Power laws in finance

My latest column in Bloomberg looks very briefly at some of the basic mathematical patterns we know about in finance. Science has a long tradition of putting data and observation first. Look very carefully at what needs to be explained -- mathematical patterns that show up consistently in the data -- and then try to build simple models able to reproduce those patterns in a natural way.

This path has great promise in economic finance, although it hasn't been pursued very far until recently. My Bloomberg column gives a sketch of what is going on, but I'd like to give a few more details here and some links.

The patterns we find in finance are statistical regularities -- broad statistical patterns which show up in all markets studied, with an impressive similarity across markets in different countries and for markets in different instruments. The first regularity is the distribution of returns over various time intervals, which has been found generically to have broad power law tails -- "fat tails" -- implying that large fluctuations up or down are much more likely than they would be if markets fluctuated in keeping with normal Gaussian statistics. Anyone who read The Black Swan knows this.

This pattern has been established in a number of studies over the past 15 years or so, mostly by physicist Eugene Stanley of Boston University and colleagues. This paper from 1999 is perhaps the most notable, as it used enormous volumes of historical data to establish the fat tailed pattern for returns over times ranging from one minute up to about 4 days. One of the most powerful things about this approach is that it doesn't begin with any far reaching assumptions about human behaviour, the structure of financial markets or anything else, but only asks -- are there patterns in the data? As the authors note:

The most challenging difficulty in the study of a financial market is that the nature of the interactions between the different elements comprising the system is unknown, as is the way in which external factors affect it. Therefore, as a starting point, one may resort to empirical studies to help uncover the regularities or “empirical laws” that may govern financial markets.    

This strategy seems promising to physicists because it has worked in building theories of complex physical systems -- liquids, gases, magnets, superconductors -- for which it is also often impossible to know anything in great detail about the interactions between the molecules and atoms within. This hasn't prevented the development of powerful theories because, as it turns out, many of the precise details at the microscopic level DO NOT influence the large scale collective properties of the system. This has inspired physicists to think that the same may be true in financial markets -- at least some of the collective behaviour we see in markets, their macroscopic behaviour, may be quite insensitive to details about human decision making, market structure and so on.

The authors of this 1999 study summarized their findings as follows:

Several points of clarification. First, the result for the power law with exponent close to 3 is a result for the cumulative distribution. That is, the probability that a return will be greater than a certain value (not just equal to that value). Second, the fact that this value lies outside of the range [0,2] means that the process generating these fluctuations isn't a simple stationary random process with an identical and independent distribution for each time period. This was the idea initially proposed by Benoit Mandelbrot on the basis of the so-called Levy Stable distributions. This study and others have established that this idea can't work -- something more complicated is going on.

That complication is also referred to in the second paragraph above. If you take the data on returns at the one minute level, and randomize the order in which it appears, then you still get the same power law tails in the distribution of returns over one minute. That's the same data. But this new time series has different returns over longer times, generated by combining sequences of the one minute returns. The distribution over longer and longer times turns out to converge slowly to a Gaussian for the randomized data, meaning that the true fat tailed distribution over longer times has its origin in some rich and complex correlations in market movements at different times (which gets wiped out by the randomization). Again, we're not just dealing with a fixed probability distribution and independent changes over different intervals.

To read more about this, see this nice review by Xavier Gabaix of MIT. It covers this and many other power laws in finance and economics.

Now, the story gets even more interesting if you look past the mere distribution of returns and study the correlations between market movements at different times. Market movements are, of course, extremely hard to predict. But it is very interesting where the unpredictability comes in.

The so-called autocorrelation of the time series of market returns decays to zero after a few minutes. This is essentially a measure of how much the return now can be used to predict a return in the future. After a few minutes, there's nothing. This is the sense in which the markets are unpredictable. However, there are levels of predictability. It was discovered in the early 1990s, and has been confirmed many times since in different markets, that the time series of volatility -- the absolute value of the market return -- has long-term correlations, a kind of long-term memory. Technically, the autocorrelation of this time series only decays to zero very slowly.

This is shown below in the following figure (from a representative paper, again from the Boston University group) which shows the autocorrelation of the return time series g(t) and also of the volatility, which is the absolute value of g(t):

Clearly, whereas the first signal shows no correlations after about 10 minutes, the second shows correlations and predictability persisting out to times as long as 10,000 minutes, which is on the order of 10 days or so.

