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.