Econofisica: alcuni tratti di una scienza ibrida

OCS
Econoﬁsica:
Alcuni tratti di una scienza ibrida

Rosario Nunzio Mantegna
Palermo University, Italy

Observatory of Complex Systems
25/2/2010
Dip. Fisica - Perugia

1

2005
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2009

Observatory of Complex Systems
http://ocs.unipa.it
25/2/2010

2

Outline
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- Some changes affecting the society and science

- The modeling of economic complex systems with
concepts and methods from physics. Three examples:

1) the filtering of information present in the

return dynamics of a stock portfolio;

2) the high frequency strategic action of economic

actors trading in a financial market;

3) the empirical evidence of specialization of market

members acting in a financial market.

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3

The IT revolution, Internet, the world wide web and its
structure (Google, Wikipedia, etc) have provided, produce
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and allow access to an enormous amount of information.
From a project of Berkley University: http://www2.sims.berkley.edu

1,000,000,000,000,000,000 bytes —
10006, or 1018

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New approaches are pursued in the scientiﬁc practice and
in the social modeling:
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Changes in the scientiﬁc practice

Some disciplines which traditionally were characterized by
a low rate of production of scientiﬁc data have rapidly
moved to a status of disciplines producing a high rate
of data and information.

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Some disciplines which changed their status

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For example biology, medical sciences and
social sciences have changed their status and today
they are characterized by a huge rate of production
of scientiﬁc data.

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From “The Economist” April 18th 2009

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In short we are entering in a new era of scientiﬁc practice
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with a huge rate of scientiﬁc data production.

Cover of the special Nature issue of September 4th, 2008
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It is however not only a matter of the size of the information
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which is produced and available. The nature of data and the
associated data mining and data interpreting procedures raise
new challenges.

In most disciplines data produced are global and of
observational nature. This is quite different from what
was the standard in the XXth century when experimental
attention was localized and experiments where highly
controlled.

We therefore need new methodological approaches and new
techniques. Often the development of these new methods and
techniques emerges in an interdisciplinary (hybrid)
environment.

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11

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Hierarchically organized complex systems

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H.A. Simon, Proc. of the American Philosophical Society 106, 467-482 (1962)

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Philip W. Anderson’s complexity manifesto
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25/2/2010
Dip. Fisica - Perugia 14

A model complex system: The ﬁnancial market
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Cross-correlation between pairs of stock
returns are well-known in financial markets

They may be quantified by
the correlation coefficient ρij
Ln P(t)

ri rj − ri rj
ρ ij =
2 2 2 2
ri − ri r − rj
j

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15

Grayscale representation of the correlation
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matrix of a portfolio of stocks

n(n-1)/2
distinct
correlation
coefﬁcients

300 stocks
traded at the
US equity
markets in
2001-2003

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16

How to analyze the complexity of a
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correlation matrix?
Clustering e.g.
Hierarchical Clustering

Super Paramagnetic Clustering

Maximum Likelihood Clustering

Sorting Point Into Neighbors

Random Matrix Theory

Correlation Based e.g. Minimum Spanning Tree (MST)
Networks Planar Maximally Filtered Graph (PMFG)

M. Tumminello, F. Lillo, R.N. Mantegna, Correlation, hierarchies, and networks in
ﬁnancial markets, Journal of Economic Bahavior & Organization (2010)

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Hierarchical clustering

25/2/2010

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AXP
MER
0.664
By starting from a correlation matrix
IBM
MER
0.617
(which is a similarity measure)
SLB
OXY
0.592
BAC
MER
0.591
AIG
IBM
BAC
AXP
MER
TXN
SLB
MOT
RD
OXY
RD
OXY
0.590
AIG
1
0.413
0.518
0.543
0.529
0.341
0.271
0.231
0.412
0.294
TXN
MOT
0.582
IBM
1
0.471
0.537
0.617
0.552
0.298
0.475
0.373
0.270
IBM
TXN
0.552
BAC
1
0.547
0.591
0.400
0.258
0.349
0.370
0.276
AXP
BAC
0.547
AXP
1
0.664
0.422
0.347
0.351
0.414
0.269
AIG
AXP
0.543
MER
1
0.533
0.344
0.462
0.440
0.318
AXP
IBM
0.537
TXN
1
0.305
0.582
0.355
0.245
SLB
RD
0.533
SLB
1
0.193
0.533
0.592
MER
TXN
0.533
MOT
1
0.258
0.166
AIG
MER
0.529
RD
1
0.590
AIG
BAC
0.518
OXY
1
IBM
MOT
0.475
MOT
MER
0.462
MER
RD
0.440
AXP
TXN
0.422
.......
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OCS
One may obtain a simpliﬁed matrix by using classical
clustering methods such us the single linkage clustering