So, its the directionality of price movements which has very little predictability, whereas the magnitude of changes follows a process with much more interesting structure. It is in the record of this volatility that one sees potentially deep links to other physical processes, including earthquakes. A particularly interesting paper is this one, again by the Boston group, quantifying several ways in which market volatility obeys several quantitative laws known from earthquake science, especially the Omori Law describing how the probability of aftershocks decays following a main earthquake. This probability decays quite simply in proportion to 1/time since the main quake, meaning that aftershocks are most likely immediately afterward, and become progressively less likely with time. Episodes of high volatility appear to follow similar behaviour quite closely.

Perhaps even better is another study, which looks at the link to earthquakes with a somewhat tighter focus. The abstract captures the content quite well:

We analyze the memory in volatility by studying volatility return intervals, defined as the time between two consecutive fluctuations larger than a given threshold, in time periods following stock market crashes. Such an aftercrash period is characterized by the Omori law, which describes the decay in the rate of aftershocks of a given size with time t by a power law with exponent close to 1. A shock followed by such a power law decay in the rate is here called Omori process. We find self-similar features in the volatility. Specifically, within the aftercrash period there are smaller shocks that themselves constitute Omori processes on smaller scales, similar to the Omori process after the large crash. We call these smaller shocks subcrashes, which are followed by their own aftershocks. We also show that the Omori law holds not only after significant market crashes as shown by Lillo and Mantegna [Phys. Rev. E 68, 016119 2003], but also after “intermediate shocks.” ...

These are only a few of the power law type regularities now known to hold for most markets, with only very minor differences between markets. An important effort is to find ways to explain these regularities in simple and plausible market models. None of these patterns can be explained by anything in the standard economic theories of markets (the EMH etc). They can of course be reproduced by suitably generating time series using various methods, but that hardly counts as explanation -- that's just using time series generators to reproduce certain kinds of data.

The promise of finding these kinds of patterns is that they may strongly constrain the types of theories to be considered for markets, by ruling out all those which do not naturally give rise to this kind of statistical behaviour. This is where data matters most in science -- by proving that certain ideas, no matter how plausible they seem, don't work. This data has already stimulated the development of a number of different avenues for building market theories which can explain the basic statistics of markets, and in so doing go well beyond the achievements of traditional economics.

I'll have more to say on that in the near future.

What moves the markets? Part II

High frequency trading makes for markets that produce enormous volumes of data. Such data make it possible to test some of the old chestnuts of market theory -- the efficient markets hypothesis, in particular -- more carefully than ever before. Studies in the past few years show quite clearly, it seems to me, that the EMH is very seriously misleading and isn't really even a good first approximation.

Let me give a little more detail. In a recent post I began a somewhat leisurely exploration of considerable evidence which contradicts the efficient markets idea. As the efficient markets hypothesis (the "weak" version, at least) claims, market prices fully reflect all publicly available information. When new information becomes available, prices respond. In the absence of new information, prices should remain more or less fixed.

Striking evidence against this view comes from studies (now almost ten or twenty years old) showing that markets often make quite dramatic movements even in the absence of any news. I looked at some older studies along these lines in the last post, but stronger evidence comes from studies using electronic news feeds and high-frequency stock data. Are sudden jumps in prices in high frequency markets linked to the arrival of new information, as the EMH says? In a word -- no!

The idea in these studies is to look for big price movements which, in a sense, "stand out" from what is typical, and then see if such movements might have been caused by some "news". A good example is this study by Armand Joulin and colleagues from 2008. Here's how they proceeded. Suppose R(t) is the minute by minute return for some stock. You might take the absolute value of these returns, average them over a couple hours and use this as a crude measure -- call it σ -- of the "typical size" of one-minute stock movements over this interval. An unusually big jump over any minute-long interval will be one for which the magnitude of R is much bigger than σ. 

To make this more specific, Joulin and colleagues defined "s jumps" as jumps for which the ratio |R/σ| > s. The value of s can be 2 or 10 or anything you like. You can look at the data for different values of s, and the first thing the data shows -- and this isn't surprising -- is a distinctive pattern for the probability of observing jumps of size s. It falls off with increasing s, meaning that larger jumps are less likely, and the mathematical form is very simple -- a power law with P(s) being proportional to s^-4, especially as s becomes large (from 2 up to 10 and beyond). This is shown in the figure below (the upper curve):

This pattern reflects the well known "fat tailed" distribution of market returns, with large returns being much more likely than they would be if the statistics followed a Gaussian curve. Translating the numbers into daily events, s jumps of size s = 4 turn out to happen about 8 times each day, while larger jumps of s = 8 occur about once every day and one-half (this is true for each stock).

