The Physics of Finance

"Unemployment is an illusion and recessions are voluntary"

2012-03-06T09:51:00.000+01:00

... I assume that most of those hearing or reading this speech at all closely are aware of the great divide that emerged in macroeconomics in the 1970s. For those who aren’t familiar with the story: in the 1930s Keynesian economics emerged as a response to depression, and by the 1950s it had come to dominate the field. There was, however, an undercurrent of dissatisfaction with that style of modeling, not so much because it fell short empirically as because it seemed intellectually incomplete. In “normal” economics we assume that prices rise or fall to match supply with demand. In Keynesian macroeconomics, however, one simply assumes that wages and perhaps prices too don’t fall in the face of high unemployment, or at least fall only slowly.

Why make this assumption? Well, because it’s what we see in reality – as confirmed once again by the experience of peripheral European countries, Portugal included, where wage declines have so far been modest even in the face of very high unemployment. But that’s an unsatisfying answer, and it was only natural that economists would try to find some deeper explanation.

The trouble is that finding that deeper explanation is hard. Keynes offered some plausible speculations that were as much sociological and psychological as purely economic – which is not to say that there’s anything wrong with invoking such factors. Modern “New Keynesians” have come up with stories in terms of the cost of changing prices, the desire of many firms to attract quality workers by paying a premium, and more. But one has to admit that it’s all pretty ad hoc; it’s more a matter of offering excuses, or if you prefer, possible rationales, for an empirical observation that we probably wouldn’t have predicted if we didn’t know it was there.

This, understandably, wasn’t satisfying to many economists. So there developed an alternative school of thought, which basically argued that the apparent “stickiness” of wages and prices in the face of unemployment was an optical illusion. Initially the story ran in terms of imperfect information; later it became a story about “real” shocks, in which unemployment was actually voluntary; that was the real business cycle approach.

And so we got the division of macroeconomics. On one side there was “saltwater” economics – people, who in America tended to be in coastal universities, who continued to view Keynes as broadly right, even though they couldn’t offer a rigorous justification for some of their assumptions. On the other side was “freshwater” – people who tended to be in inland US universities, and who went for logically complete models even if they seemed very much at odds with lived experience.

Obviously I don’t believe any of the freshwater stories, and indeed find them wildly implausible. But economists will have different ideas, and it’s OK if some of them are ones I or others dislike.

What’s not OK is what actually happened, which is that freshwater economics became a kind of cult, ignoring and ridiculing any ideas that didn’t fit its paradigm. This started very early; by 1980 Robert Lucas, one of the founders of the school, wrote approvingly of how people would giggle and whisper when facing a Keynesian. What’s remarkable about that is that this was all based on the presumption that freshwater logic would provide a plausible, workable alternative to Keynes – a presumption that was not borne out by anything that had happened in the 1970s. And in fact it never happened: over time, freshwater economics kept failing the test of empirical validity, and responded by downgrading the importance of evidence.

Read the whole thing.

Microfoundations -- fact and fiction

2012-03-05T16:12:00.000+01:00

I generally try not to write about things I know almost nothing about, but here goes. Take everything that follows here as a kind of "thinking out loud" -- a struggle to put into words my thoughts about some apparently odd ideas in macroeconomics. I say "apparently" because I don't know enough to be sure. Maybe they are all very sensible. I would greatly appreciate any further insight from anyone out there who knows.

The idea puzzling me is "microfoundations." As I understand it, the rational expectations revolution in macroeconomics, linked to the names Robert Lucas, Edward Prescott, Thomas Sargent and others, demanded that macroeconomic theories shouldn't just be built as coarse-grained effective theories operating at the macroscale and written in terms of macroscopic variables such as inflation, unemployment, etc. Rather, a good theory henceforth was to link macroeconomic outcomes back to the behaviour of the individual agents in an economy, i.e. to their microeconomics behaviour. Such as theory would have "microfoundations."

To my physicist mind, this seems entirely sensible, so far. A difficult project, no doubt but sensible. By analogy, of course, this just seems like the effort to derive thermodynamics (a macroscopic theory) from the underlying behaviours of individual particles, which is the project of statistical mechanics. Deriving theories at higher levels from behaviours at lower levels is, when possible, a natural scientific project -- it offers unification or, if it can't be carried through, points to problem areas from which new ideas are likely to come.

Now, I have also read that much of the impetus for the rational expectations movement was the famous Lucas Critique which, if I understand it correctly, says that you can't reliably base policy interventions on simple regularities observed in macroeconomic data (a historically observed tradeoff between unemployment and inflation, for example). That regularity existed, after all, in the context of the policies prevailing in the past. Change the policies and those changes, by influencing the way people act and anticipate the future, may well strongly change or destroy the regularity on which you had based your plans. Again, plausible and sensible, it seems to me.

So, I can see the attraction of theories with microfoundations -- theories, that is, giving a plausible account of how macroeconomic reality emerges out of the micro reality and actual behaviour of millions of people and firms in interaction.

Now my puzzlement. As far as can tell, the idea of "microfoundations" as actually used in macroeconomics isn't quite how I described it above, i.e. seeking to base macro theory on a plausible account of the behaviour of individuals. Rather, in economics (through the work of Lucas) it has come to mean theories in which individuals and firms are hyperrational optimizers of their utility over a span of time (they solve a complicated optimization problem over their lifetime). This no longer seems so plausible, and on this point, a commenter from Mark Thoma's blog captures my feelings on this quite clearly:

hrsaccount said...
Microfoundations would be important if there were clear evidence that they represented the truth. For example, if there had been a series of experiments demonstrating that individuals are rational and make decisions so as to maximize some measurable quantity called utility, it would be important that macro models were consistent with this and the most direct way of ensuring that would be to incorporate rational utility-maximizing households into the model.

The fact is that there is no such evidence. Microeconomics is not based on empirical evidence, and the approach used in microeconomics has no special claim to the truth. So, leaving aside the fact that the way macroeconomics uses micro (i.e., in a way that many microeconomists don't approve, ignoring aggregation issues) there's no logical reason why macro needs to even be consistent with micro.

His point seems to me very well put -- if "microfoundations" as currently interpreted don't give foundations to anything, then a theory having them has no advantage. Theories with microfoundations (as interpreted in this odd sense) have no more claim to relevance than anything else. Indeed, we might say they are even worse as they are almost certainly demonstrably inconsistent with real behaviour at the micro level.

Again, I'm not an expert on this. But I see this kind of argument breaking out over and over among economists. I often think I must have it wrong, so please if I do, someone let me know.

UPDATE

While writing the above, I happened to find and read a couple of things that clarified matters quite a bit for me. My take seems to be shared by economists as well, although I'm not sure the few things I read are representative. First, Maarten Janssen of the Tinbergen Institute published an excellent short review of the idea of microfoundations in 2008. He describes the history, but notes that key criticisms of the idea do center in the "plausibility" of the rational expectations approach. That is, including expectations in macromodels is sensible, but everything depends on how you include them:

The approaches discussed so far... all postulate rational behavior on the part of economic agents and some notion of equilibrium. If expectations are important, it is postulated that agents’ expectations concerning important variables coincide with the model’s predicted values concerning these same variables.

