Transcript Networks in Finance and Economics

DYNAMICS OF COMPLEX SYSTEMS
Self-similar phenomena and Networks
Guido Caldarelli
CNR-INFM Istituto dei Sistemi Complessi
[email protected]
3/6
•STRUCTURE OF THE COURSE
1.
SELF-SIMILARITY (ORIGIN AND NATURE OF POWER-LAWS)
2.
GRAPH THEORY AND DATA
3.
SOCIAL AND FINANCIAL NETWORKS
4.
MODELS
5.
INFORMATION TECHNOLOGY
6.
BIOLOGY
•STRUCTURE OF THE THIRD LECTURE
3.1) SOCIAL NETWORKS: E-Mails
3.2) FINANCIAL NETWORKS
3.3) STOCK CORRELATIONS
3.4) STOCK OWNERSHIPS
3.5) BOARD OF DIRECTORS
3.6) BANK LOAN NETWORKS
3.7) WORLD TRADE WEB
•3.1 SOCIAL NETWORKS: E-mails
We introduce a
model, based on
the idea of
preferential
exchange,
whose
applicability can
in principle be
extended to
other contexts.
The first data
sets available
are the mail
received by
ourselves.
Data about
traffic is very
difficult to
obtain.
•3,1 SOCIAL NETWORKS: E-mails
The data sets that we analyze are five e-mail directories coming from our
accounts and the accounts of two other colleagues. They contain 5628 e-mails
(corresponding to 393 senders) collected over roughly three years, 19219 emails (476 senders, ten years), 16102 e-mails (113 senders, three years), 13385
e-mails (516 senders, five years) and 21782 e-mails (207 senders, five years).
•3.1 SOCIAL NETWORKS: E-mails
Every agent sends an amount Mout of mails at any time-step
p( j  i ) 
The mail sent to i are proportional to those received
ki  j
k
l
It is possible to derive a rate equation for the number of mails kij
ki  j
t
 M out p( j  i)  M out
ki  j
k
l j
l
ki  j
t
 M out
ki  j
M int

ki  j
t

t 
 ki  j (t )   
 t0 
P(k )  k     1    2
l j
•3.1 SOCIAL NETWORKS: E-mails
We defined a model that reproduces the final distribution of mails.
The principle is based on PREFERENTIAL EXCHANGE.
The probability to send an e-mail to John is proportional to mails we received by John.
•3.2 FINANCIAL NETWORKS
WHICH NETWORKS ?
The cases of study where graph theory has been applied
Stock correlations
Stock ownerships
Trade between countries
Board of Directors
Inter-bank Market
NOT Network in economics (firm networks (Marro), Production
networks (Weisbuch) etc. ).
WHAT IN NETWORKS ?
Degree distribution
Community Structure
Robustness
WHY NETWORKS ?
Do we learn something? 
•3.2 FINANCIAL NETWORKS
Probably the most complex system is
human behaviour!
Even by considering only the trading
between individuals, situation seem to
be incredibly complicated.
Econophysics tries to understand the
basic “active ingredients” at the basis of
some peculiar behaviours.
For example price statistical properties
can be described through a simple
model of agents trading the same stock.
•3.3 STOCK CORRELATIONS
ri ( )  ln Pi ( )  ln Pi ( 1)
i , j 
r
Logarithmic return of stock i
ri rj  ri rj
2
j
 rj
di , j  2(1  i , j )
2
 r
i
2
 ri
2

