not report the staggering losses.  When a prominent mathematician at a major French bank was asked about this, his response was candid and logical: “We thought the pricing models were weak before the crisis, we knew they were wrong during the crisis.  Why ruin the bank by reporting numbers we knew were wrong?”.  In 2009, The Economist, a tub-thumper for Anglo-Saxon culture, reported with glee on the problems at Societe General caused by the fraud of Jerome Kerviel.  They observed that the problem with French risk managers is that their attitude is ‘That is all very well in practice, but what about the theory”.  It was precisely this belief in placing data in the context of a theory that prevented the failure of French banks, the more empirical, pragmatic Anglo-Saxon approach led to disaster.

The malevolent impact of “social physics”, aggregating a data set into a representative point on which decisions are based, ignoring the distribution, is particularly resonant for me at the moment.  SInce January I had been suffering gastric problems.  Based on my symptoms the initial assessment was that I had gallstones, a prognosis that was confirmed by ultrasound.  However, this evidence, when presented to a consultant,  was contradicted by a brute fact that it is unusual for men in my condition to be troubled by gallstones.  Within 10 days of this assessment, I was admitted to hospital as an emergency, with a gallstone blocking my bile duct,  and have had two operations in the past two weeks.  I proved not to be the “average man”.

I suspect that “on average” the consultant made the right, money saving, decision.  However in making this decision there was a risk of a greater loss, I had two complicated procedures instead of one and spent 5 nights in hospital instead of none.

Nassim Taleb might observe that my consultant was Fooled by Randomness, or had experienced a “rare event”, what Taleb calls a “Black Swan Event”.  But Taleb’s definition of a Black Swan event is novel.

The concept of the Black Swan emerged in the late thirteenth and early fourteenth centuries, at a time when scholars were introducing mathematics into physics.  The concept was part of a vigorous debate between Realists, in the Platonic sense, such as Thomas Aquinas,  who believed that by observing nature its rules and mechanisms could be gleaned.  Opposed to the Realists were Nominalists, who rejected the idea that there was a hidden Reality of nature, the rules and mechanisms were constructed by the men observing them.  Nominalists included Franciscans such as John Duns Scotus and William of Ockham, who rejected a mechanistic universe and believed if God wished to create a Black Swan, He could. The fact that only white swans had been observed did not preclude the fact that a Black Swan could exist.  Of course, the Nominalists were proved right when Black Swans were discovered in Australia.

A good financier is a Nominalist: they recognise that their models are man-made constructions that are a simple approximation of the nature of markets.  On this basis a financier does not attempt to predict the future, and is circumspect about their ability to “beat the market”.  A good financier is often distinguished by their obsession with attempting to handle the risks they might encounter rather than exploit the opportunities that might arise.  We cannot predict an earthquake but we can plan for them.

There is nothing new in suggesting that applied science should broaden its outlook to consider distributions rather than focus on expectations.  However I think finance tells us something more, we need to be sceptical about the very models that produce our distributions.

For example, the debate about climate change is not a dispute between good and evil, or between idealists and empiricists, but more a disagreement between two approaches to modelling.  It is not the data that is disputed but the modelling framework used to interpret the data.  I feel the reason the Climate Debate is so intractable is that popular science, as distinct from mainstream science as practised by the majority of scholars, cannot accommodate the idea that there can be competing theories and this is a consequence of the pervasive belief that science has a Real  basis, rather than being a social construction.  A Realist will always believe that, given enough effort, the secrets of the universe can be uncovered and the future predicted and controlled. A Nominalist is far more humble.

So none of this is new, but in the context of the Enlightenment Exchange, it is worth noting that some people argue that Quetelet, with his social physics based on the average man, put an end to the Enlightenment's focus on the rational man.

The Perils of Physics Imperialism

 My undergraduate degree was in physics, then sixteen years working in oil exploration led me to doubt the certainties that I was taught at Imperial and I became a mathematician interested in uncertainty. The domain for mathematicians interested in uncertainty are the social sciences and I have the fervour of a convert in condemning my old faith and promoting my new faith. While you might not think my beliefs are particularly relevant, I do think a contributory factor in the credit crisis was the wholesale adoption of the culture of the physical sciences by modern finance.

In December there was discussion prompted by the particle physicists Brian Cox and the comedian Robin Ince arguing that policy makers should place scientific advice at the heart of government policy making. My experience of using science to advise policy (in an industrial context) means I think this position is fundamentally mis-guided, but beyond that it relies on the belief that 'science' is indubitable and immutable. My experience is that science and mathematics is definitely not indubitable and immutable, and in fact I belief it has moved backwards in regard to understanding uncertainty in the eighteenth century and is only now recovering. This piece covers an example of what I mean. To me, the example highlights the disproportionate faith people, from all walks of life, have in what physicists say, which is usually grounded in a set of assumptions that cannot be transferred from the physical sciences into the social sciences.

Towards the end of 2012 I came across a paper 'The time resolution of the St. Petersburg paradox' published in the Philosophical Transactions of the Royal Society A on the subject of 'Handling Uncertainty in Science' by a physicist, Ole Peters. The abstract is as follows

A resolution of the St. Petersburg paradox is presented. In contrast to the standard resolution, utility is not required. Instead, the time-average performance of the lottery is computed. The final result can be phrased mathematically identically to Daniel Bernoullis resolution, which uses logarithmic utility, but is derived using a conceptually different argument. The advantage of the time resolution is the elimination of arbitrary utility functions.

The Petersburg paradox was first recorded in 1713 in correspondence between Rémond de Montmort, a gentleman mathematician, and Nicolas Bernoulli, the nephew of James Bernoulli who posthumously published his uncle's Ars Conjectandi and whose own thesis was De Usu Artis Conjectandi in Jure ('The Use of the Art of Conjecturing in Law'). It is worth mentioning that the pair, along with James, first Earl Waldegrave (ancestor of the 1990s UK Science Minister, William Waldegrave) were the first people to identify 'mixed strategy' solutions in game theory, some two hundred years before the twentieth century re-discovery of game theory. The Petersburg game is described as follows

Given a (fair ) game of Heads and Tails, A tosses a coin and gives B 1 coin if a Heads comes up, if a Tails comes up, the coin is tossed again and if a Head turns up, A gives B 2 coins, if a Tails comes up, the coin is tossed again and if a Heads comes up A gives B 4 coins. In general the coin is tossed until a Tail comes up, and, if the first Tail occurs on the nth toss, then A gives B 2^n-1 coins. How much should B offer to pay A to play the game?

The problem for Bernoulli and Montmort was that it had been established that a game should be valued by adding together all possible products of the games winnings and the chance of the winning, the mathematical expectation. There is a 1 in 2 chance of winning 1 coin, a 1 in 2 chance of getting a Head on the first toss. B wins 2 coins by there being a Head and then a Tail, with a 1 in 4 chance, there is a 1 in 8 chance of winning 4 coins and so on. In general the mathematical expectation of the game is

However, nobody would stake more than 20 coins to play the game and typically they would offer 4-6 coins to play. Mathematical expectation seemed to be wrong.

