Feature: Econophysics
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Bubble trouble
When a stock market rises unsustainably, it can create a financial bubble that sooner or later will burst.
Tobias Preis and H Eugene Stanley examine whether concepts from physics can be used to create alaw describing exactly how such crashes occur
Wild fluctuations in the stock prices and currency exchange rates of every country have had a huge impact
on the world economy and the personal fortunes of
millions of us over the last few years. These instabilities
have also had another, perhaps unintended, consequence – of thrusting the academic discipline of“econophysics” firmly into the limelight. But does afield that involves applying the concepts of statisticalphysics to economics really have anything important tocontribute to discussions about the current economiccrisis? Yes – absolutely – because finding laws describing fluctuations is the essence of statistical physics.
Physicists are not, of course, the first people to apply
statistics to economics, with mathematicians also having contributed to the field for many years. One of theirfirst significant breakthroughs came in 1900 when
Louis Bachelier, working under the tutelage of thegreat French mathematician Henri Poincaré, publishedPhysics World May 2011a PhD thesis in which he analysed real financial data.
Bachelier claimed that a histogram of the changes inshare price (measured over any period of time) forms abell-shaped curve known as a Gaussian function, withvery large fluctuations essentially never occurring. In
other words, he believed that the chances of a serious
crash occurring are almost zero. Such serious crashes
are indeed very rare, but when they do occur, theireffects can be devastating.
The model associated with Bachelier is often referred
to as the “random walk” or “drunkard’s walk” because
he assumed that stock prices go up or down randomly
by an amount that has a characteristic value. In recentyears, however, econophysicists have been able to gettheir hands on a staggering quantity of real-time financial data, including the price, volume and timestamp of
every transaction of every stock you can think of. Thanks
to this information, which is now available in huge

Tobias Preis and
H Eugene Stanley
are at the Center for
Polymer Studies,
Department ofPhysics, Boston
University, US. Preisis also at ETH Zurich,
Switzerland,e-mail mail@tobiaspreis.dePreis will be givinga free online lecture
at 4 p.m. BST on
12 May – see
physicsworld.com/
cws/channel/
multimedia for
more details

29

Feature: Econophysics

physicsworld.com

1 Rise and fallab
6520.00Δt

6510.00

price

trend #46490.00
6480.00

Δt

Δt

Δt

volume6470.00
6460.00
6450.00

volumeΔt

price6500.00

Δt

trend #2Δttrend #1Δttrend #3

2.0K1.0K

14:00

15:00

16:00

17:00

18:00

transaction by transaction(a) A small subset of the German DAX Future market comprising a few hours of trading on the afternoon of Friday 15 October 2010. The data can be analysed to obtain theprice and volume of transactions as well as the time intervals between them, all with a resolution of 10 ms. (b) This graph shows local price maxima (blue circles) and local
price minima (red circles), where these peaks and troughs are determined over a time window of 2Δt. Positive trends start at the local minima and end at the local maxima –and vice versa for negative trends – with ε = 0 assigned to the start of each trend and ε = 1 to each end. In order to study the “switching processes” between trends – both
before a trend switch and after – the subsequent volume sequences of identical length need to be considered.

financial databases, it is becoming widely accepted thatthe drunkard’s walk fails to account for the very largest
changes in stock price. These rare occurrences, which
are collectively known as the “fat tails”, correspond toevents with huge price changes that are far more common than can be explained by Bachelier’s model. However, as Bachelier’s form of a Gaussian price-change
histogram was still able to fit most of the available data,
it led some mathematicians and economists to dismiss
the fat tails as “outliers” or “tsunamis”. Being so rare,
they argued, the outliers can simply be ignored.

