The Microscopic Structure of
Financial Markets: a brief
introduction for physicists
M. Bartolozzia,b
• a Research Group, Boronia Capital, Sydney, Australia
• b Special Research Centre for the Subatomic Structure of Matter (CSSM),
University of Adelaide, Adelaide, Australia
Achievements & New
Directions in Subatomic
Physics,
Adelaide 15 – 19 Feb 2010
Outline
 (Econo)Physics
and finance
 A brief introduction to high-frequency
finance and market microstructure
 Signal detection, persistency and Antipersistency at short time scales
 Empirical features of market
microstructure and relaxation times in
the imbalance between demand and
supply
 Conclusions
What is Econophysics?
“ Econophysics is the application of typical
methods from physics to the study of the
financial markets, seen as a complex system.”
H. E. Stanley, Boston University,
Boltzmann Medal 2004:
“For his influential contribution to
several areas of statistical physics…”
However…it is nothing really that new!
E. Majorana, Scientia, Vol. 36, 58 (1942)
“On the value of the statistical laws in
physics and in social sciences.”
Physica A, Vol. 285, p. 1 (2000)
Exotic statistical physics with
applications to biology, medicine and
economics.
Physics and Finance: a very short history (I)
 R. Brown (1827)
-> Introduction of the concept of Brownian motion.
 L. Bachelier (1900) -> The concept of Brownian walk is applied to the Paris
Bourse.
 A. Einstein (1905), P. Langevin (1908), N. Wiener (1923), K. Ito (1944) ->
Development of stochastic calculus.
 B.B. Mandelbrot (1963) -> Levy or other “fat-tailed” distributions seems to closer to
empirical data that Gaussian distributions.
The ’80s -> The availability of electronic data start increasing exponentially thanks
to new technologies.
1997 -> “…the financial industry employs about the 48% of the new Ph.D. in
mathematics and physics…”, Nature, Vol. 393, 496 (1998).
1997 -> M. Scholes and R. Merton win the Nobel prize for the Black&Scholes –
Merton model for Option pricing (F. Black has passed away in the meantime).
Physics and Finance: a very short history
(II)
 1990 – Today -> Econophysics articles are published on different prestigious
journals such as Nature, Physical Review Letters, Physical Review E, Physica A,
European Physical Journal B etc…
 1999 -> The European Physical Society recognizes Econophysics as a new area
of research.
 2000 – Today
-> Different textbooks are published.
 2000 – Today
-> University courses are set up. Australia: ANU Canberra.
 2000 – Today
world.
-> Different Symposia and Conferences are held around the
Textbooks in Econophysics
J.P. Bouchaud and M. Potters – Theory of Financial Risk
and Derivative Pricing: from Statistical Physics to Risk
Management, Cambridge University Press (2003)
J. Voit – The Statistical Mechanics of Financial
Markets, Springer (2005)
R.N. Mantegna and H.E. Stanley – An Introduction
to Econophysics: Correlations and Complexity in
Finance, Cambridge University Press (2000)
High-Frequency Finance:
the state of art in automated trading
• High-frequency trading (HFT) broadly defines all the strategies which
holding period is less than one day
• In 2009 HFT accounted for about 60% of the total traded volume around
the world (and it’s growing!).
• HFT is suitable for all the liquid financial instruments (exchange rates,
equities, futures, options, fixed income)
• Why is HFT attractive?
- Relative low risk and constant steady profit.
- Low correlation with long term strategies and, therefore, allowing for
diversification.
Challenges in High-Frequency Finance
• Dealing with very large data bases (order of Tb) for backtesting
IT
• Speed of execution
• Identify a profitable strategy!
• Possible Frameworks:
- Regime Recognition
- Event Trading
- Statistical arbitrage
- Market Making
- Microstructure Trading
- Cointegration
- Machine Learning
Research
Measuring Long Memory: Hurst and his dam
The concept of Hurst exponent has been originally developed in the context of reservoir
control on the Nile river dam project: Hurst’s algorithm is known as R/S (or rescaled range)
analysis. H. Hurst, Trans. Amer. Soc. Civil Eng. 116, 770 (1951)



