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.