Private information and jumps in the market

Following my second recent post on what moves the markets, two readers posted interesting and noteworthy comments and I'd like to explore them a little. I had presented evidence in the post that many large market movements do not appear to be linked to the sudden arrival of public information in the form of news. Both comments noted that this may leave out of the picture another source of information -- private information brought into the market through the action of traders taking actions:

Anonymous said...

I don't see any mention of what might be called "trading" news, e.g. a large institutional investor or hedge fund reducing significantly its position in a given stock for reasons unrelated to the stock itself - or at least not synchronized with actual news on the underlying. The move can be linked to internal policy, or just a long-term call on the company which timing has little to do with market news, or lags them quite a bit (like an accumulation of bad news leading to a lagged reaction, for instance). These shocks are frequent even on fairly large cap stocks. They also tend to have lingering effect because the exact size of the move is never disclosed by the investor and can spread over long periods of time (i.e. days), which would explain the smaller beta. Yet this would be a case of "quantum correction", both in terms of timing and agent size, rather than a breakdown of the information hypothesis.

 and...

DR said...

Seconding the previous comment, asset price information comes in a lot more forms than simply "news stories about company X." All market actions contains information. Every time a trade occurs there's some finite probability that it's the action of an informed trader. Every time the S&P moves its a piece of information on single stock with non-zero beta. Every time the price of related companies changes it contains new information.

Both of these comments note the possibility that every single trade taking place in the market (or at least many of them) may be revealing some fragment of private information on the part of whoever makes the trade. In principle, it might be such private information hitting the market which causes large movements (the s-jumps described in the work of Joulin and colleagues).  

I think there are several things to note in this regard. The first is that, while this is a sensible and plausible idea, it shouldn't be stretched too far. Obviously, if you simply assume that all trades carry information about fundamentals, then the EMH -- interpreted in the sense that "prices move in response to new information about fundamentals" -- essentially becomes true by definition. After all, everyone agrees that trading drives markets. If all trading is assumed to reveal information, then we've simple assumed the truth of the EMH. It's a tautology.

More useful is to treat the idea as a hypothesis requiring further examination. Certainly some trades do reveal private information, as when a hedge fund suddenly buys X and sells Y, reflecting a belief based on research that Y is temporarily overvalued relative to X. Equally, some trades (as mentioned in the first comment) may reveal no information, simply being carried out for reasons having nothing to do with the value of the underlying stock. As there's no independent way -- that I know of -- to determine if a trade reveals new information or not, we're stuck with a hypothesis we cannot test.

But some research has tried to examine the matter from another angle. Again, consider large price movements -- those in the fat-tailed end of the return distribution. One proposed idea looking to private information as a cause proposes that large price movements are caused primarily by large-volume trades by big players such as hedge funds, mutual funds and the like. Some such trades might reveal new information, and some might not, but let's assume for now that most do. In a paper in Nature in 2003, Xavier Gabaix and colleagues argued that you can explain the precise form of the power law tail for the distribution of market returns -- it has an exponent very close to 3 -- from data showing that the size distribution of mutual funds follows a similar power law with an exponent of 1.05. A key assumption in their analysis is that the price impact Δp generated by a trade of volume V is roughly equal to Δp = kV^1/2.

This point of view seems to support the idea that the arrival of new private information, expressed in large trades, might account for the no-news s jumps noted in the Jouvin study. (It seems less plausible that such revealed information might account for anything as violent as the 1987 crash, or the general meltdown of 2008). But taken at face value, these arguments at least seem to be consistent with the EMH view that even many large market movements reflect changes in fundamentals. But again, this assumes that all or at least most large volume trades are driven by private information on fundamentals, which may not be the case. The authors of this study themselves don't make any claim about whether large volume trades really reflect fundamental information. Rather, they note that...

Such a theory where large individual participants move the market is consistent with the evidence that stock market movements are difficult to explain with changes in fundamental values... 

But more recent research (here and here, for example) suggest that this explanation doesn't quite hang together because the assumed relationship between large returns and large volume trades isn't correct. This analysis is fairly technical, but is based on the study of minute-by-minute NASDAQ trading and shows that, if you consider only extreme returns or extreme volumes, there is no general correlation between returns and volumes. The correlation assumed in the earlier study may be roughly correct on average, but it not true for extreme events. "Large jumps," the authors conclude, "are not induced by large trading volumes."

Indeed, as the authors of these latter studies point out, people who have valuable private information don't want it to be revealed immediately in one large lump because of the adverse market impact this entails (forcing prices to move against them). A well-known paper by Albert Kyle from 1985 showed how an informed trader with valuable private information, trading optimally, can hide his or her trading in the background of noisy, uninformed trading, supposing it exists. That may be rather too much to believe in practice, but large trades do routinely get broken up and executed as many small trades precisely to minimize impact. 

All in all, then, it seems we're left with the conclusion that public or private news does account for some large price movements, but cannot plausibly account for all of them. There are other factors. The important thing, again, is to consider what this means for the most meaningful sense of the EMH, which I take to be the view that market prices reflect fundamental values fairly accurately (because they have absorbed all relevant information and processed it correctly). The evidence suggests that prices often move quite dramatically on the basis of no new information, and that prices may be driven as a result quite far from fundamental values.