1

 (t )


t 1
t

X (t , )    (u )  
u 1


R( )  max X (t , )  min X (t , )
1t 
R / 
H
1t 
Persistency and anti-persistency in
non-stationary time series
Following the spirit of Hurst, H has been used in different contest as a benchmark
for persistency, anti-persistency or randomness in time series analysis. See E.E.
Peters “Chaos and Order in Capital Markets”, Wiley finance edition (1991), for
example.
• H=0.5
random process
• 0.5<H<1 persistency
• 0<H<0.5 anti-persistency
Hold on! Financial time series are non-stationary and
“fat”-tailed: they are “not well behaved”. What is the
consequence of this?
Hurst exponent in finance
During the years there has been a proliferation of methods to estimate the Hurst
exponent (mostly outside the financial contest):
• Rescaled Range (R/S)
• Periodgram Regression
• (m,k)-Zipf method
• Average Wavelet Coefficient Method
• ARFIMA estimation by exact maximum likelihood
• Whittle estimation
• Detrended Fluctuation Analysis
• Detrended Moving Average
However, no much attention has been dedicated to their “efficiency” when used on
non-stationary and fat-tailed data sets
For a general review:
T. Di Matteo, Quant. Finan. 7(1), 21 (2007)
How do they look like?
Hurst exponent in futures markets: is it
stationary?
The local Hurst exponent is local either in time and scale
H L (t )
Local Hurst exponent for S&P500 and DJ. L=8192 (~ 16 days). In orange a
confidence interval based on Gaussian fluctuations.
Hurst exponent in futures markets:
possible exploitation?
Time Dependent Dynamics: how
to predict the next Hurst?
Different distributions of the local Hurst exponent in three different periods (BP):
the market can be hardly stationary!
Expectation values for local Hurst
exponents
Indices
Commodities
Exchange rates
Fixed income
As the scale gets shorter we can move from H>0.5 to H<0.5. Moreover, the various
financial instrument seem to cluster in separate classes.
The EOD (and large deviations) issue
Indices
Commodities
Exchange rates
Fixed income
End of day (EOD) gaps can represent an important issue for the calculation of high
frequency quantities in finance: H is particulary sensitive to these large fluctuations
(most of the algorithm are!!)
M. Bartolozzi, C. Mellen, T. Di Matteo and T. Aste, Eur. Phys. J. B. 58 (2007) p.207-220
Modern Double Auction Markets: the Limit Order Book
Volume
----Market
Order
--(or
Market
Order
-- Limit
Order
-- limit) --
-- Cancellation--- Slippage--
gap
gap
gap
gap
Best Ask
Best Bid
Price
Tick Size:
Granularity
Mid Point Price
Bid
Ask
Limit Order Book
Volume
----Market
Order
--(or
Market
Order
-- Limit
Order
-- limit) --
-- Cancellation--- Slippage--
gap
gap
gap
gap
Best Ask
Best Bid
Price
Tick Size:
Granularity
Mid Point Price
Bid
Ask
Ultra high-frequency features of
modern double auction markets
LOB Dynamics on Large Scale (II)
Eurex Bunds Futures:
Note the Periodicity due to
the expiration of the
contract.
LOB Dynamics on Large Scale (III)
SPIFutures
LOB Asymptotic Distribution
Average distribution of the LOB for
QQQ and SPY. Exchange traded founds
that track Nasdaq and S&P500
Cumulative distribution of incoming
orders. The dashed line is a power law
with exponent 1.
M. Potters and J.-P. Bouchaud, Physica A, Vol. 324, p. 133 (2003)
Relaxation in the limit order book
imbalance
How long it takes for the limit order book to get back from an “out-of-equilibrium”
position to a new “meta-stable” equilibrium?
~
(t ) 
Nb
Na
i 1
Nb
j 1
Na
i 1
j 1
b
a
V
(
t
)

V
 i  j (t )
~
  [1,1]
b
a
V
(
t
)

V
 i  j (t )
Is the time required for a
volume imbalance to “relax”
back to the “meta-equilibrium”
position.
For the analysis we used tick-by-tick snapshots of the order book from 13/07/2004 to
08/12/2006 for the DAX (~5*107 and N=10) and the SPI (~6*106 and N=5).
Empirical PDFs of relaxation times
  0.1
  0