The Petersburg game is generally thought to have been resolved by Nicolas' cousin Daniel Bernoulli, who is today known for his contributions to physics, in a paper he presented to the Imperial Academy of Sciences of St Petersburg (hence its name) in 1731 (it was published in 1738). Economists (good reviews can be found in Basset or Cowen and High) see Daniel's solution as being the earliest expression of the mathematical conception of utility, Daniel writes that "Any gain, however small, brings always a utility (emolumentum) inversely proportional to the whole wealth" and described this in terms of a differential equation, which was the rage at the time in physics, with u being utility and x being wealth, and k a constant, the equation is

The solution to this equation is that u = k log(x), that utility is the logarithm of wealth. On this basis Daniel was able to calculate a reasonable value of the game, which was dependent on B's initial wealth.

Daniel's solution appealed to nineteenth century economists as utility theory took hold of the discipline following Bentham, Mill, Walras and Menger. In particular, logarithmic utility would be given a biological, as distinct from mathematical, justification with the identification of the Weber-Fechner law that showed that the brain responded to stimuli in a logarithmic fashion. By the late twentieth century the recieved wisdom, captured in Peter Bernstein's Against the Gods (p 107) was that the paradox appears, is solved by Daniel in 1731/1738, it was briefly mentioned by Keynes in a Treatise on Probability and was picked up by von Neumann and Morgenstern.

Peter's highlights the role of fairness in the development of early probability, but in saying "was no universal agreement on the relevant concept of fairness" is mistaken: the concept was understood, the mathematical formulation was problematic. But when it comes to the issue of the Petersburg game, he falls in step with the mainstream account of the story. However, his main thrust is in the direction of ergodicity. The word ergodic was coined by the physicists Boltzmann in relation to his study of gases and derives from the Greek for 'work' and 'way'. Today, in physics, it relates the average time a gas particle is in some point of space (which can be the phase space, not necessarily a simple location) with the relative size of the space. However, this is a manifestation of a more general mathematical concept, that a random dynamical system is ergodic if it will eventually have a stable distribution independent of the initial state. For example, if a deck of cards is shuffled enough times, we would expect the eventual distribution to be uniform, whatever the initial state of the deck was (i.e the ace of spades is equally likely to be at any point in the deck). Generally speaking a dynamical system with constant coefficients is ergodic.

Peters argues that the problem with the Petersburg Game is that the system is not ergodic, in that the classical expectation of considering all possible winnings along with their space (chance of occurring) was not the same as the time averaged expectation. This is a physicists view of ergodicity, the association of time averages and space averages.  From the perspective of mathematics, the game is clearly ergodic, it is a stable system with unchanging parameters and in this respect the game is also ergodic in the terms that economists would understand.  Peters' interpretation of the game challenges a physicists understanding of ergodicity but does not really address economists' concerns in the area.Peters then analyses the game in the context of time, with the game being an investment in an asset that offers a high return along with the chance of bankruptcy. On this basis he derives and expression that equates to Daniel's logarithmic utility and goes on to criticise Menger's objections to Daniel's solution.

My objection to Peters' paper is not that it is wrong, it isn't, but its assumption that the paper presents a new novel resolution of the Paradox because he has a naive understanding of the history and importance of the Game.

The most detailed scholarly analysis of the game can be found in an article, "The Saint Petersburg Paradox 1713-1937" by the French philosopher and historian of science Gerard Jorland. The article is difficult to read, because it only exists in a book and is French philosophy written in English.

Jorland argues that the paradox was resolved in the late eighteenth century by Condorcet. Jorland comes to this conclusion by discussing the various attempts to resolve the problem which he categorises into two explanations that emerged as soon as the problem was identified. Nicolas Bernoulli, following his uncle's lead, argued that events with very low chance were 'morally impossible', chances less than 1 in 10,000 could be ignored. A colleague of Nicolas', Cramer, argued that while "Mathematicians value money in proportion to its quantity, commonsense men proportion it to its use" (i.e. utility) and introduced the concept of moral expectation, rather than summing the products of gains and probability, the gains should be raised to the power of the probability and then multiplied, formally

and so the logarithm of the moral expectation is the mathematical expectation of the log utility of the game.

Throughout the eighteenth century French (mathematicians) debated whether the Petersburg game should be resolved by taking a lower bound on probabilities, following Nicolas, or an upper bound on gains, following Cramer, as Daniel did. The issue was resolved in 1785 by Condorcet who took the view that the expectation did not 'really' exist: if someone has a 1 in 2 chance of winning 2 and a 1 in 2 chance of winning 0, the mathematical expectation is 1, but they would never win 1. In the Petersburg game the only way someone will win an infinite amount is if the game continued for an infinite number of rounds, which in reality is impossible. Today, this  is incorporated into stochastic control in finance by the incorporation of the transversality condition  that the discounted value of payoffs at infinite time should be zero.

This view, that infinite payoffs are meaningless, became standard in the nineteenth century, and and developed into the position that mathematical expectation is only valid for repeatable events, when the law of large numbers can come into play. Laplace, who did not believe in randomness, only ignorance, accepted the idea of moral expectation and realised that it equated to mathematical expectation if the number of possible payoffs was infinite (and the division of risks could be infinite). This was the basis of insurance, individuals have a finite number of risks and so employ moral expectation where as insurance companies, with a portfolio of risks, employ mathematical expectation. The physicist EmanuelCzuber noted in 1882 that classical mathematical expectation was meaningless for single events.

Peters approaches his solution to the problem by observing that

To the individual who decides whether to purchase a ticket in the lottery, it is irrelevant how he may fare in a parallel universe. Huygens (or Fermats) ensemble average is thus not immediately relevant to the problem.

This point was appreciated by Condorcet, Laplace and Czuber, it is not new.

Peters actually employs Cramer (and Daniel Bernoulli's) idea of moral expectation by considering the time evolution of the game. He assumes that there is a 'growth rate' for each round of the game r such that the value of the game increases by e^r each round. This is a cheat, acknowledged by Peters to facilitate the comparison with Bernoulli. In fact, a more appropriate formulation was presented by Durrand in 1957 and more recently by Szelkey and Richards.

Peters argues that Menger's criticism of Daniel Bernoulli's resolution of the Petersburg Game in the context of logarithmic utility was wrong. Menger argued that a Petersburg Game could be constructed that offered infinite payoff even using moral expectation/logarithmic utility, and so the only resolution of the Paradox was to use bounded utility functions, not just logarithmic utility functions. It is not clear why Peters thinks Menger was wrong, this is not surprising since Menger was not wrong, if B wins e²ⁿ instead of 2ⁿ logarithmic utility does not solve the problem. What Peters fails to appreciate was that Menger was conforming to the resolution provided by Condorcet, the game cannot proceed for an infinite number of rounds, which is were the resolution to the Paradox really lies.