Expect the unexpected
Most physicists will find it incredulous that somethingthat does not fit into a theory should be dismissed asmerely an outlier. And econophysicists share that viewtoo. But econophysicists have another trait – a reluctance to construct a theory before analysing as much
relevant data as exist. Collecting significant amountsof data in traditional areas of physics often requires
years of painstaking labour, but nowadays vast quantities of financial data are available, often free of charge.
In the late 1990s Paramaswaram Gopikrishnan andVasiliki Plerou, who were then graduate students working at Boston University in the US, decided to analyse
every transaction of every single stock in the major USmarkets. At that time, the analysis of such huge data
sets was not as commonplace as it is today and required
a significant upgrade to their university’s computer
system to complete the task. Using the extra computerresources, the two students constructed a histogramthat displayed the number of times the stock market
changed by a certain amount, plotted as a function ofthat amount. They did this by analysing 1000 different
stocks each consisting of 200 000 data points.
What Gopikrishnan and Plerou found was that largetransactions are more common than they had expected,
with the tail of their histogram not being Gaussian butfollowing an “inverse quartic power law”. This law
30means that if there is a probability p of, say, a $5 pricechange occurring, then the probability of a $10 pricechange is p/24, i.e. p/16. This inverse quartic law excels
at describing the probability of very rare events, suchas those occurring once every few decades. Events corresponding to 100 standard deviations, for example,
have a probability of about 10–350 with a Gaussianmodel, but a far more realistic likelihood of 10–8 (i.e.
one in a hundred million) with the inverse quartic law.
This latter value is confirmed by real data. For example,Gopikrishnan and Plerou typically found two events
corresponding to 100 standard deviations in a databaseof 200 million entries.
Although the inverse quartic law accounts for theseincredibly rare events, as well as for more commonplace events, it is still not popular with some economists– particularly those who demand a complete theory
that explains the causes before accepting empirical
data. Physicists are not, however, accustomed to waiting for a fully formed theory before reporting newresults. After all, the theoretical basis of many physicalphenomena is often not understood for years after anexperimental observation – and sometimes not even atall. To see this, you only have to look at the huge number of papers streaming out about high-temperature
superconductivity despite the fact that we have noaccepted theory for this phenomenon at all.

Shock and panic
The discovery of the inverse quartic law was a consequence of an intense study of a huge number of financial-transaction data. However, the law had one bigflaw: it did not pay any regard to when the transactions
took place. Individual price changes, in other words,
were simply collected in the bins of a histogram and all“time ordering” of the data was lost. But can we findlaws to describe this time ordering? It is a crucial question for econophysicists seeking to understand stockmarket behaviour, which rises and then suddenly falls
Physics World May 2011

Feature: Econophysics

physicsworld.com

2 Peaks and troughs
ba

1.0
τ*v*

1.4
1.2

0.9
0.8

1.0
0.2
0.6
1.0
1.4
1.8
renormalized time ε between successive extrema

0.20.61.0
1.4
1.8
renormalized time ε between successive extrema
dcβ+v = –0.15

0.00
log10 τ∗log10 v∗

0.20

0.10
β–v = –0.07–0.05

β–v = 0.09

–0.10

β+v = 0.12

0.00
–2.8

–2.4

–2.0
–1.6
log10 |ε – 1|

–1.2

–1.6

–1.2
–0.8
log10 |ε – 1|

–0.4

Whenever a stock market switches from a rising to a falling trend or vice versa, (a) the volume, v*, of transactions peaks, while (b) the time between
transactions, τ*, reaches a minimum. These aggregated volume and “inter-trade” times were obtained by averaging all trends in data from theGerman DAX Future stock market, with the “renormalized time” on the x-axis running from the start of a trend (ε = 0) to the end (ε = 1). Price maxima
and price minima coincide with peaks in the volume and dips in the inter-trade time. Analysis also reveals a power-law behaviour as seen from(c) this log–log plot of averaged transaction volumes versus ε both before reaching a switching point (ε < 1, circles) and after reaching a switching
point (ε > 1, triangles). Power-law behaviour is also revealed by (d) this log–log plot of the averaged inter-trade times. β is the value of the slope.