And he mentions several branches of research criticizing this view and testing it, in particular, testing whether in a decentralized economy economic agents may learn over time to have expectations that are consistent with those that are assumed by the rational expectations hypothesis:

The general conclusion of this literature is that due to the feedback from expectations to economic behavior, the outcomes of an economic model with learning agents do not converge to the rational expectations solution. It then follows that the microfoundations literature mentioned so far has not really succeeded in deriving all macroeconomic propositions from fundamental hypotheses on the behavior of individual agents. The requirements of methodological individualism have thus not been satisfied by the microfoundations literature that has pre-dominantly presumed that individuals behave rationally...

I cannot say I'm surprised. So we're left with theories that only go one short step toward the idea of microfoundations, and, in my view, can't claim they have given microfoundations to anything -- the use of the word in these models is totally unwarranted, and I think way overstates what they achieve.

I think much the same point of view is expressed by Michael Woodford, himself a big name in macro modelling. In a response to an essay by John Kay critical of modern macroeconomics and its unrealistic assumptions, Woodford in a roundabout way eventually says, well, yes, I agree:

It has been standard for at least the past three decades to use models in which not only does the model give a complete description of a hypothetical world, and not only is this description one in which outcomes follow from rational behavior on the part of the decision makers in the model, but the decision makers in the model are assumed to understand the world in exactly the way it is represented in the model. More precisely, in making predictions about the consequences of their actions (a necessary component of an accounting for their behavior in terms of rational choice), they are assumed to make exactly the predictions that the model implies are correct (conditional on the information available to them in their personal situation).
This postulate of “rational expectations,” as it is commonly though rather misleadingly known, is the crucial theoretical assumption behind such doctrines as “efficient markets” in asset pricing theory and “Ricardian equivalence” in macroeconomics. It is often presented as if it were a simple consequence of an aspiration to internal consistency in one’s model and/or explanation of people’s choices in terms of individual rationality, but in fact it is not a necessary implication of these methodological commitments. It does not follow from the fact that one believes in the validity of one’s own model and that one believes that people can be assumed to make rational choices that they must be assumed to make the choices that would be seen to be correct by someone who (like the economist) believes in the validity of the predictions of that model. Still less would it follow, if the economist herself accepts the necessity of entertaining the possibility of a variety of possible models, that the only models that she should consider are ones in each of which everyone in the economy is assumed to understand the correctness of that particular model, rather than entertaining beliefs that might (for example) be consistent with one of the other models in the set that she herself regards as possibly correct.

So I feel that my suspicions and objections aren't misplaced, despite my vast ignorance. One other excellent article I recommend is this one from 2011 in which Woodford details the history of modern macroeconomics over the past century. Nothing I've read has given such a complete and clearly explained exposition, while it seems being balanced along the way (or so it seems, to my physicist's eyes).

How markets become efficient (answer: they don't)

2012-02-15T15:22:00.003+01:00

A staggering amount of effort has been spent -- and wasted -- exploring the idea of market efficiency. The notoriously malleable efficient markets hypothesis (EMH) claims (in its weakest form) that markets are "information efficient" -- market movements are unpredictable because smart investors keep them that way. They should quickly -- even, "instantaneously" in some statements -- pounce on any predictable pattern in the market, and by profiting will act to wipe out that pattern.

I've written too many times (here, here, here, for example) about the masses of evidence against this idea. It's not that predictable patterns don't attract investors who often act in ways that tend to wipe out those patterns through arbitrage. Part of the problem is that investors often act in ways that amplify the pattern (following trends, for example). Moreover, there are fundamental limits to arbitrage -- "the markets can stay irrational longer than you can stay solvent." Still, the EMH stumbles onward like a zombie -- dead, proven incorrect and misleading, yet still taking center place in the way many people think of markets.

I found an illuminating new perspective on the matter in this recent paper by Doyne Farmer and Spyros Skouras, which explore analogies between finance and ecology. This analogy is itself deeply suggestive. They note, for example, how the interactions between hedge funds can be useful viewed in ecological terms -- funds sometimes act as direct competitors (the profits of one reducing opportunities for another), and in other cases as predator and prey or as symbiotic partners. But I want to look specifically at an effort they make to give a rough estimate of the timescale over which the actions of sophisticated arbitragers might reasonably be expected to wipe out a new predictable pattern in the market. That is, if the market for whatever reason is temporarily inefficient -- showing a predictable pattern -- how quickly should it be returned to efficiency? How long is the time to relaxation that the EMH claims is "instantaneous" or close to it?

The gist of their idea is very simple. Before you can exploit a predictable pattern, you first have to identify it. If you're going to invest money trading against it, you need to be fairly sure you've identified a real pattern, not just a statistical fluke. If you're going to invest somebody else's money, you have to convince them. This takes some time. The stronger the pattern, the more it stands out and the less time it should take to be sure. Weaker signals will be hidden by more noise, and reliable identification will take longer. Looking at how much time it should take to get good statistics should give an order of magnitude of how long a pattern should persist before any smart investor can begin reliably trading against it and (perhaps) erasing it.

Here's the specific argument, expressed using the Sharpe ratio (ratio of expected return to standard deviation of a strategy exploiting the pattern):

This makes obvious intuitive sense. If S is very large, making the pattern obvious, more deterministic and easier to exploit, then the time over which it might be expected to vanish is smaller. Truly obvious patterns can be expected to vanish quickly. But if S is small, the timescale for identification and exploitation grows.

As Farmer and Skouras note, successful investment strategies often have Sharpe ratios of about S = 1, so this gives a result of about 10 years. [This is the result if one makes the analysis on an annual timescale, with the Sharpe ratio calculated on a yearly basis. If we're talking about fast algorithmic trading, then the analysis takes place on a shorter timescale.]

So, 10 years is the order of magnitude estimate -- which is a rather peculiar interpretation of the word "instantaneous." Perhaps that word should be replaced in the EMH with "very slowly," although that somewhat dampens the appeal of the idea: "The EMH asserts that sophisticated investors will very slowly identify and exploit any inefficiencies in the market, tending the erase those inefficiencies over a few decades or so." Given that new inefficiencies can be expected the arise in the mean time, you might as well call this more plausible hypothesis the PIMH: the perpetually inefficient markets hypothesis.

And their estimate, Farmer and Skouras point out, is actually optimistic:

We should stress that this estimate is based on idealized assumptions, such as log-normal returns – heavy tails, autocorrelations, and other effects will tend to make the timescale even longer.... As a given inefficiency is exploited, it will become weaker and the Sharpe ratio of investment strategies associated with it drops. As the Sharpe ratio becomes smaller the fluctuations in its returns become bigger, which can generate uncertainty about whether or not the strategy is still viable. This slows down the approach to inefficiency even more.

Of course, as I mentioned above, this analysis depends on timescale. Take t in years and we're thinking about predictable patterns emerging on the usual investment horizon of a year or longer, patterns exploited by hedge funds and mutual funds of the more traditional (not high frequency) kind. Here we see that the time to expect predictable patterns to be wiped out is very long indeed. If 10 years is the order of magnitude, then it's likely some of these patterns persist for several decades -- getting up to the time of a typical investing career. Hardcore supporters of the EMH should learn to speak more honestly: "We have every reason to expect that predictable market inefficiencies should be wiped out fairly quickly, at least on the timescale of a human investment career."