In fact the Comte de Buffon (of Buffon's needle) did a very un-French thing and resolved the Paradox empirically.  He asked a young boy to conduct 2,048 experiments of the Game and tabulated the results.  He found that the experimental results closely matched the theoretical results based on the Binomial Distribution and that the total payout (to B) of the 2,048 games was a little over 10,057 coins  resulting in a fair price for the Game of around 5, close to the original observation of Nicolas Bernoulli and Montmort (in fact if you consider 2ⁿ trials of the game, the expected value of the game is n/2, and so if you consider events of chance less than 1 in 10,000 to be "morally impossible", this corresponds to it being "impossible" to see 14 heads in a row, you would value the game around 6.5-7).

Buffon's result employs ensemble averages to arrive at a sensible answer.  There seems little value in Peters approach employing contemporary ideas in physics.

Peters is challenging the concept of ergodicity in finance, ergodicity was introduced into finance from physics (Mirowski has written on this) along side utility theory. This was done in the context of post-Laplacian science, when there was no such thing as randomness, only a lack of information. Pre-Laplacian, and pre-Smithian ideas relating mathematics to the uncertainties of finance were lost. Peter is not just re-inventing the wheel by presenting his resolution of th ePetersburg papradox, his attack on Menger disguises the real solution to the problem, which is that mathematical expectation is not really relevant to economic affairs, because there is n stability in the dynamical system (a lack or ergodicity) and parallel universes don't really exist.

Peters paper has been well received, particularly by those who believe the solutions to the problems of finance lie in physics or in heterodox economics. Unfortunately, the well regarded actuarial consultancy Towers Watson has also succumbed to the allure of physics, shame on them.

Food and Finance

It seems, from the cheap seats at least, that the Eurozone crisis has abated, so now the media is filled with a food crisis. For anyone not following the story, despite food being an important issue of public discourse, following the emergence of BSE twenty years ago and the GMO debate, it turns out things are not as transparent as one would hope and when Europeans have been paying for processed beef they have ended up eating dead horses.  This is no great surprise to someone who has occasionally bought a burger for £1.50.

I started thinking about the relationship between the food crisis and finance when a non-academic colleague asked me to explain the difference between a new Master's in Financial Engineering we have introduced and our existing programmes in Risk Management and Financial Mathematics.  My explanation was as follows

In the past, investors did the equivalent of buying tomatoes, onions, meat and pasta.  Today financial engineers sell investors ready made lasagne.  Risk managers make sure it is correctly labelled (the traffic lights/no horsemeat instead of beef) while financial mathematicians put a price on the lasagne.

This got me thinking about the failures of food regulation and financial regulation.   The changes in finance that have occurred over the past forty years mirror those in food.  In the 1960s investors had a limited choice of where they could put there money - the job of an investment manager at a life insurance company involved buying government bonds, and if you were particularly daring, Ford or Exxon stock. When Bretton-Woods collapsed in 1971 this benign environment became perilous, exchange rates fluctuated and governments responded by adjusting their interest rates while commodity prices became volatile.  The work of fund managers ceased to be sedate and became something of a high wire juggling act.  To make the juggling act easier, hedge funds, and then banks, started processing the volatile raw assets into investment vehicles designed to meet the needs of the fund manager.

Pretty much contemporaneous with these developments in finance, the UK food industry was making similar innovations.  As society changed and housewives went out to work the food industry started selling processed ready-meals.  Apart from convenience, ready meals are sometimes seen as being important in developing consumer tastes - the British menu is today much more varied than the traditional "meat and two veg" of the sixties.  The downside has been concerns as to the nutritional value of ready meals, and as a result there has been more and more regulation on the industry, particularly relating to labelling.

My problem is that despite significant effort (and expenditure) by regulators, the consumer does not seem to have been well protected.  This statement could apply to either food or finance.

In the aftermath of the Credit Crisis, the then Science Minister, Lord Drayson asked me to collate views from leading mathematicians working in finance what the causes of the crisis were. One academic (asked to be anonymous)  commented as follows

I was involved in a meeting to discuss new financial regulation. As a mathematician, I had anticipated that the discussion would be on the robustness of the underlying models being used. In fact the discussion focused on the processes to ensure financial institutions complied with the letter of the regulation.

What the academic observed was that as a result of detailed regulations, bankers stopped "thinking" about their models and processes and resorted to a "box ticking" exercise.  The regulations were so detailed that they responded by focusing on the legal issues around the regulations and not the substance  of their business.  One wonders if something similar has happened in the food industry.  If we pass the "traffic lights" all is well with our ready meal.

This issue was raised by Andy Haldane in his "The Dog and the Frisbee" paper presented last year when he argued that "less is more" in banking regulation.  If the regulations are too detailed, bankers can lose sight of the key issues. However, Haldane's arguments do not square with the views of his new boss, the Bank's Governor designate, Mark Carney.

In the conversations I had with bankers who were involved in Credit Derivatives after the crisis, the point was made that the emphasis in the business was "knowing the components" of a structured product (a ready meal) rather than having a whizz-bang pricing model (efficient food processing).  One suspects the food industry will start taking a closer look at its supply chain as well, now.

Just as EU bureaucrats discuss the Food Crisis, their colleagues are proposing new regulations on finance.  The Financial Transaction Tax that the EU are looking to implement has three objectives:

• To tackle fragmentation of the Single Market that an uncoordinated patchwork of national financial transaction taxes would create;
• To ensure that the financial sector makes a fair and substantial contribution to public finances and covering the cost of the crisis, particularly as it is currently under-taxed compared to other sectors;
• To create appropriate disincentives for financial transactions which do not contribute to the efficiency of financial markets or to the real economy

The EU food labelling regulations share the first objective, but Europe does not tax salt-sugar-fat in processed foods and does not seek to hinder the development of innovative food products, even if they do not contribute to the nutritional well being of EU citizens.

Food and financial regulation are extremely important to the well being of Europe's citizens.  While Food Regulation will be developed in public, and as a result I do not expect the EU to implement a Food Transaction Tax.  However Financial Regulation is rarely considered outside the specialist media.  As a result, I worry that ill considered regulations will be implemented that do more harm than good.

Why some physicists shouldn't do finance

In January 2011 the most prestigious British science journal, Nature, published a paper, Systemic risk in banking ecosystems, co-authored by Lord May, past President of the even more prestigious Royal Society, an eminent Professor of biology at (one time or another)  the Universities of Oxford, London (Imperial) and  Princeton and past Chief Scientific advisor to the UK Government, and Dr Andy Haldane, an economist who is the Bank of England's Executive Director of Financial Stability.  The paper was picked up by the media and widely publicised, the BBC's science correspondent heralding its publication and the BBC then broadcast a radio programme on the interaction of biology, complexity and banking.