AIG
IBM
BAC
AXP
MER
TXN
SLB
MOT
RD
OXY

AIG
1
0.543
0.543
0.543
0.543
0.543
0.440
0.543
0.440
0.440

IBM
1
0.591
0.617
0.617
0.552
0.440
0.552
0.440
0.440

BAC
1
0.591
0.591
0.552
0.440
0.552
0.440
0.440

AXP
1
0.664
0.552
0.440
0.552
0.440
0.440

MER
1
0.552
0.440
0.552
0.440
0.440

TXN
1
0.440
0.582
0.440
0.440

SLB
1
0.440
0.590
0.592

MOT
1
0.440
0.440

RD
1
0.590

OXY
1

C<SL

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OCS
Or, for example, the average linkage clustering

AIG
IBM
BAC
AXP
MER
TXN
SLB
MOT
RD
OXY

AIG
1
0.501
0.501
0.501
0.501
0.412
0.308
0.412
0.308
0.308

IBM
1
0.536
0.577
0.577
0.412
0.308
0.412
0.308
0.308

BAC
1
0.536
0.536
0.412
0.308
0.412
0.308
0.308

AXP
1
0.664
0.412
0.308
0.412
0.308
0.308

MER
1
0.412
0.308
0.412
0.308
0.308

TXN
1
0.308
0.582
0.308
0.308

SLB
1
0.308
0.562
0.591

MOT
1
0.308
0.308

RD
1
0.562

OXY
1

C<AL

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Hierarchical clustering output in a typical case
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N = 100 (NYSE) daily returns 1995 -1998 C < = (ρ< )
ij

ρ < = ρα k
ij

€ where
αk
is the ﬁrst
€ node where
elements
€ i and j merge
together
Average Linkage Cluster Analysis

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Filtered matrix
N = 300 (NYSE); daily returns 2001- 2003

€

C < from ALCA C

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Correlation based networks

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Correlation based networks
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i
( i, j, ρij )
wij=ρij
j
1 3 0.90
 
 1 0.13 0.90 0.81
  1 4 081 
0.13 1 0.57 0.34  3 4 0.71
C = → S = 
0.90 0.57 1 0.71
  2 3 0.57
 0.81 0.34 0.71 1  2 4 0.34 
 
1 2 0.13
Correlation Matrix (C)
Sorted List of Links (S)
†R.N. Mantegna, Eur. Phys. J. B 11, 193-197 (1999)
€
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Minimum Spanning Tree
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Deﬁne a similarity measure between the elements of the system

Construct the list S by ordering similarities in decreasing order

Starting from the ﬁrst
element of S,
add the corresponding link
if and only if
the graph is still a Forest or a Tree

Minimum Spannig
Tree
(MST)

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Correlation based tree(s)
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For the single
linkage clustering
procedure the
correlation based
tree is the minimum
spanning tree

Correlation based trees and hierarchical trees do
NOT carry the same amount of information.

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Minimum Spanning Tree
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N=100 (NYSE)
daily returns
1995-1998
T=1011

G. Bonanno, F. Lillo and R.N.M., Quant. Fin. 1, 96 (2001)
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MST and Planar Maximally Filtered Graph
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Deﬁne a similarity measure between the elements of the system

Construct the list S by ordering similarities in decreasing order

element of S,
element of S,
if and only if
if and only if

the graph is still a Forest or a Tree

the graph is still Planar (g=0)

Minimum Spannig Planar Maximally
Tree
Filtered Graph
MST
PMFG

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Planar Maximally Filtered Graph
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N=100 (NYSE)
daily returns
1995-1998
T=1011