All in all, this way of estimating the time for relaxation back to the "efficient equilibrium" suggests that the relaxation is anything but fast, and often very slow. The EMH may be right that there likely aren't any obvious patterns, but more subtle predictable patterns will likely persist for long periods of time, even while they present real profit opportunities. The market is not in equilibrium. And with no mechanism to prevent them, new predictable patterns and "inefficiencies" should be emerging all the time.

What's going on in Greece

2012-02-14T17:31:00.000+01:00

Numerian, one of the most insightful financial bloggers I know, gets to the bottom of the markets' surge in the wake of the Greek passage of the new "austerity" measures:

For now, stock markets are happy because they get their periodic injection of heroin – oops, make that “liquidity” – to keep the game going. The game is the one we have all lived through our whole lives – the one where capitalism continues to grow by taking on more and more debt, until now every country is at the point where only the government is big enough to take on the enormous amounts of new debt necessary to keep paying principal and interest on the old debt. At least some countries are: the United States, the UK, Germany, France. Greece of course lost that privilege several years ago, and now even big borrowers like Italy are allowed into the markets only for very short term maturities.

The game, in short, is about over, choking to death on too much debt, kept on a resuscitator by politicians and central bankers who know the public has no way to stop them from raising trillions of dollars or euros with new bond issues. Only the market can stop an out-of-control debtor. Greece has found that out, Italy and Spain and Portugal are close to finding that out, and the UK and the United States are on the list, being no more virtuous than the Greeks. Once the government can no longer borrow, default in some form is inevitable, and austerity follows. The Greeks have austerity handed to them by the Germans; everyone else will be able to choose their own forms of austerity, as different economic and social forces fight with each other in a country that has run out of borrowing capacity and must live off the taxes it is able to raise.

If the stock market had a long term view, it would think about these things. It would look at Greece as a combination horror story and warning sign. Instead, the stock market lives for the day only, and for now the debt binge continues, and the fix of easy credit is being pumped into the financial system once more. Let the celebrations continue.

Approaching the singularity -- in global finance

2012-02-13T16:47:00.000+01:00

In a new paper on trends in high-frequency trading, Neil Johnson and colleagues note that:

... a new dedicated transatlantic cable is being built just to shave 5 milliseconds off transatlantic communication times between US and UK traders, while a new purpose-built chip iX-eCute is being launched which prepares trades in 740 nanoseconds ...

This just illustrates the technological arms race underway as firms try to out-compete each other to gain an edge through speed. None of the players in this market worries too much about what this arms race might mean for the longer term systemic stability of market; it's just race ahead and hope for the best. I've written before (here and here) about some analyses (notably from Andrew Haldane of the Bank of England) suggesting that this race is generally increasing market volatility and will likely lead to disaster in one form or another.

We may be getting close. If Johnson and his colleagues are correct, the markets are already showing signs of having already made a transition into a machine-dominated phase in which humans have little control.

Most readers here will probably know about futurist Ray Kurzweil's prediction of the approaching "singularity" -- the idea that as our technology becomes increasingly intelligent it will at some point create self-sustaining positive feedback loops that drive explosively faster science and development leading to a kind of super-intelligence residing in machines. Humans will be out of the loop and left behind. Given that so much of the future vision of computing now centers on bio-inspired computing -- computers that operate more along the lines of living organisms, being able do things like self-repair and adaptation, true reproduction, etc, -- it's easy to imagine this super-intelligence again ultimately being strongly biological in form, while also exploiting technologies that earlier evolution was unable to harness (superconductivity, quantum computing, etc.). In that case -- again, if you believe this conjecture has some merit -- it could turn out ironically that all of our computing technology will act as a kind of mid-wife aiding a transition from Homo sapiens to some future non-human but super-intelligent species.

But forget that. Think singularity, but in the smaller world of the markets. Johnson and his colleagues ask the question of whether today's high-frequency markets are moving toward a boundary of speed where human intervention and control is effectively impossible:

The downside of society’s continuing drive toward larger, faster, and more interconnected socio-technical systems such as global financial markets, is that future catastrophes may be less easy to forsee and manage -- as witnessed by the recent emergence of financial flash-crashes. In traditional human-machine systems, real-time human intervention may be possible if the undesired changes occur within typical human reaction times. However,... in many areas of human activity, the quickest that someone can notice such a cue and physically react, is approximately 1000 milliseconds (1 second)

Obviously, most trading now happens much faster than this. Is this worrying? With the authors, let's look at the data.

In the period from 2006-2011, they found (looking at many stocks on multiple exchanges) that there were about 18,500 specific episodes in which markets, in less than 1.5 seconds, either 1. ticked down at least 10 times in a row, dropping by more than 0.8% or 2. ticked up at least 10 times in a row, rising by more than 0.8%. The figure below shows two typical events, a crash and a spike (upward), both lasting only 25 ms.

Apparently, these very brief and momentary downward crashes or upward spikes -- the authors refer to them as "fractures" or "Black Swan events" -- are about equally likely. And they become more likely as one goes to shorter time intervals:

... our data set shows a far greater tendency for these financial fractures to occur, within a given duration time-window, as we move to smaller timescales, e.g. 100-200ms has approximately ten times more than 900-1000ms.

But they also find something much more significant. They studied the distribution of these events by size, and considered if this distribution changes when looking at events taking place on different timescales. The data suggests that it does. For times above about 0.8 seconds or so, the distribution closely fits a power law, in agreement with countless other studies of market returns on times of one second or longer. For times shorter than about 0.8 seconds, the distribution begins to depart from the power law form. (It's NOT that it becomes more Gaussian, but it does become something else that is not a power law.) The conclusion is that something significant happens in the market when we reach times going below 1 second -- roughly the timescale of human action.

Ok. Now for the punchline -- an effort to understand how this transition might happen. In my last blog post I wrote about the Minority Game -- a simple model of a market in which adaptive agents attempt to profit by using a variety of different strategies. It reproduces the realistic statistics of real markets, despite its simplicity. I expect that some people may wonder if this model can really be useful in exploring real markets. If so, this new work by Johnson and colleagues offers a powerful example of how valuable the minority game can be in action.

Their hypothesis is that the observed transition in market dynamics below one second reflects "a new fundamental transition from a mixed phase of humans and machines, in which humans have time to assess information and act, to an ultrafast all-machine phase in which machines dictate price changes." They explore this in a model that...

...considers an ecology of N heterogenous agents (machines and/or humans) who repeatedly compete to win in a competition for limited resources. Each agent possesses s > 1 strategies. An agent only participates if it has a strategy that has performed sufficiently well in the recent past. It uses its best strategy at a given timestep. The agents sit watching a common source of information, e.g. recent price movements encoded as a bit-string of length M, and act on potentially profitable patterns they observe.

This is just the minority game as I described it a few days ago. One of the truly significant lessons emerging from its study is that we should expect markets to have two fundamentally distinct phases of dynamics depending on the parameter α=P/N, where P is the number of different past histories the agents can perceive, and N is the number of agents in the game. [P=2^M if the agents use bit strings of length M in forming their strategies]. If α is small, then there are lots of players relative to the number of different market histories they can perceive. If α is big, then there are many different possible histories relative to only a few people. These two extremes lead to very different market behaviour.