May and Haldane is an important contribution summarising a programme of research that the Bank of England had been developing in collaboration with Lord May.  However, just under a third of the paper, the section entitled Potential causes of an initial shock focuses on a paper, Eroding market stability by proliferation of financial instruments  written by three physicists, Caccioli, Marsili and Vivo, and published in the European Physics Journal B in 2009.  May and Haldane describe the paper as

A sophisticated and important analysis of a major flaw in the pricing of derivatives.

The purpose of this post is to describe why the analysis undertaken by the physicists Caccioli, Marsili and Vivo is deeply flawed and goes on to discuss how damaging "science based policy advice" can, consequently,  be deeply flawed if it lacks a scientific base but is propagated by scientific authority.  This is presented as a concrete counter-example to the opinions offered  Brian Cox and Robin Ince on the need to place science at the pinnacle of policy advice.

Caccioli et al (I discuss this published paper, a pre-print is publicly available on arXiv) argue that

Arbitrage Pricing Theory (APT), the theoretical basis for the development of financial instruments, with a dynamical picture of an interacting market, in a simple setting. The proliferation of financial instruments apparently provides more means for risk diversification, making the market more efficient and complete. In the simple market of interacting traders discussed here, the proliferation of financial instruments erodes systemic stability and it drives the market to a critical state characterized by large susceptibility, strong fluctuations and enhanced correlations among risks. This suggests that the hypothesis of APT may not be compatible with a stable market dynamics. In this perspective, market stability acquires the properties of a common good, which suggests that appropriate measures should be introduced in derivative markets, to preserve stability.

 By APT the authors mean what mathematicians understand as the Fundamental Theorem of Asset Pricing (FTAP), which I have discussed at length.  The issue that Caccioli et al focuses on, and it is an important point that is generally overlooked in economic and financial discussions of the FTAP, is that of market incompleteness.

Central to the FTAP is the following idea: How should you value a bet on roulette before the wheel is spun? In a more general setting, think about a world that jumps from "now" to a future that could take on, randomly, one of K states  (in the roulette example K is 36+1 in Europe and 36+2 in the US).  The problem finance has is how to price an asset, now, that has a different value in each of the K states, given that we do not know what state will come up.  At the core of the FTAP is a simple mathematical result, we can solve a system of simultaneous equations with  K unkowns if we have K equations involving the unkowns.  This is Cramer's Rule, a mathematical result taught to school kids that has nothing to do with economic or financial theory: if you reject it you may as well reject 2+2=4.

Say the number of assets in a market is N, and each of the N assets has a specific payout in each of the K states of the future world, then if N=K we say that the market is complete.  If N>K it means that there are (N-K) "redundant" assets.  What this means, as a result of Cramer's Rule, is that these (N-K) redundant assets can be perfectly replicated by constructing a portfolio of the N=K 'primal' assets. In this case, on the basis that we know the initial prices of these N assets we know precisely the prices of the (N-K) remaining assets.  Along with this precision comes a cost: we cannot make any profits in the market, the fact that the market is riskless means it cannot generate any profits for the market.  This is the basis of the "academic" practice of derivative pricing.

In practice, markets are "incomplete", meaning that N is less than K, assets cannot be priced precisely, there is uncertainty and are risks in the market, which consequently offers (some) profit opportunities. Some economists believe that market incompleteness is about information incompleteness, and so, just as a physicist might repeat an experiment more and more times to get a more and more precise estimate of a parameter, they believe that increasing the number of assets traded in a market brings that market closer to completeness.  They treat incompleteness as an epistemological problem, based on limited knowledge, rather than an ontological problem, that the market is either complete or not : one is either a virgin or not, according to conventional science.

Caccioli et al are, rightly, concerned with the fallacy that increasing assets in a market will lead to market completeness.  However, rather than writing a short and trivial paper describing why the fallacy is stupid, they seem to buy into the notion that increasing assets in a market could lead to completeness.  And in fact they study a finite state market with K=64 which will become complete as soon as the number of assets hits N=64.

I am uploading to SSRN a technical paper describing the numerous conceptual and technical problems I have with Caccioli et al, but here is a non-technical summary.

• Caccioli et al do not price assets using the FTAP, rather they assume that financial institutions seek to maximise profit by offering assets where demand is high.  The model is naive  and despite the claim that the model is dynamic and interacting, it does not attempt to identify a price that matches supply and demand.
• Their key result, reported in Haldane and May, is that when N/K>4, the market becomes unstable and volatility explodes.  This is Haldane and May's Potential cause of an initial shock.

I do not expect a lay audience to appreciate the fact immediately that if N/K>4 the market is complete, and around 3K of the assets are redundant.  All prices are known precisely in the market, so there is no volatility, and the returns to financial institutions offering instruments are zero.  

To offer an analogy, the physicists Caccioli et al are arguing that the solar system will become unstable if the planets are moving at 4 times the speed of light.  Well a teenager could fantasise about such a result, but policy advisors should not worry too much about it.

The problem seems to be that Caccioli et al do not seem to understand APT/FTAP and take an economic fallacy to show that under a simplistic model of bankers being predatory profit maximisers, the financial system is unstable.    The authors bear good scientific reputations, with one, Marsili, being on the editorial board of the European Physics Journal B yet one wonders how such a paper saw the light of day?

Science is littered with examples of rubbish papers going through the peer review process, the MMR controversy originated in a Lacent paper, but the "theory" around science is these errors are quickly corrected by "science".  However in the case of Caccioli et al, the error was propagated by even more prestigious scientists, and disseminated, through the BBC, into wider society.

I have a great deal of respect for Lord May (though I might disagree with him, as Denis Mollison has, on the role of uncertainty in biological models) and he should be commended for engaging in finance.  He cannot be expected to understand the details of the FTAP.  The same could not be said for Dr Haldane.

Haldane (and May?) make the following statement

Caccioli and colleagues note that APT makes several conventional assumptions upon which everything else depends: "perfect competition, market liquidity, no-arbitrage and market completeness". Crucially, this adds up to the implicit assumption that trading activity has no feedback on the dynamical behaviour of markets. And indeed, in the APT-fuelled boomtime that preceded the bust, APT seemed to be very successful. In its imaginary world, market failures are caused by regulatory carelessness, resulting in a focus on creating institutional arrangements that seek to guarantee the premises upon which APT is based. To the contrary, Caccioli and colleagues argued that APT is not a ‘theory’ in the sense habitually used in the sciences, but rather a set of idealized assumptions on which financial engineering is based; that is, APT is part of the problem itself.

Let's dissect this statement in detail.

