Wednesday 21 September 2011

Who was the first Quant?

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?

Friday 2 September 2011

Maths and the markets

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.