Johnson and colleagues suggests that the transition between these regimes is just what shows up in the statistics around the one second threshold. They first argue that the regime for large α (many strategies per agent) should be associated with the trading regime above one second, where both people and machines take part. Why? As they suggest,

We associate this regime (see Fig. 3) with a market in which both humans and machines are dictating prices, and hence timescales above the transition (>1s), for these reasons: The presence of humans actively trading -- and hence their ‘free will’ together with the myriad ways in which they can manually override algorithms -- means that the effective number (i.e. α > 1). Moreover α > 1 implies m is large, hence there are more pieces of information available which suggests longer timescales... in this α > 1 regime, the average number of agents per strategy is less than 1, hence any crowding effects due to agents coincidentally using the same strategy will be small. This lack of crowding leads our model to predict that any large price movements arising for α > 1 will be rare and take place over a longer duration – exactly as observed in our data for timescales above 1000ms. Indeed, our model’s price output (e.g. Fig. 3, right-hand panel) reproduces the stylized facts associated with financial markets over longer timescales, including a power-law distribution.

What they're getting at here is that crowding in the space of strategies, by creating strong correlations in the strategies of different agents, should tend to make large market movements more likely. After all, if lots of agents come to use the very same strategy, they will all trade the same way at the same time. In this regime above one second, with humans and machine, they suggests there shouldn't be much crowding; the dynamics here do give a power law distribution of movements, but it is what is found in all markets in this regime.

In contrast, they suggest that the sub one second regime should be associated the the α < 1 phase of the minority game:

Our association of the α < 1 regime with an all-machine phase is consistent with the fact that trading algorithms in the sub-second regime need to be executable extremely quickly and hence be relatively simple, without calling on much memory concerning past information: α < 1 regime with an all-machine phase is consistent with the fact that trading algorithms in the sub-second regime need to be executable extremely quickly and hence be relatively simple, without calling on much memory concerning past information: Hence M will be small, so the total number of strategies will be small and therefore... α < 1. Our model also predicts that the size distribution for the black swans in this ultrafast regime (α < 1) should not have a power law since changes of all sizes do not appear – this is again consistent with the results in Fig. 2.

And...

Our model undergoes a transition around α = 1 to a regime characterized by significant strategy crowding and hence large fluctuations. The price output for α < 1 (Fig. 3, left-hand panel) shows frequent abrupt changes due to agents moving as unintentional groups into particular strategies. Our model therefore predicts a rapidly increasing number of ultrafast black swan events as we move to smaller α and hence smaller subsecond timescales – as observed in our data.

The authors go on to quantify this transition in a little more detail. In particular, they calculate in the simple minority game model the standard deviation of the price fluctuations. In the regime α < 1 this turns out to be roughly proportional to the number N of agents in the market. In contrast, it goes in proportion only to the square root of N in the α > 1 regime. Hence, the model predicts a sharp increase in the size of market fluctuations when entering the machine dominated phase below one second.

The paper as a whole takes a bit of time to get your head around, but it is, I think, a beautiful example of how a simple model that explores some of the rich dynamics of how strategies interact in a market can give rise to some deep insights. The analysis suggests, first, that the high frequency markets have moved past "the singularity," their dynamics having become fundamentally different -- uncoupled from the control, or at least strong influence, of human trading. It also suggests, second, that the change in dynamics derives directly from the crowding of strategies that operate on very short timescales, this crowding caused by the need for relative simplicity in these strategies.

This kind of analysis really should have some bearing on the consideration of potential new regulations on HFT. But that's another big topic. Quite aside from practical matters, the paper shows how valuable perspectives and toy models like the minority game might be.

minority games

2012-02-06T16:50:00.001+01:00

I have an essay just published in Bloomberg. The essay is quite short and I wanted to give interested readers a few more details and links to further reading on the subject: how to build simple yet plausible models of financial markets. Hence, this post.

Traditional models of markets in economics work from the idea of equilibrium. What happens in the market is assumed to reflect the interplay of two things: 1. the decisions of countless market participants who act more or less rationally in their best interests, their actions collectively bringing the market toward a balance in which stocks, bonds, or other instruments take their realistic "fundamental" values (or at least something close to those values), and 2. external shocks that hit the market in the form of new pieces of information concerning businesses, political events, changes in regulations, technological discoveries and so on. Without shocks from outside, these models imply, the market would settle down (very rapidly it is assumed) into an unchanging state. Everything that happens in the market, this perspective asserts, can be traced to new information perturbing that balance.

I've written before (here, for example, and here) about the considerable evidence against this picture. This includes many large market movements that occur in the absence of any apparent "shock" to the market, and "excess volatility" -- a general tendency for market prices to move more than they should by any rational reckoning of their true value. Many studies have established strong mathematical regularities in market fluctuations that reflect this excess volatility, including the fat tailed statistics of market returns and the long memory of returns -- the way markets absorb information, empirically, doesn't seem to be fast at all, but involves a long slow process of digestion stretching over a decade and more.

So, how to build models that go beyond this equilibrium picture? If the simple notion of equilibrium is too simple to explain market reality, why not non-equilibrium models? That is, models of markets in which the individuals' actions, even in the absence of outside shocks, never lead the market to settle down into some equilibrium balance. Rather, people might instead perpetually change their ideas and strategies, interacting in a way that gives the market a rich internal dynamics, much as the weather. Studies of very simple two player strategic games actually show that this kind of outcome should be expected in any case in which the decisions facing individuals are quite complex and not open to simple rational solution -- that is, in other words, in most cases (and obviously in the case of any real world market).

Science often makes progress by posing toy models that capture the logical essence of some problem. By struggling with it, people learn new ways of thinking. In my Bloomberg column I touched on one of the toy models that has recently inspired non-equilibrium models of markets. This is the so-called El Farol Bar problem invented by economist Brian Arthur in1994. The original paper is still well worth a read; it's one of those papers that puts forth an idea so sound and plausible that it seems obvious, and it is hard to believe that its ideas were revolutionary and against everything in mainstream economic habit. To this day, this remains true. But enough background.

My Bloomberg column gives a rough introduction to the idea of the bar problem -- it is an archetype of a situation in which it is not possible for people to make decisions through rational deliberation, but rather have to take a more practical approach, forming hypotheses and theories, and learning from experience. The result is a dis-equilibrium situation that continues to evolve in time, never settling into an equilibrium. There are plenty of articles giving more detailed synopses of the model and how it works.

But the essence of the bar model has since then been explored in much greater detail through another model -- the so-called minority game. It's just the bar model stripped down to be as simple as possible while still preserving the essential element that no one in the game can "figure it out." There is no strictly rational way to play; intelligent play requires perpetual adaptation and learning. The minority game is probably the simplest mathematical model of a market in this sense. It is, of course, too simple to be anything more than a toy model. But it is, at least, a toy model looks a lot more like real markets than does the EMH or anything else in economics. And the benefit of its relative simplicity is that it is possible to see how various fundamental factors influence how it works, without thousands of other potential complications that obscure matters.

So --what is the minority game?

Arthur assumed that people would enjoy the bar is 60% of less attended, and would be miserable if more than 60% attended. The minority game simplifies the cutoff to 50%, giving symmetry to the problem -- now the people who do well on any week are those who act so as to be in the minority (going when most others stay home, and vice versa). The minority game also dispenses with the story of the bar. Just let each N agents on each play choose either 1 or -1, with those in the minority gaining some payoff (let N be odd). Now let them play many times. Each agent just goes along trying each time to be in the minority. How does one do that?