• APT makes several conventional assumptions upon which everything else depends: "perfect competition, market liquidity, no-arbitrage and market completeness".   No-arbitrage is essential in the FTAP, but completeness is conditional and will depend on perfect competition (lack of frictions) and market liquidity.  These are "conventional" in the sense that they are presented to finance and economics undergraduates, but I teach second year maths students about the problem of incompleteness. Caccioli and colleagues do not have a good enough grasp of the FTAP to comment on its structure.
•  the implicit assumption that trading activity has no feedback on the dynamical behaviour of markets Neither does the model employed by Caccioli et al, their asset prices do not adjust to balance supply and demand, the most basic example of feedback in markets.  Pot calling kettle black?
•  the APT-fuelled boomtime that preceded the bust Is the BoE claiming that the "boomtime" was a consequence of a mathematical model, was lax interest rate policy and market oversight not more significant? Did the collapse of Bretton woods and fixed exchange rates, stable interest rates and cartel determined commodity prices not have any impact on the development of the derivatives markets that the FTAP was developed in response to?
• market failures are caused by regulatory carelessness, resulting in a focus on creating institutional arrangements that seek to guarantee the premises upon which APT is based OK so th eregulation was lax, but it was all the fault of those pesky mathematicians leading us off the straight and narrow.  Begs the question, how do you earn your money?  Are we forgetting about the fact that the regulators could have argued that Credit Default Swaps, (re-)introduced into the markets in the mid-1990s could have been treated as insurance contracts, and so would not have been tradeable assets, but he regulators chose to let them pass.  Are we forgetting the pleas by mathematicians like Phillipe Artzner and Freddy Delbaen  or Michael Gordy that fell on deaf ears by regulators and policy makers, see my comment in an earlier post

The problem with modern finance is not in the mathematical models, but in that the models were an end in themselves and not a means for developing a consensus, understanding, knowledge about finance. Banks employed geniuses to develop these models in house that they kept secret, or, they bought black boxes that had been created by geniuses elsewhere.  When mathematicians, such as Phillipe Artzner and Freddy Delbaen  or Michael Gordy, shone a light on the some of the leading industry models, their illumination was blocked by the towering geniuses, the "masters of the universe", working in banking.

One has the impression that the regulator is looking for a justification for the failure in regulation, and Caccioli et al provides an escape clause: it was the failure of FTAP. The irony is that my interpretation of the FTAP is that it is rooted in reciprocity and emerges out of a philosophical tradition that aimed at establishing Justice rather than to maximise profit, the framework underpinning Caccioli et al, and which I believe is at the heart of the stability problem.  But maths is an easier target than mainstream philosophy.

This sorry tale is really about a failure of the model of science that the likes of Brian Cox and Robin Ince adhere to.  The Cox-Ince model should prevent the opinions of regulators, that "it was not my fault", or physicists that "bankers are evil" impacting policy advice.  But the duo of Haldane and May stand as clear exemplars of the brute fact that we are all human and opinion does trump an abstract notion of the purity of science.

This issue is at the heart of the model of Caccioli et al is that it  ignores the basic lack of human objectivity and layers abstract physical analogy on physical analogy rather than consider the market as a social structure.  Furthermore, the mess the regulators got themselves into is essentially adhering to the model of science promoted by Cox&Ince, that it is indubitable.  Social scientists, on the whole, are far more circumspect, science is socially constructed, and so the problems of subjectivity, and the shakiness of science, need to be taken seriously.

Magic, markets and models of science

There is a well developed theory that a key impetus for the development of European science in the seventeenth century  was magical thinking, developed and promoted through the sixteenth century by the likes of Paracelsus, John Dee and Emperor Rudolph II.  While there is little doubt that Hermeticism and Alchemy had a significant influence on the development of natural philosophy, magical thinking cannot explain the uniqueness of the scientific developments in Europe in the 1600s, since magic is a feature of all cultures, notably China.  But this factual observation is tempered with an opinion, that good science is open where as magic is hidden and secretive, this is a central theme in Mauss' A General Theory of Magic.

An alternative, minority, theory for the foundations of modern science is in European financial practice.  I prefer this theory because, by their very nature markets are social, collaborative, open, forums (those queasy about markets might wish to consider my view).  Evidence for the significance of financial practice in the development of science comes in the fact that Copernicus was trained in financial mathematics and wrote on money before he wrote on cosmology, the Merchant Adventurer Thomas Gresham was a more influential contemporary of Dee, despite relative number of contemporary biographies of the two Elizabethians, who laid the basis of the Royal Society with the establishment of Gresham College. Simon Stevin was trained in finance and founded the influential Dutch Mathematical School that inspired Descartes and performed many of the experiments that Galileo is famous for.  While Newton's interest in magic has been promoted, the fact that he spent half his life running the Mint is often overlooked.

Furthermore, magic was most influential in central Europe, centred on Rudolph's court, while the scientific revolution was centred in western Europe, by the likes of Huygens and Bernoulli who were as likely to work on financial problems as physical ones.  Finally, my observation of good financial practitioners is that, contrary to popular belief, they do not believe they can control the markets, rather they have to navigate through its intrinsic uncertainties using the best tools available - specifically mathematics.  This contrasts with the image of the magician controlling nature.

To develop this point, there has been significant criticism of the use of mathematics in the the lead upto the Credit Crisis, that is still affecting all our lives. In particular, the use of a technique known as Value at Risk, to measure the riskiness of investments and the Gaussian Copula, the "formula that killed Wall Street".  Now, the fact is that both these techniques emerged out of the investment bank, J.P. Morgan, and the problem in blaming these mathematical techniques for the Credit Crisis is that J.P. Morgan did not engage in trading Credit Default Swaps on Mortgage Backed Securities, and did not hold the alchemical Collatorallised Debt Obligations of Mortgage Backed Securities.  The reason?  J.P. Morgan having developed the techniques understood them and could not reconcile the model results with their "opinions" of the market.  The "science" was being tempered by "opinion" (all this is covered in Gillian Tett's book Fool's Gold, or  STS/HPS types might like this).

While, at first sight, these observations might not seem relevant to my criticism of the article by Cox&Ince, I see the two as being linked. Fundamentally, my concern, and it might be unjustified, is that in advocating the primacy of "science" people are endorsing a kind of group think that leads to the sorts of disasters like the Credit Crisis.  This might be seen as a bit melodramatic but lets consider two examples, climate change and evolution.

My understanding is that there is little academic dispute about most of the raw data  relating to the current climate (modulo the hockey-stick): temperatures are rising and there has been an increase in CO2 in the atmosphere. The academic debate is to what extent man-made emissions of CO2 are causing the rise in temperature, and this is a question of model choice, and since models are made by humans they are, unfortunately for Cox&Ince, social constructs.  In November I listened with interest to Andy Kerr talk about plans to bring academics on both sides of the climate debate to discuss the model issues in order to move to a better consensus.  This is good science, but the science is not changing as a consequence of new data but as a consequence of human deliberation.