~ ]
P( )  exp[  / 
P ( )  

The distribution seem to be well fitted by the stretched exponential: this is actually
similar to what happens in many physical systems (k=0.4 in the inset)!
F. Alvarez, A. Alegria and J. Colmenero, Phys. Rev. B 44, 7306 (1991)
Possible explanations
Kohlrausch-WilliamsWatts equation
(KWW)


~
exp[ ( /  ) ]   e



  ( )d 
0
 ( )
Debye distribution
•Dielectric polarization decay
•Quasi-elastic neutron scattering
•Kinetic relaxations...etc
Theoretical justification :: The Levy stable distribution is a solution of the
previous equation for some parameter values.
Moreover, the KWW equation can be solved numerically once the value of
α as been fixed.
Summary and conclusion (I)

Implementation of HFT systems requires a lot of
care in terms of signal detection. High-frequency
data are not usually “user-friendly”.

Application of Hurst analysis for high-frequency
(1 min) futures contracts: emergence of
different dynamics related to the scale of
observation and the particular financial
instrument.

Practical Issues: - Non-stationarity – Spikes (EOD)
M. Bartolozzi, C. Mellen, T. Di Matteo and T. Aste, Eur. Phys. J. B. 58 (2007) p.207-220
Discussion and conclusion (II)
 Introduction to the “quark structure of finance”:
the limit order book
 Relaxation
times in demand and supply are
characterized by stretched exponential
distributions, analogously to those of many
physical systems…
M. Bartolozzi et. al., Proc. of SPIE, vol. 6802,680203, (2008)
http://aps.arxiv.org/abs/0712.2910,
The Microscopic Structure of
Financial Markets: a brief
introduction for physicists
M. Bartolozzia,b
• a Research Group, Boronia Capital, Sydney, Australia
• b Special Research Centre for the Subatomic Structure of Matter (CSSM),
University of Adelaide, Adelaide, Australia
Achievements & New
Directions in Subatomic
Physics,
Adelaide 15 – 19 Feb 2010
Financial Data
Price Index
P(t )
Logarithmic Price Returns
R(t )  ln[ P(t  1)]  ln[ P(t )]
Volatility
v(t )  R(t )
Standard & Poor 500 (S&P500) data set
from 3/1/1950 to 18/7/2003. N=13468.
DFA and Gaussian increments (I)
How accurate the DFA estimator really is? First we test it against well behaved
data, that is with fractional Gaussian increments generated via a wavelet
construction method proposed by Mayer and Sellan (Appl. And Comp.
Harmonic Anal., Vol. 3(4), p. 377 (1996)).
Ensemble average
fractional Brownian motion
DFA-1
DFA-2
L=1024
DFA and Gaussian increments (II)
DFA-2
H = 0.2
H = 0.3
H = 0.4
H = 0.5
H = 0.6
H = 0.7
H = 0.8
L=1024
0.22 (4)
0.31 (4)
0.40 (4)
0.50 (5)
0.60 (6)
0.70 (6)
0.79 (7)
In this case we have bias just for strong
correlations and the errors are power law
distributed with exp ~ 0.36.
DFA and Levy increments…a legend of “fat tails”
The distribution of i.i.d. variable DOES NOT converge only to Gaussian distribution:
there exist another important family of stable distributions with “fat”-tails: they are
called Levy distributions
lim L ( y, ) 
y
  (0,2)
1
y
1
α=2 Gaussian
The Levy distributions can be RESCALED
to collapse in a single distribution L* ->
Levy distributions are self-affine (fractal)
distributions
Mantegna & Stanley, Nature 376, p.46 (1995)
L ( y, )   1/  L ( 1/ y,1)   1/ L ( 1/  y)
A self-affine process (statistically):
f (  x )   ( ) f ( x )
Self-affine processes and Levy distributions
A mono-fractal self-affine process is defined as (fractional Brownian motion, for
example)
x(t )   ( ) x(t )
 ( )  H
The distribution of these processes at a certain scale t can be rescaled
t
P( x(t ), t )
G ( y , ) 
  t
G ( x(t ), t )
G ( y, ) 
 y