Of course, everything hinges on how we suppose the agents play. Following Arthur, the typical approach is to suppose that they use some kind of trial and error learning. The record of outcomes of the game is given by some sequence of 1s and -1s -- these giving the winning choices that would have been in the minority in each instance. For example, the last 5 outcomes might be 1,1,-1,-1,1. A "strategy" in the minority game associates every such recent history with a choice of what to do next -- a choice of 1 or -1. A strategy may say that IF (1,1,-1,-1,1) THEN choose 1, or instead IF (1,1,-1,-1,1) THEN choose -1. If we suppose that agents look back to the past M outcomes in trying to forecast the future (this is often referred to as their "brain size"), then there are P=2^M different histories that the agents can resolve or discern. A strategy, mathematically speaking, is a map or association that assigns a choice 1 or -1 to each and every one of the possible P=2^M different histories. A strategy is a plan for action, if you will, an IF...THEN statement of what to do for every possible sequence of happenings (at the level the agents can perceive).

Now, since each possible sequence can be associated with 1 or -1, there are actually 2^P or 2^{2^M} different possible strategies. Even with M as low as 5, this is already 2³² which is well over one trillion different possible strategies. It grows extremely fast as M gets bigger.

The way the minority game is played is that, at the outset, each one of the players gets assigned a number S of these strategies at random. You can think of this as their intellectual inheritance. It's a collection of S different "ways to think" about how to predict the world. (The interesting properties of the model do not depend on this choice.) Agents use these strategies to play the game, and keep track of which strategies seem to work well, and which ones don't. This is generally done through some simple algorithm wherein each strategy in an agent's bag gains a point when it makes a good prediction, and gets docked a point if it predicts incorrectly. On each play, the agents look into their bag and play the strategy with the highest score. This is like a person mulling over the various theories in their head, and using the one that seems to have been most successful in the past.

So that's it. You have a collection of agents using simple learning dynamics to try to predict the future based on the past. They try at each moment to choose what most others will not choose. Clearly no one strategy can emerge as the winner, because if it did and everyone started using it then they would all be in the majority, and all be losers. Successful strategies, therefore, sow the seeds of their own demise. This is what makes the game interesting. But wait -- how is this linked to markets?

Qualitatively, the minority game is a little like a market. It's a situation in which each person tries to profit by choosing the right strategy, but where what is "right" is determined by the collective actions of everyone together. There is no "right" independent of what others do. The minority game, like any market, is a game involving an enormous and complex ecology of interacting strategies. Except there is no price. A minor addition can change that, however.

We get the simplest crude model of a market if we think of those choosing 1 as "buyers" and those choosing -1 as "sellers" of some stock. Naturally, more buyers than sellers drives a price up, and an imbalance the other way drives it down. Mathematically, you can take the change in the logarithm of the price (the return) at time t as being proportional to Number(buyers) - Number(sellers). The resulting price has a highly erratic behaviour that very roughly resembles the movements of financial markets. In particular, even without any external information hitting the market, no shocks from outside or perturbations of any kind, the price fluctuates up and down unpredictably merely through the perpetual evolution of the agents' trading strategies. One day, apparently caused by nothing, the price may suddenly drop by 10% -- all because many agents chose to sell just as a part of trying to predict what was about to happen next.

This post has already gone on long enough. For lots of detail, I highly recommend this excellent review of the minority game by Tobias Galla, Giancarlo Mosetti and Yi-Cheng Zhang, which I highly recommend (Zhang is one of the co-inventors of the minority game). There is a great deal more to say, but two points are most important for now:

1. This simplest version of a minority-game market does NOT give rise to the so-called "stylized facts" of real markets -- the fat tailed statistics of return distributions, and the long memory and persistence of market volatility. In fact, the returns for this simplest minority game market are Gaussian, at least when the number of players is large. However, this model is only a beginning, and these stylized facts emerge quite naturally as soon as one includes a few more realistic features. For example, in this version of the game, every agent must either choose 1 (buy) or -1 (sell) at each moment. Hence, the volume of trading is always the same; it is forced to be fixed, which is of course completely unnatural.

An easy way to relax this is to give the agents the ability also to abstain -- they might just hold what they have, neither buying or selling. This can be included with some measure of confidence, whereby an agent doesn't trade unless one of its strategies has been very successful in the recent past. This immediately leads to a market with fluctuations in volume, and this market does turn out to have stylized facts much like those of real markets, with fat tailed returns and long memory. Another way to allow for a fluctuating volume is to let the agents accrue wealth over time, and let the volume of their trading grow in proportion to that wealth. Again, this is an obvious step toward reality, and again leads to market fluctuations with many of the rich statistical features seen in real markets, even in such simple models.

2. The other important point is that the minority game is so simple that it can be solved with some analytical techniques borrowed from physics. The details aren't so important, but the results show a fascinating behaviour in the market with a kind of phase transition separating two distinct regimes of behaviour. The picture that emerges gives a beautiful coarse grain understanding of markets as ecologies of interacting agents.

The basic story is that one key parameter in the minority game determines its overall character. The parameter is α=P/N, where N is the number of agents in the game. If this is small, then there are lots of players relative to the number of different market histories they can perceive. If this is big, then there are many different possible histories relative to only a few people. These two extremes lead to very different market behaviour and it's not hard to see why.

The figure below (from the Galla et al paper) shows what is effectively the market predictability as a function of this parameter α=P/N. In other words, this shows how much a past record of prices can be used to make accurate predictions of future prices. For large α, this predictability is positive, and it grows as α grows. Why? In principle, this kind of predictability ought to make for easy success in the game. But only if the agents' have enough flexibility in their behaviour to pounce on the pattern and exploit it. In this regime, it happens frequently that patterns emerge in the prices and yet no agent has the intellectual capacity to exploit it. Recall that each individual is given a random set of strategies to work with. With very few people in the market, that means very few strategies relative to the many possible market histories on which the agents make their decisions. In short, if the diversity of agents and their behavioural strategies is insufficient to let them jump on all possible patterns, the market retains a degree of predictability.

For the opposite limit, α small, things look very different. Now we have an oversupply of agents and strategies in play relative to the number of discernible histories. Now it is very likely that any predictable pattern will suit some agent's bag of strategies, and that agent will jump on it, profit, and their activity will tend to wipe out that pattern. This phase looks rather like the efficient market -- such a rich diversity of agents with different skills and mindsets that any predictable patterns are immediately washed away.

One of the most interesting conjectures emerging from this picture is that real markets are probably something like "marginally efficient", at least much of the time. If fully efficient and hence unpredictable (many agents in the market, low α), then it's very hard to make profits in the market. This should make the market less attractive and some people should leave to do something else. But as the number of people falls, this pushed the market toward higher α and into the inefficient phase in which the market becomes predictable. This should in turn attract more people into the market to profit from easy patterns. In this simple picture, the market should tend to evolved toward a value of α where the predictability of prices is low but not zero, making it hard but not impossible to find patterns in the market. This is, judging in a loose non-scientific way, just about where markets often seem to be.

Works on the minority game now number on the thousands, and I only wanted to get across the most basic points. It's a beautiful simple model that captures a great deal of the qualitative character of markets, and with further changes can be taken step by step toward more sophisticated models of markets. There's no reason to stick to idea that payoff go to those in the minority. The game has generalized, for example, to majority type games (where trend following pays off) or mixed majority-minority games (where market reversion and trend following both play a role). A few steps further and you can forget the game itself and simply model markets as systems in which people interact with various trading strategies, each buying and selling, and where prices move through their collective action. Lots of models study a simple mixture of fundamentalists, who try to trade on the base of real information about stocks, and trend followers who ride the momentum. Computing power and imagination present the only limits.