The other great bug-bear of science is evolution.  As a mathematician I admire Alan Turing, Turing's most significant work (in terms of citations and impact), undertaken whilst at Manchester at the time of his suicide, was on morphogenisis and this was inspired by Sir D'Arcy Wentworth Thompson's On Growth and Form, written to counterbalance the dominance of Darwinian evolution and the emphasis on survival of the fittest. Biological structuralists have come under attack from neo-Darwinists for introducing  metaphysical (i.e. mathematical) explanations for phenomena.  Who is being un-scientific here, the mathematicians or the neo-Darwinists?

Paul Nurse, the President of the Royal Society whom Jack Stilgoe has associated with Cox&Ince, has argued that "keep science as far as is possible from political, ideological and religious influence".  I think Nurse is mistaken here for two reasons.

Firstly, in the thirteenth century the Dominicans, such as Thomas Aquinas and Albert the Great, began Europe's development of science by developing the idea that God could not interfere with nature at will: the deity designed the cosmos, like a machine, but once set in motion they could not interfere with it.  The Franciscans, such as John Duns Scotus and William of Ockham, took a different view: God was not constrained by nature.  Of course today only a fool would agree with Scotus and Ockham.  Or would they? The idea of a Black Swan was developed at this time, the empirical rationalist Dominicans argued that, since no Black Swan had been observed they were not possible.  The fideist Fransciscans disagreed, God (and nature) could be capricious and produce a Black Swan if they wanted to, and of course they did.  (The Black Swan of Nassim Taleb is different, it is a rare event and in finance we would call Taleb's swan an epistemological problem, Scotus' an ontological problem, and yes finance practitioners are talking in these terms).

The point is, understanding the theological arguments about Black Swans help in our understanding of the contemporary debate.  A good scientist, in my view, needs to be humble and accept that they do not, and cannot, know everything.  This was central to good financial practice, until "science" became influential through economics, physics and biology.

This leads me on to the second reason why I think Paul Nurse is wrong, by separating themselves from outside influences scientists as susceptible to loosing their humility, or at least of becoming trapped in an ivory tower.  This is precisely what happened in finance at those inbstutions that failed, unlike J.P. Morgan. I am of the opinion that the consequences for society could be as dire if science is not integrated with political, ideological and religious influences.  Argument should be suppressed if you think you'll lose the argument, but is benign otherwise, something I have discovered as a parent.

Cox&Ince begin their article by describing the significant improvements in the human condition over the past two to three hundred years, which they ascribe to the "scientific method".  This brash statement cannot, scientifically, be offered as fact.  As a counter example I offer the case of the Soviet Union/Russia.  Few would argue that Russia's scientific achievements between 1950 and 2010 were not comparable to those of the UK, France or Germany, but these achievements have not resulted in the quality of life and freedoms enjoyed by western Europeans.  Indeed there are many western nations that cannot match Russia's scientific achievement yet are none the less more attractive places to live, Belgium for example.

The argument that science leads to wealth is an opinion not supported by fact, an argument originating in Thomas Gresham's less successful, but more famous, nephew, Francis Bacon.  Good science is a consequence of good political structures, and good political structures are closely associated with sound financial practice, and without these social structures there would not have been the technological advances that we enjoy today.  This is why I choose Athena as my avatar, acknowledging the precedence of the civilising god over the technological god, Hephaestus.  Sorry, am I mixing religion and science here?

Promoting the view that scientists hold a particularly important position in the context of political decision making is dangerous, because despite their best intentions scientists are frequently wrong and are rarely able to distinguish what is true from what is false (the issue about the failure of the Law of Excluded Middle in my last post).  A technocracy is just as much an oligarchy as a monarchy, theocracy, plutocracy or stratocracy  and is as likely to kill the goose that lays the golden egg as any other system of government that denies dissent.

A historical example of scientists being wrong.  Galileo was convicted by the Catholic Inquisition for publishing Dialogue Concerning the Two Chief World Systems in 1632.  Galileo's original title for the Dialogue was Dialogue on the Ebb and Flow of the Sea, because, as a  consequence of the Copernican theory and mathematics, Galileo argued that there would be one tide a day.  When he sent the book to the Church for approval, he was told to change the title because every European sailor  knew that there were two tides a day.  For the Church of the time, built on Aristotle, Galileo's use of mathematics to describe reality was not just philosophically wrong, it also resulted in absurd conclusions.  The  historian, Harold Brown explains

 Galileo's attempt to account for the tides as a result of the combined daily and annual motion of the earth, and his belief that this argument provided a physical proof that the earth moves, stands as something of an embarrassment.

Are we to consign Galileo to the set of non-scientists and commend the Church for his prosecution, I doubt this would go down well with the likes of Cox&Ince.

The point is scientists must "commit to an evaluative framework", this is what Galileo has done, nailing his colours to mathematics and the Copernican model.  The phrase "commit to an evaluative framework" comes out of one of the rare scientific studies of the behaviour of traders, by Daniel Beunza and David Stark, who note that

the trader is emotionally distant from any particular trade, to be able to take a position, the trader must be strongly attached to an evaluative principle and its affiliated instruments

This description of a trader seems to fit that of a scientist.  However, what is understood in finance, for example the traders at J.P. Morgan using VaR and the Gaussian Copula, and was understood by Duns Scotus and Ockham, is that faith in the evaluative framework should not be blind, the scientist must be open to persuasion that they may be wrong.  The view taken by Cox and Ince is that this can only occur after new data has appeared, but the vacuousness of this position is given by the example of the Black Swan and the way that science can avoid these lapses in prudence is by being open.

It might seem to be splitting hairs to argue in favour of science originating out of finance rather than magic, but I am of the opinion that creation myths have a critical role in how cultures view themselves.  If science believes it emerges out of magic it will be forever associated with secret knowledge that enables the magician to control nature and convert base metal into gold.  If we regard science as originating out of markets constructed social instruments, then it is natural that we think of science as being "just another" social construction and it is made more human, and possibly more relevant.  Simultaneously, and this is my ultimate objective, we shall start observing markets from a scientific perspective, rather than having them hidden from public oversight by a veil of mystery and obscure incantations.

Now if I have been a bit obtuse, let me be more explicit.  I worry about the views around the belief  that "science" has a special status that makes it difficult to challenge was a contributory factor in the Credit Crisis.  I will give an anecdote that a physical scientist would dismiss but social scientists would consider as evidence.  After the collapse of British and American banks there was anger in the British and American banking community that the French had been engaged in exactly the same practices as the failed Anglo-Saxon banks, but had weathered the storm.  I was party to a private conversation where a prominent mathematician who works for a prestigious French bank was asked about this complaint.  The continentally trained mathematician pointed out that "we thought the models were probably wrong before the Crisis, we knew they were wrong during the Crisis, the view was the solvency of the bank should not be compromised on the basis of bad models".  The UK/US banks can be seen as believing in the models before and during the Crisis, as a result, Lehmans, Bear Sterns BoS, Northern Rock and RBS no longer exist as independent entities where as Paribas and SocGen do.  In a way, this aspect of the Credit Crisis can be seen as empirical evidence for a failure of Analytic Philosophy  and a triumph for Continental Philosophy.