 y 
1
1

P
,1 
P 
 ( )   ( )   ( )   ( ) 
Fractal and Levy processes are
statistically self-affine and they
scale in the same way with
For λ=τ
H 
1

dx
P( x, t )  y ,
dy
 ( )  H   H
DFA and Levy… (II)
For Levy i.i.d. data α > 1 (H>0.5)
The “fat”-tails matter!
For the algorithm: Samorodnitzky and Taqqu, Stable non-Gaussian Random Process (2004)
H=0.6
H=0.7
H=0.8
L=1024
L=4096 L=16384
0.56 (8)
0.63 (10)
0.68 (12)
0.58 (7) 0.57 (5)
0.64 (8) 0.65 (7)
0.70 (10) 0.71 (8)
An Hurst exponent different from 0.5 does not mean necessary prsistency or antipersistency. Large, self-similar, increments enhance the “diffusion” of the time seires
and the scaling of the variance goes “as if” there exist serial correlations.
In practice, “fat tails” can enhance the value of H!
“Tails” in Futures Data
We test the relevance of the large fluctuations in data sets of futures data. In
particular we report an example on a small window with Bunds data. In this case we
cannot make ensamble average: concept of local Hurst exponent -> HL(t)
1 min Bunds futures (Jan 2003/Dec 2004)
L=1024, slide 10 min (~ 2 days)
The large fluctuations enhance the value of H.

 r (t )  g (t )


r (t )   g (t )
if
if
r (t )  0
r (t )  0
Limit orders and Levy distributions
Possible drawback with the previous interpretation: why a stable Debye distribution
should exist after all? The framework of stochastic processes comes to help in this
case. In fact the KWW is satisfied for
  L ,
Levy distribution with parameters:
  (0,1)

   1
NOTE: Levy distributions are fixed points for the addition of i.i.d. random
variables, xi, whose individual distributions are asymptotically a power law,
Pi(xi) ≈ |xi|-1-μ with 0<μ<2, that is, with infinite variance.
Moreover, the distribution of
relaxation times depends on
the initial imbalance
L , 1  L , 1 ( )
Herding effects?
Decrease of
heterogeneity?
Average relaxation time as a
function of the imbalance
For 0.1< k <0.6 the relation is basically linear.
~  1 
   
  
~


Some Credits
• Boronia Capital Research Group. In particular, the Order Book
group which includes: C. Mellen, F. Chan, D. Fussell, M. Fender
and D. Oliver.
http://www.boroniacapital.com.au
• The econophysics group at ANU and, in particular, T. Di Matteo
and T. Aste.
http://wwwrsphysse.anu.edu.au/~tdm110/econophysics.html
Persistency and anti-persistency in
non-stationary time series
Following the spirit of Hurst, H has been used in different contest as a benchmark
for persistency, anti-persistency or randomness in time series analysis. See E.E.
Peters “Chaos and Order in Capital Markets”, Wiley finance edition (1991), for
example.
• H=0.5
random process
• 0.5<H<1 persistency
• 0<H<0.5 anti-persistency
Hold on! Financial time series are non-stationary and
“fat”-tailed: they are “not well behaved”. What is the
consequence of this?
First we check the performance of the algorithm against Gaussian and Levy
increments, afterwards we apply our results to the analysis of different futures
indices
The Detrended Fluctuation Analysis: an
algorithm for non-stationary time
series


• The time series is divided in M=N/
non-overlapping boxes of equal length

• For each box we calculate the local
polynomial trend and then the
fluctuations around this trend are
calculated. The DFA is usually denoted
as DFA-p according to the polynomial
used for the detrending
• As the final step we
average the fluctuations at
each scale. Then the Hurst
exponent, H, is calculated via
the following scaling relation
Scale
Time (t)


2
1
i
i
F ( , p) 
x (t )  y , p (t )

 tithbox
i
F ( , p)
M

H
Some average estimates of the errors
Hurst exponent in futures markets
(III)
Distribution of local Hurst
exponents: emerging patterns (I)
Distribution of local Hurst
exponents: emerging patterns (II)