I'm going to explore some of these more sophisticated models in future, but the minority game remains extremely important precisely because of its simplicity. That simplicity allows full analysis leading to the picture of two phases shown above. But this is already considerably more complex than market models based on assumed fully rational behaviour. It's crude, very much so, but a small step in the right direction.

Markets -- increasingly complex dynamics over the past decade

2012-01-24T19:55:00.000+01:00

Didier Sornette is among the most creative scientists I know, and always seems to come up with an approach to problems that is more or less orthogonal to what anyone has done before. In a paper just out (as a preprint), he and Vladimir Filimonov offer a really novel analysis on the old question about whether market movements are caused by A. external influences such as news (exogenous causes) or B. influences internal to the market itself such as emotions, avalanches of belief and opinion, etc. (endogenous causes). This matter, of course, touches directly on the infamous efficient markets hypothesis, which insists on interpretation A (all A, no B).

I've written before (here and here, for example) about various studies trying to match up news feeds with big market moves to see if the latter can be explained by the former. Generally, the evidence suggests no, implying some mixture of A and B. Sornette and Filimonov now take a very different approach, which is an attempt to use mathematics to directly measure how much of the dynamics of a time series can be attributed to endogenous, internal causes. The mathematical technique is itself interesting. If it can be trusted, then the results suggest that markets in the past decade have become much more strongly driven by internal, endogenous dynamics than they were before. As the authors point out, this could well reflect the explosion of algorithmic trading, as computers interact with one another in lots of complex feedback loops.

The authors envision their technique as a device for measuring the amount of "reflexivity" in the market, referring to the term used by George Soros to describe how human perceptions and misperceptions interact in the market to drive changes. This is a fascinating idea if it can be done. Here's how it works. Sornette and Filiminov model price time series as being generated by a statistical "point process" -- the idea is to generate price dynamics by modelling the arrival of actual buy and sell orders in the market. The simplest way to do this is to use a Poisson process, with equal probability at all times. This gives a random time series of price movements, but an unrealistic one that lacks the most interesting properties of real markets -- fat tails in the distribution of returns, and long term memory in the volatility (and also volume fluctuations). To get realistic time series, it's possible to let the arrival of buy and sell orders have strong correlations in time, as they in fact do in real markets. This technique is referred to as a "self-excited Hawkes model."

Another way to put this is as follows. In an ordinary Poisson process, the average number of events striking in an interval dt (say, 1 second) is a constant, λ. In the richer process with correlations, this will now be a function of time λ(t). The key to the analysis here is expressing this quantity (essentially, the rate of buy and sell orders hitting to book around time t) as the sum of two very different processes -- 1. a background contribution due to external events such as news, which drive the market, and 2. a feedback contribution coming from the tendency for orders now to have consequences, leading to further orders in the future. The result is eq.(1) of the paper:

Here the first term on the right is the background (which drives the exogenous dynamics) and the second term is the feedback, with h being some function that reflects the likelihood that an event at time t_igenerates another one at time t later. The first term creates a steady stream of events, the second one creates events which create events which create events, a branching stream of further consequences.

Now, the task of fitting time series generated by such processes to real financial data is more involved and relies on some standard maximum likelihood techniques. The authors also assume for simplicity that the function h has an exponential form (events tend to cause others soon after, and less so with increasing time). The key parameter emerging out of such fits is n, which can be interpreted as the fraction of events of endogenous origin, or in effect, the fraction of market activity due to internal dynamics. The statistical fit also estimates μ, this being the background level of exogenous shocks, which also rises and falls with time. Sornette and Filimonov use data on E-mini futures on a second by second basis over about 12 years to run the analysis, the key results of which come out in the figure below.

The four parts going downward show volume and price, and then the estimated background and the fraction of events caused by internal dynamics, n. The most interesting feature is the general rise in n over the decade showing an increasing influence of internal dynamics, or events which causes further events through internal market mechanisms. In contrast, the background of exogenous shocks -- information driven dynamics -- remains more constant (except with a spike around the time of the Lehman Bros collapse). From this the authors offer a few comments:

The first important observation is that, since 2002, n has been consistently above 0.6 and, since 2007, between 0.7 and 0.8 with spikes at 0.9. These values translate directly into the conclusion that, since 2007, more than 70% of the price moves are endogenous in nature, i.e., are not due to exogenous news but result from positive feedbacks from past price moves. The second remarkable fact is the existence of four market regimes over the period 1998-2010:
(i) In the period from Q1-1998 to Q2-2000, the final run-up of the dotcom bubble is associated with a stationary branching ratio n fluctuation around 0.3.
(ii) From Q3-2000 to Q3-2002, n increases from 0.3 to 0.6. This regime corresponds to the succession of rallies and panics that characterized the aftermath of the burst of the dot-com bubble and an economic recession.
(iii) From Q4-2002 to Q4-2006, one can observe a slow increase of n from 0.6 to 0.7. This period corresponds to the “glorious years” of the twin real-estate bubble, financial product CDO and CDS bubbles, stock market bubble and commodity bubbles.
(iv) After Q1-2007 the branching ratio stabilized between 0.7 and 0.8 corresponding to the start of the problems of the subprime financial crisis (first alert in Feb. 2007), whose aftershocks are still resonating at the time of writing.

It should be emphasized that the analysis here is only sensitive to endogenous dynamics over timescales of only around 10 minutes or less. This stems from some assumptions necessary to deal with the highly non-stationary character of the data, as trading volume has exploded over the decade. Hence, the lower values of n earlier in the decade could reflect the failure of this analysis to detect important endogenous feedbacks operating on longer timescales (in the burst of the dot-com bubble, for example).

Now for what is perhaps the most fascinating thing coming out of this paper -- the idea that this analysis may be able to distinguish big markets movements caused by real news or other fundamental changes from those more akin to bubbles and caused purely by human behaviour (panics and the like) or algorithmic feedbacks. Sornette and Filimonov looked at two specific events, on 27 April and 6 May 2010, where markets moved suddenly and in a dramatic way. The first was caused by S&P downgrading Greece's debt rating, the second is, of course, the Flash Crash. The same analysis using their method shows strikingly different results for these two events:

The "branching ratio" here is just the n we've been talking about -- the fraction of market dynamics caused by internal dynamics. The most significant finding is that while the first event of 27 April showed absolutely no change in this value, expected given the apparently clear origin of this event in external information, the 6 May Flash Crash shows a sudden spike in the internal dynamics. Hence, based on this example, it appears that this parameter n acts like a flag, identifying events caused by powerful internal feedbacks. As the authors put it,

The top four panels of Fig. 3 show that the two extreme events of April 27 and May 6, 2010 have similar price drops and volume of transactions. In particular, we find that the volume was multiplied by 4.7 for April 27, 2010 and by 5.3 times for May 6, 2010 in comparison with the 95% quantile of the previous days’ volume. The main difference lies in the trading rates and in the branching ratio. Indeed, the event of April 27, 2010 can be classified according to our calibration of the Hawkes model as a pure exogenous event, since the branching ratio n (fig. 3D1) does not exhibit any statistically significant change compared with previous and later periods. In contrast, for the May 6, 2010 flash crash, one can observe a statistically significant increase of the level of endogeneity n (fig. 3D2). At the peak, n reaches 95% from a previous average level of 72%, which means that, at the peak (14:45 EST), more than 95% of the trading was due to endogenous triggering effects rather than genuine news.