Despite the fact that I have spent an hour or so writing this piece, and you have spent time reading it, I do not think we should worry too much about the article by Cox&Ince.  Neither of the authors is qualified to inform policy makers, and I suspect the likes of Andy Kerr advising on climate change, the scientists advising DEFRA on GMOs or the badger cull, or even my modest contribution to DECC's modelling efforts, believe that policy makers should be so confident of accepting scientific advice without it being tempered by political considerations.  Cox & Ince, perhaps, should focus more on the distinction between science and entertainment rather than the relationship between science and politics.

Science, politics, mathematics and finance

I went "offline" over the Christmas break and so missed the fallout of Brian Cox and Robin Ince's article in the New Statesman Politicians must not elevate mere opinion over science.  The essence of C&I's piece is that science exists as an "adjudicator above opinion", but C&I's "science" is narrowly defined, "science is a process, a series of structures that allow us, in as unbiased a way as possible, to test our assertions against Nature", essentially science is the set of phenomena that does not involve human interaction. They go on to say that

Science is the framework within which we reach conclusions about the natural world. These conclusions are always preliminary, always open to revision, but they are the best we can do. It is not logical to challenge the findings of science unless there are specific, evidence-based reasons for doing so. Elected politicians are free to disregard its findings and recommendations. Indeed, there may be good reasons for doing so. But they must understand in detail what they are disregarding, and be prepared to explain with precision why they chose to do so. It is not acceptable to see science as one among many acceptable “views”. Science is the only way we have of exploring nature, and nature exists outside of human structures.

 This is a well structured argument.  The five sentences are hard to disagree with, the paragraph gets our agreement, then it hits us with more controversial comments that to disregard scientific advice requires precise explanation.  The final clause is telling "nature exists outside of human structures".

The problem I have with this assessment is that C&I claim science is essential in areas such as climate change, vaccines, GMOs and evolution.  Now, apart from evolution, all these topics involve human interaction with nature: the issue about climate change is whether it is human induced; vaccines and GMOs are essentially   compounds synthesised by humans and placed into "nature".   While nature exists outside human structures, vaccines, GMOs and human-induced climate change cannot.

Brian Cox has Tweeted "Reviewed criticism of @robinince and my Christmas New Statesman, and concluded none is scientifically valid - so still time to get it :)".  This gets to the heart of my problem with the C&I argument - they define what is "scientific" in such a way that it becomes impossible to argue a case against them "scientifically".

As a mathematician working in relation to finance, why should I care?  Between 2006-2011 I was the "RCUK's Academic Fellow in Financial Mathematics", the Academic Fellowship scheme was initiated by the UK government in the aftermath of the MMR vaccine fiasco and the problems with the introduction of GMOs to establish 800 scientists who would act as advocates for their discipline in the event that discipline became a subject of the news.

In September 2008 I contacted the RCUK, my funders, about "what should I do", given that the world was in the grip of the Credit Crisis, the RCUK advised I contact the Science Media Centre,  a publicly funded “venture working to promote the voices, stories and views of the scientific community to the national news media when science is in the headlines”  to facilitate communication between mathematicians and science journalists on the credit crisis.  I sent an e-mail to the SMC but did not here anything back.  After I had been asked to appear on the BBC's Newsnight programme to discuss the science behind the Crisis, I phoned the SMC about my earlier contact.  The SMC had decided that the Credit Crisis was not a science story, and therefore beyond their remit. This was really annoying because a highly political Press Officer, Fiona Fox, with no scientific credentials was telling me, someone who had a Physics BSc, had worked in a technology based industry, had a PhD in applied probability and was an established academic,  what science was.  The following February, I was contacted by the office of the Science Minister who asked me to collate the views of mathematicians n the Credit Crisis, they asked if I had any support from the SMC and I recounted the tale.  Later that day I was contacted by the SMC to sort things out - some four months after the damage had been done.

This experience with Ms Fox revealed to me some serious issues with British science.  There are a hard-core of scientists, a vocal minority, who are convinced science is undervalued, because it is under-funded and the country is not run as a technocracy.  They often complain that they are not taken seriously by society, while simultaneously withdrawing from addressing issues that are of concern to society: true science is about cosmology or particle physics, not about obesity or poverty.

What is perplexing is that, as many sociologists have pointed out, is that, for the likes of C&I,  the one area that should not be open to rational examination (i.e. science) is their beloved "science".

Now lets get to the maths and finance.  C&I are famous for being the presenters of the award winning BBC Radio Show The Infinite Monkey Cage.  The title is an oblique reference to the idea that if you collect an infinite number of monkeys and typewriters  they will eventually come up with the works of Shakespeare.  To be precise, they will almost surely (a.s.) come up with the works of Shakespeare.  The term a.s. is mathematical, a physicist like Cox would not worry about it, and it means that it is not certain that the monkeys would come up with the works, just that it is highly probable - the two are not the same as any financial practitioner is regularly reminded.

This distinction between a.s and certainty plays an important role in the philosophy of mathematics, in particular intuitionism or constructivism.  L.E.J. Bouwer argued that  statements like "there are a sequence of 100 9's in the decimal expansion of pi" (which as an irrational number has an infinite number of digits) can neither be proved to be true or false, in mathematics we cannot rely on the Law of Excluded Middle that assumes propositions are either true of false. If we cannot rely on truth/falsity of mathematical statements that apply to continuous phenomena (which involve infinite sets), then how can we rely on scientific statements to be true or false?  The issue is that many of the scientists who worry about public disregard for science often do not undertake a rational assessment of the "scientific method" that they are so reliant on, and dismiss any questioning of it as "un-scientific" because it implicitly considers science as a human construct (which is the essence of  my paper on Ethics and Finance).  The story of Brouwer and the LEM is particularly pertinent here because Brouwer took a constructivist approach because he was a Marxist - it was a political act.

Brouwer was not isolated in his distrust of non-constructivist theories, Poincare, Borel and Lebesgue were equally circumspect of the approach.  Again, what is pertinent here is that both Poincare and Borel were mathematicians who took an active role in political life and contemporary culture, they did not withdraw into academic cloisters and complain about society's disinterest in science.