The latter papers do propose another mechanism as the driver of routine large market movements. This is a more mechanical process centering on the natural dynamics of orders in the order book. I'll explore this in detail some other time. For now, just a taster from this paper, which describes the key idea:

So what is left to explain the seemingly spontaneous large price jumps? We believe that the explanation comes from the fact that markets, even when they are ‘liquid’, operate in a regime of vanishing liquidity, and therefore are in a self-organized critical state [31]. On electronic markets, the total volume available in the order book is, at any instant of time, a tiny fraction of the stock capitalisation, say 10⁻⁵ −10⁻⁴ (see e.g. [15]). Liquidity providers take the risk of being “picked off”, i.e. selling just before a big upwards move or vice versa, and therefore place limit orders quite cautiously, and tend to cancel these orders as soon as uncertainty signals appear. Such signals may simply be due to natural fluctuations in the order flow, which may lead, in some cases, to a catastrophic decay in liquidity, and therefore price jumps. There is indeed evidence that large price jumps are due to local liquidity dry outs.

Why game theory is often useless...

Economic theory relies very heavily on the notion of equilibrium. This is true in any model for competitive equilibrium -- exploring how exchange can in principle lead to an optimal allocation of resources -- or more generally in the context of game theory, which explores stable Nash equilibria in strategic games.

One thing physicists find wholly unsatisfying about equilibrium in either case is economists' near total neglect of the crucial problem of whether the agents in such models might ever plausibly find an equilibrium. You can assume perfectly rational agents and prove the existence of an equilibrium, but this may be an irrelevant mathematical exercise. Realistic agents with finite reasoning powers might never be able to learn their way to such a solution.

More likely, at least in many cases, is that less-than-perfectly rational agents, even if they're quite clever at learning, may never find their way to a neat Nash equilibrium solution, but instead go on changing and adapting and responding to one another in a way that leads to ongoing chaos. Naively, this would seem especially likely in any situation -- think financial markets, or any economy as a whole -- in which the number of possible strategies is enormous and it is simply impossible to "solve the problem" of what to do through perfect rational reflection (no one plays chess by working out the Nash equilibrium).

A brilliant illustration of this insight comes in a new paper by Tobias Galla and Doyne Farmer. This is the first study I've seen (though there may well be others) which addresses this matter of the relevance of equilibrium in complex, high-dimensional games in a  generic way. The conclusion is as important as it is intuitively reasonable:

Here we show that if the players use a standard approach to learning, for complicated games there is a large parameter regime in which one should expect complex dynamics. By this we mean that the players never converge to a fixed strategy. Instead their strategies continually vary as each player responds to past conditions and attempts to do better than the other players. The trajectories in the strategy space display high-dimensional chaos, suggesting that for most intents and purposes the behavior is essentially random, and the future evolution is inherently unpredictable.

In other words, in games of sufficient complexity, the insights coming from equilibrium analyses just don't tell you much. If the agents learn in a plausible way, they never find any equilibrium at all, and the evolution of strategic behaviours simply carries on indefinitely. The system remains out of equilibrium.

A little more detail. Their basic approach is to consider general two player games between, say, Alice and Bob. Let each of the two players have N possible strategies to choose from. The payoff matrices for any such game are NxN matrice (one for each player) giving the payoffs they get for each pair of strategies being played. The cute idea in this analysis is to choose the game randomly by selecting the elements of the payoff matrices for both Alice and Bob from a normal distribution centered on zero. The authors simply choose a game and simulate play as the two players learn through experience -- playing strategies from their repertoire of N possibilities more frequently if those strategies give good results.

With N = 50, the results show clearly that many games do not ever settle into any kind of stable behaviour. Rather, no equilibrium is ever found. The typical dynamics is reflected in the figure below, which shows the difference in payoffs to the two players (Alice's - Bob's) over time. Even though the two agents work hard to learn the optimal strategies, the complexity of the game prevents their success, and the game shows no signs whatsoever of settling down:

As the authors note, this kind of rich, complex, ongoing dynamics looks quite similar to what one sees in real systems such as financial markets (the time series above exhibits clustered volatility, as do market fluctuations). There are periods of relative calm punctured by bouts of extreme volatility. Yet there's nothing intervening here -- no "shocks" to the system -- which would create these changes. It all comes from perfectly natural internal dynamics. And this is in a game with N = 50 strategies. It seems likely things will only grow more chaotic and less likely to settle down if N is larger than 50, as in the real world, or if the number of players grows beyond two.

Hence, I see this as a rather profound demonstration of the likely irrelevance of equilibrium analyses coming from game theory to complex real world settings. Dynamics really matters and cannot be theorized out of existence, however hard economists may try. As the paper concludes:

Our results suggest that under many circumstances it is more useful to abandon the tools of classic game theory in favor of those of dynamical systems. It also suggests that many behaviors that have attracted considerable interest, such as clustered volatility in nancial markets, may simply be specific examples of a highly generic phenomenon, and should be expected to occur in a wide variety of different situations.