Now the question is -- are these jumps linked to the announcement of some new information? To test this idea, Joulin and colleagues looked at various news feeds including feeds from Dow Jones and Reuters covering about 900 stocks. These can be automatically scanned for mention of any specific company, and then compared to price movements for that company. The first thing they found is that, on average, a new piece of news arrives for a company about once every 3 days. Given that a stock on average experiences one jump every day and one-half, this immediately implies an imbalance between the number of stock movements and the number of news items. There's not enough news to cause the jumps observed. Stocks move -- indeed, jump -- too frequently.

Conclusion: News sometimes but not always causes market movements, and significant market movements are sometimes but not always caused by news. The EMH is wrong, unless you want to make further excuses that there could have been news that caused the movement, and we just don't recognize it or haven't yet figured out what it is. But that seems like simply positing the existence of further epicycles.

But another part of the Joulin et al. study is even more interesting. Having found a way to divide price jumps into two categories: A) those caused by news (clearly linked to some item in a news feed) and B) those unrelated to any news, it is then possible to look for any systematic differences in the way the market settled down after such a jump. The data show that the volatility of prices, just after a jump, becomes quite high; it then relaxes over time back to the average volatility before the jump. But the relaxation works differently depending on whether the jump was of type A or B: caused by news or not caused by news. The figure below shows how the volatility relaxes back to the norm first for jumps linked to news, and second to jumps not linked to news. The later shows a much slower relaxation:

As the authors comment on this figure,

In both cases, we find (Figure 5) that the relaxation of the excess-volatility follows a power-law in time σ(t) − σ(∞) ∝ t− β (see also [22, 23]). The exponent of the decay is, however, markedly different in the two cases: for news jumps, we find β ≈ 1, whereas for endogenous jumps one has β ≈ 1/2. Our results are compatible with those of [22], who find β ≈ 0.35.

Of course, β ≈ 1/2 implies a much slower relaxation back to the norm (whatever that is!) than does β ≈ 1. Hence, it seems that the market takes a longer time to get back to normal after a no-news jump, whereas it goes back to normal quite quickly after a news-related jump.

No one knows why this should be, but Joulin and colleagues made the quite sensible speculation that a jump clearly related to news is not really surprising, and certainly not unnerving. It's understandable, and traders and investors can decide what they think it means and get on with their usual business. In contrast, a no-news event -- think of the Flash Crash, for example -- is very different. It is a real shock and presents a lingering unexplained mystery. It is unnerving and makes investors uneasy. The resulting uncertainty registers in high volatility.

What I've written here only scratches the surface of this study. For example, one might object that lots of news isn't just linked to the fate of one company, but pertains to larger macroeconomic factors. It may not even mention a specific company but point to a likely rise in the prices of oil or semiconductors, changes influencing whole sectors of the economy and many stocks all at once. Joulin and colleagues tried to take this into account by looking for correlated jumps in the prices of multiple stocks, and indeed changes driven by this kind of news do show up quite frequently. But even accounting for this more broad-based kind of news, they still found that a large fraction of the price movements of individual stocks do not appear to be linked to anything coming in through news feeds. As they concluded in the paper:

Our main result is indeed that most large jumps... are not related to any broadcasted news, even if we extend the notion of ‘news’ to a (possibly endogenous) collective market or sector jump. We find that the volatility pattern around jumps and around news is quite different, confirming that these are distinct market phenomena [17]. We also provide direct evidence that large transaction volumes are not responsible for large price jumps, as also shown in [30]. We conjecture that most price jumps are in fact due to endogenous liquidity micro-crises [19], induced by order flow fluctuations in a situation close to vanishing outstanding liquidity.

Their suggestion in the final sentence is intriguing and may suggest the roots of a theory going far beyond the EMH. I've touched before on early work developing this theory, but there is much more to be said. In any event, however, data emerging from high-frequency markets backs up everything found before -- markets often make violent movements which have no link to news. Markets do not just respond to new information. Like the weather, they have a rich -- and as yet mostly unstudied -- internal dynamics.