M. Tumminello, T. Di Matteo, T. Aste and R.N.M., PNAS USA 102, 10421 (2005)

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Network of crimes of a large set of
Swedish suspects

• Christofer Edling, (Jacobs University Bremen)
• Fredrik Liljeros (Stockholm, Sweden)
• Jerzy Sarnecki (Stockholm, Sweden)
• Michele Tumminello (Palermo)

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Sex offences
involving a
child
274 crimes
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are in the
largest
Human
crime
trafﬁcking
community
and
procuring
Illegal
immigration

Misuse
of ofﬁce

Environmental
offences
Tax offences

Work environment act
A large network comprising 330 crimes with 14 530 links
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Price formation in a double auction ﬁnancial
market

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A detailed representation of book dynamics in a short
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Price formation in a double auction market
period of time

- sell limit
AZN price (pence)

orders

- buy limit
orders
○ sell market
orders

x buy market
orders

time (s)
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Research challenges
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Price formation and
liquidity disclosure
in platform based
competing markets.

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Price formation in a ﬁnancial market

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Examples of databases:

Rebuild order book

Open book and TAQ

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We♩ have investigated the conditional spread decay
G(τ | Δ) and the relation between permanent ( I ) and
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immediate ( Δmo) price impact

♩A.Ponzi, F. Lillo, R.N.Mantegna, Market reaction to a bid-ask spread change: A power-law relaxation
dynamics; PRE 80, 016112 (2009).
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The permanent impact is a fraction of the immediate
price impact
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Data obtained by investigating 71 highly liquid stocks of the LSE

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The conditional spread decay is a power-law decay
suggesting the existence of a strategic placement
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of limit and market orders

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Indeed the rate of orders arriving into the market
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is a function of the value of the spread.

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Order placement also affects “time to ﬁll”♩
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♩Z. Eisler, J. Kertesz, F. Lillo and R.N. Mantegna, Diffusive behavior and the modeling of
characteristic times in limit order executions, Quantitative Finance 9, 547 (2009).

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OCS

GSK

AZN
First passage time
LLOY
SHEL
λ ≈ −1.5 VOD
Time to ﬁll

λ ≈ −2
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€

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Empirically detected resulting strategies

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Conceptual challenges
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The most basic assumption of idealized systems
used in economic theory.

In mainstream economics, the economic actor is described in terms
of a representative agent, which is:

- fully rational;

- has access to all available information;

- is able to process all information instantly and without errors.

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Equilibrium theory is a milestone of classic economic theory.
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However it is a static description.
Steve Smale, Mathematical Problems for the Next Century,
August 7, 1998, Second Version
“ Problem 8: Introduction of dynamics into economic theory.
The following problem is not one of pure mathematics, but lies on the interface of
economics and mathematics. It has been solved only in quite limited situations.
Extend the mathematical model of general equilibrium theory to include price
adjustments. There is a (static) theory of equilibrium prices in economics starting
with Walras and ﬁrmly grounded in the work of Arrow and Debreu (see Debreu, 1959).
For the simplest case of one market this amounts to the equation supply equals
demand and a natural dynamics is easily found (Samuelson, 1971). For several markets,
the situation is complex. ............
Problem 8 asks for a dynamical model, whose states are price vectors (perhaps enlarged
to include other economic variables). This theory should be compatible with the existing
equilibrium theory. A most desirable feature is to have the time development of prices
determined by the individual actions of economic agents.
I worked on this problem for several years, feeling that it was the main problem of
economic theory (Smale, 1976). See also (Smale, 1981a) for background.”
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Heterogeneity at the micro level
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The Journal of Political Economy, Vol. 109, No. 4 (Aug., 2001), pp. 673-74

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An empirical analysis of heterogeneous trading
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behavior: The Spanish stock market
 Market members are credit entities and investment firms
which are members of the stock exchange and are entitled
to trade in the market.

 Approx 200 market members at the BME (350/250 at the NYSE)
 We only study approximately 80 because:
 Not all the members trade during the whole period
 We have only chosen those members whose activity is
continuous

Snapshot of our database

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Market members vs agents
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Market members (MMs)
are not agents. A market
member may act on
behalf of many different
agents.

This could be due either
because a MM acts as an
intermediary or because
a MM is doing client
trading.