Do you believe it? It sounds plausible to me. I guess the thing that would be good to see is some thorough tests of the method applied to time series for which we know the origin of the dynamics. That is, take something like a chaotic oscillator and drive it with some external noise with a controllable level and see if this method generally gives reliable results in teasing out how much of what happens is driven by the noise, and how much by the internal dynamics. Perhaps this has already been done in some of the papers describing the development of the self-excited Hawkes model. I'll try to check on this.

In any event, I think this is certainly a provocative and interesting new approach to this old question of internal vs external dynamics in markets. And I actually don't find the result surprising at all that n has increased markedly over the past decade. This is precisely what one would expect as trading moves over to algorithms that react to what other algorithms do on a sub second basis.

Corporate psychopaths

2012-01-19T15:18:00.000+01:00

Unmissable by way of Money Science, how the financial crisis emerged from an industry of business and finance that actively recruits social psychopaths due to their superior skills in morally unconstrained decision making. The original article is an essay by my fellow Bloomberg columnist William Cohan:

It took a relatively obscure former British academic to propagate a theory of the financial crisis that would confirm what many people suspected all along: The “corporate psychopaths” at the helm of our financial institutions are to blame.

Clive R. Boddy, most recently a professor at the Nottingham Business School at Nottingham Trent University, says psychopaths are the 1 percent of “people who, perhaps due to physical factors to do with abnormal brain connectivity and chemistry” lack a “conscience, have few emotions and display an inability to have any feelings, sympathy or empathy for other people.”

As a result, Boddy argues in a recent issue of the Journal of Business Ethics, such people are “extraordinarily cold, much more calculating and ruthless towards others than most people are and therefore a menace to the companies they work for and to society.”

Money science also links to this article by Brian Basham, also describing Boddy's work. Listen to this:

Cut to a pleasantly warm evening in Bahrain. My companion, a senior UK investment banker and I, are discussing the most successful banking types we know and what makes them tick. I argue that they often conform to the characteristics displayed by social psychopaths. To my surprise, my friend agrees.

He then makes an astonishing confession: "At one major investment bank for which I worked, we used psychometric testing to recruit social psychopaths because their characteristics exactly suited them to senior corporate finance roles."

Here was one of the biggest investment banks in the world seeking psychopaths as recruits.

The Money science article goes on to give a surprisingly long list of recent books and papers which have explored the role of psychopathology in finance and other corporations. All heart warming stuff, further evidence, I guess, that we really do live in highly civilized and well organized societies.

Natural models of markets

2012-01-17T15:16:00.000+01:00

I've written quite a bit about the shortcomings of traditional economic models of markets. Most notably, such models -- typically characterized by rational agents the actions of whom lead (by assumption) to a market equilibrium of some sort -- generally fail to explain basic dynamical features of real markets. These include, most prominently, 1) a pronounced tendency in all markets to large price fluctuations reflected in "fat tailed" distributions of returns, and 2) vigorous and persistent volatility with delicate long-memory features, which is closely linked to the way that episodes of volatility "cluster" in the market.

These phenomena show up in specific mathematical signatures in the statistics of price movements, and these signatures present some of the most obvious details that any decent model of markets ought to reproduce quite naturally. Models capable of doing this have only emerged in the past two decades, and only succeed by making a clear break with the neo-classical tradition of equilibrium. This short review from 2006 by economist Blake LeBaron gives a nice introduction to the motivation for this kind of work, which very much follows in the tradition of natural science (as opposed to much of modern economics) in pursuing explanations in models with make plausible assumptions and explore the conclusions which follow from them.

This is a topic I'll be writing about more in the next few months (in connection with a longer term project I'm working on). Le Baron's review is a little old (5 years) and needs updating, but he describes some crucial points very effectively. A few highlights are worth mentioning:

Le Baron first outlines some of what he calls the "major puzzles of financial markets," which are directly linked to the things I just mentioned. First is simply the existence of pronounced volatility:

Volatility is the most obvious and probably the most important puzzle in finance. Why do financial prices and foreign exchange rates move around so much relative to other macro series both on a short term and long term basis? The difficulty of overall financial volatility was first demonstrated in Shiller (1981), and an update is in Shiller (2003). The issue has been that it is difficult to find financial or macro economic fundamentals that move around enough to justify the large swings observed in financial markets. As a potential policy problem, and an issue for long range investors, this might be the most important puzzle faced by financial modelers.

As Le Baron notes, lots of models (ARCH, GARCH, etc) have been produced which reproduce the mathematical character of volatility but without even attempting to understanding its origin. Moreover, most models in traditional economics, by sticking to the view that individuals are more or less identical and have rational expectations, simply shy away from the very phenomenon needing to be modeled:

The persistence of volatility in many financial markets has led to an entire industry of models, and is an area of intense interest both in academic and commercial areas. However, although there is a lot of empirical activity, the underlying microeconomic motives for volatility persistence are still not well understood. There are very few models which have even tackled this problem. This is probably due to the fact that in a homogeneous agent framework this is simply a very difficult problem.

The second puzzle of markets that Le Baron lists is what I mentioned first above: the fat tails of market returns or "excess kurtosis" in the statistical lingo:

Financial returns at relatively high frequencies (less than one month) are not normally distributed. There is not much of a strong theoretical reason that they need to be, but the hope has often been that some form of the central limit theorem should drive returns close to normality when aggregated over time. Recently, a new field, Econophysics, has appeared which stresses that returns also have additional structure that can be described using power laws. The determination and testing of power laws remains a somewhat open area, and the set of processes that generate acceptable power law pictures is also not well understood.

Aside from the long memory of volatility and fat tailed returns, he also mentions the rich dynamics of trading volume, which is equally as interesting as those of prices. Unfortunately, when it comes to the dynamics of volume, Le Baron notes, "Most traditional financial models remain completely silent."

Le Baron rightly points out that these rich dynamics in time are almost certainly linked directly to two things: 1) the fact that different market participants have different expectations at any moment (indeed, it is such differences which drive trading) and 2) that these differences undergo perpetual evolution through time. This suggests that a decent explanation of the origin of these so-called "stylised facts" (and others like them) will likely emerge from models which attempt to follow and capture something about the dynamics of beliefs and expectations in a population of diverse interacting agents:

Many of the most puzzling results from finance deal with problems of behavioral heterogeneity, and the dynamics of heterogeneity. The study of market heterogeneity as a kind of complicated dynamic state variable that needs to be modeled is probably one of the defining features of agent-based models. Empirical features such as trading volume are directly related to the amount of heterogeneity in the market, and demand models that can speak to this issue. Other empirical features are probably indirectly related. Large moves, excess kurtosis, and market crashes all probably stem from some type of strategy correlation that keeps the law of large numbers from functioning well across the market. These changing patterns can only be explored in a framework that allows agent strategies to adapt and adjust over time...