Difficulties with learning...

I just finished reading this wonderful short review of game theory (many thanks to ivansml for pointing this out to me) and its applications and limitations by Martin Shubik. It's a little old -- it appeared in the journal Complexity in 1998 -- but offers a very broad perspective which I think still holds today. Game theory in the pure sense generally views agents as coming to their strategies through rational calculation; this perspective has had huge influence in economics, especially in the context of relatively simple games with few players and not too many possible strategies. This part of game theory is well developed, although Shubik suggests there are probably many surprises left to learn.

Where the article really comes alive, however, is in considering the limitations to this strictly rational approach in games of greater complexity. In physics, the problem of two rigid bodies in gravitational interaction can be solved exactly (ignoring radiation, of course), but you get generic chaos as soon as you have three bodies or more. The same is true, Shubik argues, in game theory. Extend the number of players above three and as the number of possible permutations of strategies proliferates it is no longer plausible to assume that agents act rationally. The decision problems become too complex. One might still try to search for optimal N player solutions as a guide to what might be possible, but the rational agent approach isn't likely to be profitable as a guide to the likely behaviour and dynamics in such complex games. I highly recommend Shubik's short article to anyone interested in game theory, and especially its application to real world problems where people (or other agents) really can't hope to act on the basis of rational calculation, but instead have to use heuristics, follow hunches, and learn adaptively as they go.

Some of the points Shubik raises find perfect illustration in a recent study (I posted on it here) of typical dynamics in two-player games when the number of possible strategies gets large. Choose the structure of the games at random and the most likely outcome is a rich ongoing evolution of strategic behaviour which never settles down into any equilibrium. But these games do seem to show characteristic dynamical behaviour such as "punctuated equilibrium" -- long periods of relative quiescence which get broken apart sporadically by episodes of tumultuous change -- and clustered volatility -- the natural clustering together of periods of high variability. These qualitative aspects appear to be generic features of the non-equilibrium dynamics of complex games. Interesting that they show up generically in markets as well.

When problems are too complex -- which is typically the case -- we try to learn and adapt rather than "solving" the problem in any sense. Our learning itself may also never settle down into any stable form, but continually change as we find something that works well for a time, and then suddenly find it fails and we need to learn again.

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.

High-frequency trading, the downside -- Part II

In this post I'm going to look a little further at Andrew Haldane's recent Bank of England speech on high-frequency trading. In Part I of this post I explored the first part of the speech which looked at evidence that HFT has indeed lowered bid-ask spreads over the past decade, but also seems to have brought about an increase in volatility. Not surprisingly, one measure doesn't even begin to tell the story of how HFT is changing the markets. Haldane explores this further in the second part of the speech, but also considers in a little more detail where this volatility comes from.

In well known study back in 1999, physicist Parameswaran Gopikrishnan and colleagues (from Gene Stanley's group in Boston) undertook what was then the most detailed look at market fluctuations (using data from the S&P Index in this case) over periods ranging from 1 minute up to 1 month. This early study established a finding which (I believe) has now been replicated across many markets -- market returns over timescales from 1 minute up to about 4 days all followed a fat-tailed power law distribution with exponent α close to 3. This study found that the return distribution became more Gaussian for times longer than about 4 days. Hence, there seems to be rich self-similarity and fractal structure to market returns on times down to 1 around second.

What about shorter times? I haven't followed this story for a few years. It turns out that in 2007, Eisler and Kertesz looked at a different set of data -- for total transactions on the NYSE between 2000 and 2002 -- and found that the behaviour at short times (less than 60 minutes) was more Gaussian. This is reflected in the so-called Hurst exponent H having an estimated value close to 0.5. Roughly speaking, the Hurst exponent describes -- based on empirical estimates -- how rapidly a time series tends to wander away from its current value with increasing time. Calculate the root mean square deviation over a time interval T and for a Gaussian random walk (Brownian motion) this should grow in proportion to T to the power H= 1/2. A Hurst exponent higher than 1/2 indicates some kind of interesting persistent correlations in movements.