Now the finance.  C&I base their science on observation, data, and the predictive models constructed on the basis of the data.  However there appears to be an assumption that "science" will come up with the right models, modulo the approximation problem, given the data.  However this approach makes some omissions: what data is collected and why (science does not work by collecting reams of data in the hope something will drop out), data analysis is subjective (is climate data a hockey stick or a bath - see McIntyre&McKitrick, what does the data say?), models are human constructions.

Making these observations does not seem relevant to C&I, but they are crucial in  modern finance,  an arena of people competing to select and interpret data and develop the best models.  It is a microcosm of good science, and for this reason it should be taken more seriously by the scientific establishment.  Not least because modern finance is more relevant, and therefore more interesting, to the public than cosmology or theoretical physics.

The Problem with the Foresight Report on Computer Trading

It turns out that Paul Wilmott shares my views on the BIS Foresight  Future of Computer  Trading in the Financial Markets.  I have made the point that the Report is running the risk of appearing as what Roger Pielke Jr describes as "stealth advocacy", scientific research presenting itself as neutral technical analysis while it is in fact advocating a specific course of action.

My main issue with the Report is captured in its observation that (p 140, ss8.2)

In financial markets, the ideal is to discover the ‘fundamental’ price of assets

This statement is economically controversial, I do not think Keynes would agree with it.  Intuitively, if you have ever bought a house, did that house have a 'fundamental' price? Rather than develop these more philosophical points I shall focus on the mathematical objections to the statement after highlighting how this assumption drives the Report's overall conclusions.

If there exists a fundamental price, then the markets are attempting to solve an epistemological problem, and wish to identify the true price in a mass of noisy data.  This means that financial stability is defined as (p 19 n1)

the lack of extreme movements in asset prices over short time periods

and is closely related to volatility (p 19 n2)

variability of an asset’s price over time

which is a bad thing, so the report looks for evidence of HFT increasing volatility (which it does not find, but the FT cites research that suggests there is a link between increased volatility and HFT).  

The definition does mean, however, that the pathological behaviour described in Figs 4.2-4.6 of the Report is not market instability, they are localised transient effects that are quickly corrected.  The analogy is that it is OK for an aeroplane to go out of control, providing it does not crash, I do not think the aeronautical industry would allow itself to be run on this basis.

The problem is with the Report's treatment of High Frequency Trading.  The Report recognises that this aspect is controversial, particularly with pension managers and conventional fund managers.  Because it takes an approach that will see HFT as beneficial, because it increases information flows that help resolve the epistemological problem,  the concerns of those not involved in HFT, like the pension funds,  are counter-balanced/nullified. 

My concern is that in taking the epistemological approach to markets, it is inevitable that HFT is beneficial.  Just as if you model  credit default as some form of contagious agent, dynamical systems analysis will point to a solution involving a few, large, well defended financial institutions (i.e. how you deal with mad cow disease), while modelling the banking system as a communications network, loans are 'packets' that move around the network, a distributed system like the internet involving many small institutions with lots of connections, is best.  The result, by and large, depends on how you approach the problem.  The Bank of England seem to be moving to the Internet model of banking, rather than the Contagion model. 

The mathematical objection to the approach the Report takes is rooted in the Fundamental Theorem of Asset Prices, which indicates that a 'true' price of an asset only exists in an idealised situation, the second statement of the Theorem is explicit in stating that in actual markets, a unique, 'fundamental'  price is impossible to identify, there is a financial Heisenberg uncertainty principle going on.  The problem of a price is ontological, not epistemological.  I suspect this is one way of summarising the points Wilmott made to the enquiry, because it is such a dominant theme in contemporary mathematical finance.

The line the Report takes is explainable in terms of the background of the bulk of the contributors to the evidence based, they are in the main from dynamical systems (rooted in ergodic systems) and computer science (data and rules to manipulate data).  There are only a few who contributed to the evidence base whom I recognise would be familiar with the FTAP (Cont, Shied, Avellaneda, Mitra, Jaimungal, Cvitanic out of over 200, 2 of whom declare a commercial interest.  There were 11 alone at the Complex Systems workshop).

However, the Report, in taking a very particular approach  narrows the scope of discussion and leads the reader of the Report to a conclusion that is heavily dependent on the assumption that markets solve an epistemological problem.  The fact that there is such a bias towards market insiders on the High Level Stakeholder Group, there is no representation from pension funds but the ISDA, a lobby group for derivatives traders that gave us the infamous Potts opinion, are there,  lays the report open to the accusation that it is stealth advocacy.  Given that the public have been angered by the apparent privatisation of profits and socialisation of losses by banks in the past, appearing to side with the proprietary traders over pension fund managers, seems a very short-sighted approach to take.
I think these observations are compounded by the fact that the Report could have been clearer in distinguishing the various impacts computers are having on markets, rather than a focus on addressing HFT within this context described above.  The Report's Executive Summary opens with

A key message: despite commonly held negative perceptions, the available evidence indicates that high  frequency trading (HFT) and algorithmic trading (AT) may have several beneficial effects on markets.  However, HFT/AT may cause instabilities in financial markets in specific circumstances. This Project has shown that carefully chosen regulatory measures can help to address concerns in the shorter term. 

This distinction is lost in discussing the benefits of Computer Based Trading (the combination of the two) as improved liquidity, reduction in transaction costs and improved market efficiency.  

The risks are observed collapses in liquidity and instability.  When discussing instability the observation is made that HFT does not appear to increase volatility, but there are issues about stability, which the report describes in terms of non-linearities, incomplete information and what the report calls 'normalisation of deviance' but sociologists describe as 'counter-performativity' (markets follow a model and then discover the model is wrong).  

HFT is highlighted in regard to market abuse, which the report argues there is no evidence.  However the distinction between AT, usually conducted for agency trading, and HFT, usually conducted by proprietary traders, in their contribution to the risks is not really developed.  Can we explore this issue?

The report defines 'liquidity' (p 19, n3) as 

the ability to buy or sell an asset without greatly affecting its price. The more liquid the market, the smaller the price impact of sales or purchases

AT has made a significant contribution to this, particularly through the work developing out of Chriss and Almgren's pioneering research in optimally executing large trades.  The report's definition of liquidity is somewhat biased , a different definition associates liquidity with the 'depth' of the market, the ability to actually buy or sell an asset in the market.  HFT will not increase depth, but it may present a mirage of improved liquidity as assets are churned by proprietary traders.  This is associated with the Flash Crash, p 56. 

Liquidity is related to what the Report describes as  (p 19, n4)

price discovery ... the market process whereby new information is impounded into asset prices.

Clearly if trades are executed quickly, price discovery improves.  However there are issues with mechanistic approaches to news processing (Derwent Capital Markets' closure).

The Report notes that there has been a reduction in transaction costs since the introduction of CBT, but this cannot be ascribed to HFT and the FT cites 2012 research that suggests the reduction in costs occurred before HFT emerged.

Overall I think there is a good argument that the Report is far from being "pure science" and lacks the balance of "honest brokerage".