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Data
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We investigate 4 highly capitalized stocks: Telefonica
(TEF), Banco Bilbao Vizcaya Argentaria (BBVA),
Banco Santander Central Hispano (SAN) and
Repsol (REP)

The investigated period is 2001-2004

We investigate market dynamics by focusing on the
trading of each selected stock separately for each
available calendar year.

By doing so we have up to 4x4 distinct sets of results
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Investigated variable
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 Inventory variation = the value (i.e. price times volume)
of an asset exchanged as a buyer minus the value exchanged as a
seller in a given time interval.
t +τ
v i ( t ) ≡ ∑εi ( s) pi ( s)Vi ( s)
s= t
sign
+1 for buys
price
volume
-1 for sells
In this talk, we investigate the τ = 1 trading day
€ 25/2/2010

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Correlation between MMs’ inventory variation

min=-0.53
max=0.75
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Correlation matrix of MM inventory variation
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Is the cross correlation matrix of MM
inventory variation carrying information
about the market dynamics?
A random null hypotesis can be tested by
using Random Matrix Theory

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Eigenvalue spectrum
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1st eigenvalue

Shuffling threshold
2nd eigenvalue

RMT threshold

The first eigenvalue is not compatible with random trading and is
therefore carrying information about the collective dynamics of firms.
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Origin of collective behavior
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• Which is the meaning of the largest eigenvalue of the correlation
matrix of inventory variation?

• Principal Factor Analysis suggests that there is a factor which is
driving the inventory variation of many firms

• The presence of the collective behavior is not due to the fact that
some firms are buying and other are selling (shuffling experiment)

• Rather it suggests that there are groups of firms having
systematically the same position in the market as the other
members of the group they belong to.

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The factor driving inventory variation is
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significantly correlated with price return

Correlation between the factor and price return ranges between
0.47 and 0.74, being statistically significant at 99% confidence in
all 16 sets
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A taxonomy of market members

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Uncategorized
“noisy” ﬁrms
“trending” MMs
“reversing” MMs
(ex: momentum
(ex: contrarian
traders)
traders)

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A closer look on all ﬁrms in the 4x4 sets

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variation with stock return
Correlation of inventory

Block bootstrap validation
Block bootstrap validation

Lillo, Moro, Vaglica and RNM, New Journal of Physics, 10 (2008) 043019
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BBVA 2003
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Inventory variation R
correlation matrix
obtained by sorting
the MMs in the rows
and columns
according to their U
correlation of
inventory variation T
with price return

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The taxonomy is rather stable over the years

Categorization of active MMs for the Telefonica Stock

TEF 2001 2002 2003 2004

Reversing 43 39 42 37

Uncategorized 28 31 31 29

Trending 11 10 8 6

Total 82 80 81 72

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Specialization is stable over the years

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P(Y|X) is the probability that a MM of the group X
switches to group Y in the next year (data for
Telefonica stock averaged over 3 years)
X
Reversing Uncategorized Trending

Reversing 0.71 0.19 0.03

Uncategorized 0.16 0.62 0.35
Y

Trending 0.02 0.07 0.44

Exited 0.11 0.12 0.18

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Conclusions
OCS

 There is a growing role of the observational
approach in several scientiﬁc disciplines.

 Economics and social sciences are among
the disciplines with a high rate of data
production.

 In the modeling of ﬁnancial markets it is
essential to consider the intrinsic
heterogeneity of the economic actors.

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"The only hero able to cut off Medusa's
head is Perseus, who flies with winged
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sandals. ..... . To cut off Medusa's head
without being turned to stone, Perseus
supports himself on the very lightest of
things, the winds and the clouds, and
fixes his gaze upon what can be revealed
only by indirect vision, an image caught
in a mirror. I am immediately tempted to
see this myth as an allegory on the poet's
relationship to the world, a lesson in the
method to follow when writing."
Michelangelo Merisi
da Caravaggio
Italo Calvino, Six Memos for the Next Millennium
Head of Medusa (1598)
Vintage Books, Random House, New York 1988
Uffizi gallery
OCS website: http://ocs.unipa.it
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Econofisica: alcuni tratti di una scienza ibrida

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