However, as Haldane notes, Reginald Smith last year showed that stock movements over short times since around 2005 have begun showing more fat-tailed behaviour with H above 0.5. That paper shows a number of figures showing H rising gradually over the period 2002-2009 from 0.5 to around 0.6 (with considerable  fluctuation on top of the trend). This rise means that the market on short times has increasingly violent excursions, as Haldane's chart 11 below illustrates with several simulations of time series having different Hurst exponents:

The increasing wildness of market movements has direct implications for the risks facing HFT market makers, and hence, the size of the bid-ask spread reflecting the premium they charge. As Haldane notes, the risk a market maker faces -- in holding stocks which may lose value or in encountering counterparties with superior information about true prices -- grows with the likely size of price excursions over any time period. And this size is directly linked to the Hurst exponent.

Hence, in increasingly volatile markets, HFTs become less able to provide liquidity to the market precisely because they have to protect themselves:

This has implications for the dynamics of bid-ask spreads, and hence liquidity, among HFT firms. During a market crash, the volatility of prices (σ) is likely to spike. From equation (1), fractality heightens the risk sensitivity of HFT bid-ask spreads to such a volatility event. In other words, liquidity under stress is likely to prove less resilient. This is because one extreme event, one flood or drought on the Nile, is more likely to be followed by a second, a third and a fourth. Reorganising that greater risk, market makers’ insurance premium will rise accordingly.

This is the HFT inventory problem. But the information problem for HFT market-makers in situations of stress is in many ways even more acute. Price dynamics are the fruits of trader interaction or, more accurately, algorithmic interaction. These interactions will be close to impossible for an individual trader to observe or understand. This algorithmic risk is not new. In 2003, a US trading firm became insolvent in 16 seconds when an employee inadvertently turned an algorithm on. It took the company 47 minutes to realise it had gone bust.

Since then, things have stepped up several gears. For a 14-second period during the Flash Crash, algorithmic interactions caused 27,000 contracts of the S&P 500 E-mini futures contracts to change hands. Yet, in net terms, only 200 contracts were purchased. HFT algorithms were automatically offloading contracts in a frenetic, and in net terms fruitless, game of pass-the-parcel. The result was a magnification of the fat tail in stock prices due to fire-sale forced machine selling.

These algorithmic interactions, and the uncertainty they create, will magnify the effect on spreads of a market event. Pricing becomes near-impossible and with it the making of markets. During the Flash Crash, Accenture shares traded at 1 cent, and Sotheby’s at $99,999.99, because these were the lowest and highest quotes admissible by HFT market-makers consistent with fulfilling their obligations. Bid-ask spreads did not just widen, they ballooned. Liquidity entered a void. That trades were executed at these “stub quotes” demonstrated algorithms were running on autopilot with liquidity spent. Prices were not just information inefficient; they were dislocated to the point where they had no information content whatsoever.

This simply follow from the natural dynamics of the market, and the situation market makers find themselves in. If they want to profit, if they want to survive, they need to manage their risks, and these risks grow rapidly in times of high volatility. Their response is quite understandable -- to leave the market, or least charge much more for their service. 

Individually this is all quite rational, yet the systemic effects aren't likely to benefit anyone. The situation, Haldane notes, resembles a Tragedy of the Commons in which individually rational actions lead to a collective disaster, fantasies about the Invisible Hand notwithstanding:

If the way to make money is to make markets, and the way to market markets is to make haste, the result is likely to be a race – an arms race to zero latency. Competitive forces will generate incentives to break the speed barrier, as this is the passport to lower spreads which is in turn the passport to making markets. This arms race to zero is precisely what has played out in financial markets over the past few years.

Arms races rarely have a winner. This one may be no exception. In the trading sphere, there is a risk the individually optimising actions of participants generate an outcome for the system which benefits no-one – a latter-day “tragedy of the commons”. How so? Because speed increases the risk of feasts and famines in market liquidity. HFT contribute to the feast through lower bid-ask spreads. But they also contribute to the famine if their liquidity provision is fickle in situations of stress.

Haldane then goes on to explore what might be done to counter these trends. I'll finish with a third post on this part of the speech very soon. 

But what is perhaps most interesting in all this is how much of Haldane's speech refers to recent work done by physicists -- Janos Kertesz, Jean-Philippe Bouchaud, Gene Stanley, Doyne Farmer and others -- rather than studies more in the style of neo-classical efficiency theory. It's encouraging to see that at least one very senior banking authority is taking this stuff seriously.