in a manner that cries out for an explanation.
To analyse the time behaviour of financial data, what
the current authors have done is to analyse massive
data sets comprising three fluctuating quantities. These
are: the price of each transaction; the number of exchanged shares in each transaction (i.e. the volume);
and the time between each transaction and the next.
Our goal was to find out if there are general laws todescribe market behaviour near the “switching points”
in the data – these are either local minima where theshare price falls before starting to rise again (alsoknown as an “uptrend”) or local maxima where theprice peaks before falling (a “downtrend”). In otherwords, are there regularities either just before or justafter a switching point?
To find out, we analysed a time series of the German
DAX Future stock market, which trades “futureshares” of the top 30 German companies trading on
the Frankfurt Stock Exchange (2010 J. Stat. Phys. 138431). It comprises 14 million transactions recordedwith an incredible resolution of only 10 ms, whichallows us to analyse in detail all “microtrends” occurring. We also looked at “macrotrends” over periodsof up to 100 days by analysing 2.6 million daily closing
prices of stocks in the S&P 500 share index in the US.
What we first needed to do, however, was find a way oflooking at all the data at once, without having to worry
about the fact that the time between switching points
can vary greatly from minutes to hours. Our solutionwas to “renormalize” the data by simply assigning atime ε = 0 to the start of an uptrend and ε = 1 to theend of an uptrend (figure 1). We also applied thismethodology to negative trends, where ε = 0 and ε = 1
Physics World May 2011

correspond to the start and end of a downtrend, respectively. Taken together, the two analyses let us lookat positive and negative trends ranging in time overnine orders of magnitude (i.e. from 10–2 s to 107 s).
Financial markets and their participants are wellknown to be hugely complex systems in which individual traders interact with one another through various
mechanisms. For instance, traders are explicitly awareof all price fluctuations, and use this information whenplacing their individual orders – trying to guess when
the price has a local maximum so they can sell at thebest possible time, and trying to guess when the priceis at a local minimum so they can buy at the best possible time. Although people have never quite been sureif this “panic hypothesis” is really true, we found plentyof evidence to back that view up. In particular, ouranalysis revealed that the volume of each transaction
increases dramatically as the end of a trend is reached(figure 2a), while the time interval between each transaction drops (figure 2b). In other words, as prices start
to rise or fall, stock is sold more frequently and in larger
chunks. Traders become tense and panic because they
are scared of missing a trend switch.
Based on the results of both databases, we also foundto our surprise that there is a unique empirical powerlaw quantifying both transaction volumes and intertrade times in all the financial markets we analysed.
Even more surprising, we found the same exponent ontimescales varying over nine orders of magnitude, from
10 ms to 100 days (figure 2c and d). In other words, theformation of increasing and decreasing trends – bubbles and crashes – is scale-free in that the same law
holds over up to at least nine orders of magnitude.

As prices
start to rise orfall, traders
become tenseand panic
because they
are scaredof missing atrend switch
31

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Scale-free market fluctuations Data from the German DAX Future stock-market
index, with each curve showing just a tiny part of a data set consisting of 14 million
transactions. The curves reveal how the rise and fall of share prices behave in a“self-similar” way, looking the same over whatever timescale you choose to pick –in this case over intervals of 1 day (top), 1 hour, 10 minutes and 1 minute (bottom).
Local maxima and minima, or “switching points”, are marked as blue and red circles.

Respect the data
Our results are consistent with the idea that the wellknown catastrophic bubbles that occur over large timescales – such as the global financial crashes of 1929 and2008 – may not be outliers. Instead, they are single dramatic representatives caused by the formation of increasing and decreasing trends on timescales varying
over nine orders of magnitude from milliseconds toyears. Just as the minor, tiny earthquakes that occur all
the time turn out to be connected to the largest and mostdramatic quakes, so it may be possible to learn more
about the characteristics of dramatic crashes by studying
the gargantuan amount of data concerning tiny crashes.
On typical trading days, economists who stick withtraditional Gaussian models will have nothing to lose.
However, they open themselves up to a false sense ofsecurity not unlike those who ignore very small earthquakes: unless you work in a seismic detection centre,
you will be oblivious to the minor quakes occurring all
the time. But huge earthquakes, when they do happen,can cause devastation. The 2008 Sichuan earthquake
in China, for example, claimed nearly 70 000 lives,
while, at the time of writing, more than 27 000 people
have either died or remain unaccounted for after theearthquake in Japan in March.
Our experience might lead us to believe, naively, thata histogram of earthquake magnitudes will have abroad maximum for everyday earthquakes and a small
sharp maximum for devastating earthquakes. Researchers studying actual seismic data, in fact, do notfind this kind of “bimodal distribution” but rather apower law – the Gutenberg–Richter law. Yet while governments respect the real possibility of extremely rarehuge earthquakes by requiring that buildings should beappropriately designed and constructed, it is not clear
if policy makers equally respect the possibility of ex■tremely rare huge financial shocks.
Physics World May 2011