I think this point about the failure of the law of large numbers is, while obvious, still worth emphasizing. Something in the market ruins the simple picture of normal statistics, which would emerge from independent factors driving prices changes. Somehow the actions of different market participants must come to be correlated, thereby leading to non-normal fat tailed returns and long memory effects. In principle, of course, there may be many mechanisms contributing to this lack of independence (some participants following trends could be enough, for example).

Now, I doubt anyone actually working in finance would find this observation anything but banal. Yet many economists seem determined to continue modelling markets as if this were not the case. I don't know enough economic history to know why the homogeneous rational expectations view has such a following, but it seems very weird to me indeed.

The rest of Le Baron's review explores an example of the kind of model -- an agent based model for a market very much out of equilibrium -- which reproduces the above features quite naturally, at least qualitatively. The idea is simply to respect the fact that agents in a market are different and change their behaviour over time, adapting to what happens in the market. There is no presupposition that market prices must settle down to an equilibrium; the market does what it does as people interact and trade and try to profit as well as they can. The simple model explored here is one developed by Le Baron in 2002, but shares basic features with many other models developed by others. Agents can choose between a risky asset (a stock) and a risk free bond, and they use a variety of information to make their trading strategies:

Agents chose over a set of portfolio strategies that map current asset market information into a recommended portfolio fraction of wealth in the risky asset. This fraction can vary from zero to one since short selling and borrowing are not allowed. Information includes lagged returns, dividend price ratios, and several trend indicators. Agents must evaluate rules using past performance, and it is in this dimension where they are assumed to be heterogeneous. Agents use differing amounts of past information to evaluate rules. In other words, they have different memory lengths when it comes to evaluating strategies. Some agents use 30 years worth of data, while others might use only 6 months. In this way this model implements to behavioral features. First, agents are clearly boundedly rational in that they do not attempt to determine the entire
state space of the economy, which would be unwieldy if they attempted this. Also, they are assumed to have “small sample bias” since they don’t all choose to use as much data as possible.

The paper is an easy read so I won't get into much detail, but what emerges from the interaction of learning and adapting agents in a setting like this is immediately much more realistic and interesting than anything coming from traditional models. For example, the figure below shows the price of the risky asset as a function of time. In the model, there is a true equilibrium price which is made to fluctuate in a normal, Gaussian way (this price being linked to dividends). The actual price in the market is rarely at this equilibrium, but instead has large fluctuations around it, being some times far too high and at others too low, and often moving very rapidly from one point to another.

Now there are some features of this time series that don't look realistic. There seems to be a periodicity of sorts, for example. But this is a very simple model and the exciting thing is what it gets right -- easily giving a model of a market which never settles down to a prices, has persisting volatility, fat tails in returns, rich dynamics for trading volume and so on. Details can be found in the paper.

As Le Baron concludes the review,

Agent-based models make more progress than other frameworks in explaining these features due to that fact that at their core is a world of people who process information differently, and try hard to continually adjust and adapt their behavior over time. This market may never reach anything that looks like an equilibrium efficient market, but it is in a continual struggle toward this. The range of facts these models explain, and the robustness of their explanations to different structures and parameters, is impressive. At the moment, no other models can capture this many facts with this kind of simplicity and style.

As I said, that was 5 years ago. The final statement remains very much true, however. I think it is becoming increasingly clear, even to most economists, that agent-based models present currently the best technique to go beyond the restricting and unrealistic assumption of equilibrium economics and to build models of markets which get their basic behaviour right.

I intend this post just to be a beginning of an exploration of what is currently happening in the development of more realistic non-equilibrium market models. The field is currently exploding and there are many interesting questions to tackle. For example, models of this kind often involve quite a large number of parameters describing agents strategies and so forth. Yet many models work very similarly despite large differences in such parameters. An open question is whether it may be possible to develop a kind of map of the space of possible models, showing a set of classes into which different models fall. Physicist Luciano Pietronero has done some interesting work in this direction.

In any event, I find it hard to understand why anyone today would go on studying homogeneous rational equilibrium, at least in application to the financial markets.

Rational -- by definition and ideology

2012-01-09T15:45:00.000+01:00

I've been doing some background reading on rationality in economics, and came across this fairly unique perspective offered by economist Duncan Foley. It's from 2003. What sets it apart from most other reviews of the role of the rationality assumption in economics, is that Foley tries to trace the history of this approach as it emerged out of the tradition of Hobbes and Locke in political philosophy. As Foley notes, the idea of rationality is in many ways beyond question for most economists, and not at all an empirical matter:

...an orientation toward situating explanations of economic phenomena in relation to rationality has increasingly become the touchstone by which mainstream economists identify themselves and recognize each other. This is not so much a question of adherence to any particular conception of rationality, but of taking rationality of individual behavior as the unquestioned starting point of economic analysis.

It has become so, he asserts, because this way of thinking has emerged from the "just so" story developed by Hobbes, Locke and others which allegedly explains how property rights and political institutions solve problems arising from the anarchic struggle of man against man in the original state of nature. They place reason at the core of this project, and essentially use this story to explain why things are as they are -- this is the rational world and the only way things can be, if we are to avoid the chaos of anarchy. In essence, the rationality assumption is part of a propaganda campaign. Foley:

A hallmark of these [rationally designed] institutions is that they are in themselves in principle democratic and egalitarian (everyone has an equal right to vote or to hold property) but lead
inexorably to sharp inequalities in economic well-being. It is not hard to see that an economic science whose philosophical starting point was not rational individual action would create an embarrassing discord with this political tradition. The whole point of the Hobbes-Locke “discourse” (to use the jargon of post-modernism) is to rationalize the existing inequalities of power and economic well-being that arise from the institutions of modern society as being unavoidable consequences of the interaction of naturally constituted rational individuals
confronting each other as equals, given the natural and unalterable conditions of human existence. Economic science has a place in this grand project only insofar as it can relate itself to the same philosophical foundations.

I think there's a strong current of truth here. There is in today's economic theory a standing presumption that people should be modelled as rational decision makers (optimizers), and the argument often seems to boil down (in some disguised form) to "we must, because if we do not, we will not be able to prove theorems about equilibrium and its efficiency." This is of course too strong, and research programs in behavioural economics, information asymmetries and so on seem to be working to correct this, but the effort required reflects how much resistance there is to such change and how much intellectual inertia still resides in the idea of thorough-going rationality. Foley suggests that efforts to bring more realistic perspectives such as bounded rationality into core theory have been resisted precisely because they cannot be used to justify the just-so story of the efficient equilibrium:

...in its pragmatic focus on understanding and explaining how people actually behave in modern society, bounded rationality loses contact with the underlying project of rationalizing the institutions of modern society. For example, there really is no logical place in the discourse of bounded rationality for the Fundamental Theorems of Welfare Economics that purport to establish a connection between competitive market equilibrium and an efficient allocation of resources.

I think he may largely be right. If so, this would go a long way to explaining why economics has persisted with such a narrow set of theoretical concepts for such a long time. Maybe it's not actually trying to explain and understand the world at all, but to rationalize why it is OK that it is as we see it. And that's not encouraging for those of us hoping it will change in a big way:

It will not be easy to create a social science that transcends the antinomies and limitations of rational-actor theory. Certainly we cannot depend on the “usual” processes of scientific self-criticism to accomplish much in this direction. No accumulation of its empirical anomalies, or demonstration of its logical inadequacies will somehow magically dispel the power of rational-actor theory, because its power does not rest in the last instance on the adequacy of its
explanations or the consistency of its logic.