Correlation between returns
(averaged on trading days)
Distance between stocks i, j
•3.3 STOCK CORRELATIONS
Spanning Trees,
Bonanno et al. Physical Review E 68 046130 (2003).
Correlation based minimal spanning trees of real data from daily stock returns of 1071 stocks for the 12-year
period 1987-1998 (3030 trading days). The node colour is based on Standard Industrial Classification system.
The correspondence is:
red for mining
green for transportation,,
electric,gas and sanitary services
black for retail trade
cyan for construction
light blue for public
administration
purple for finance and insurance
yellow for manufacturing
magenta for wholesale trade
orange for service industries
•3.3 STOCK CORRELATIONS
Real Data from NYSE
Correlation based minimal spanning trees of real data from daily stock returns of 1071 stocks for the 12-year period
1987-1998 (3030 trading days). The node colour is based on Standard Industrial Classification system.
The correspondence is:
red for mining
green for transportation, communications,
electric,gas and sanitary services
black for retail trade
cyan for construction
light blue for public
administration
purple for finance and insurance
yellow for manufacturing
magenta for wholesale trade
orange for service industries
“Topology of correlation based” G. Bonanno, G. C. , F. Lillo, R. Mantegna.PRE E 68 046130 (2003).
•3.3 STOCK CORRELATIONS
Data from Capital Asset Pricing Model
In the model it is supposed that returns follow
ri (t )  i  i rM (t )   i (t )
ri(t) = return of stock i
rM(t) = return of market (Standard & Poor’s)
i,i = real parameters
i, = noise term with 0 mean
Correlation based minimal spanning trees of of an artificial market composed by of 1071 stocks according to
the one factor model.
The node colour is based on Standard Industrial Classification system. The correspondence is:
red for mining
green for transportation, communications,
electric,gas and sanitary services
black for retail trade
cyan for construction
light blue for public
administration
purple for finance and insurance
yellow for manufacturing
magenta for wholesale trade
orange for service industries
•3.3 STOCK CORRELATIONS
Without going in much detail about degree distribution or clustering of the two graphs
We can conclude that:
the topology of MST for the real and an artificial market are greatly different.
Real market properties are not reproduced by simple random models
•3.3 STOCK CORRELATIONS
Graph from Threshold, Onnela et al. Eur. Phys. J. B 38, 353–362 (2004).
•3.3 STOCK CORRELATIONS
Planar Maximally Filtered Graph,
Tumminello et al. PNAS 102, 10421 (2005).
•3.4 STOCK OWNERSHIPS
Stock Ownership
Garlaschelli et al. Physica A, 350 491 (2005).
•3.4 STOCK OWNERSHIPS
The degree distribution is fat-tailed
•3.4 STOCK OWNERSHIPS
It is not only the topology that matters.
In this case as in many other graphs the weight of the link is crucial
w
2
ij
SI 
For every stock i you compute this quantity.
The sum runs over the different holders
•
If there is one dominating holder SI tends to one
•
If all the holders have a similar part SI tends to 1/N
j


  wij 


 j

HI ( j )  
i
2
2
ij
w


  wil 
 l

2
For every guy j you compute this quantity.
The sum at the denominator runs over the different holders of i
Then you sum on the different stocks in the portfolio
This gives a measure of the number of stocks controlled
•3.4 STOCK OWNERSHIPS
•3.5 BOARD OF DIRECTORS
multiple Interlock
Bipartite Graph:
• two different kind of graphs
• edges between different
groups.
Projection
Link vertices of the same group
Consider Weight !
Weight
Stock Ownership
Various, see Ref in Battiston et al. EPJB, 350 491 (2005).
•3.5 BOARD OF DIRECTORS
•3.6 BANK LOAN NETWORKS
Banks exchange
money
overnightly, in
order to meet
the customer
needs of liquidity
as well as ECB
requirements
Inter-Bank Loans
De Masi et al. In preparation.
•3.6 BANK LOAN NETWORKS
•EUROPEAN CENTRAL BANK provides LIQUIDITY to European
Banks, through weekly auctions.
•EVERY BANK must DEPOSIT to NATIONAL CENTRAL BANK the
2% of all deposits and debts issued in the last two years. This reserves are
supposed to help in the case of liquidity shocks
•2% value fluctuates in time and it is recomputed every month.
Banks sell and buy liquidity to adjust their liquidity needs and at the
same time tend to reduce the value of reserve.
We report here the analysis on 196 Italian banks (plus 18 banks from
abroad who interact with them) who did 85202 transaction in 2000.
•3.6 BANK LOAN NETWORKS
Vertices are banks and edges a single loan
•3.6 BANK LOAN NETWORKS
Most of the activity starts before the 24 the day where the 2% is computed.
So dynamics and therefore shape is originated by ECB policy
•3.6 BANK LOAN NETWORKS
Actually the
banks form
different groups
roughly related
to their “size”
when considering
the average
volume of money
exchanged.
•3.7 WORLD TRADE WEB
World Trade Web
The system is
composed by 179
countries
connected by
trade channels
exchanged in the
year 2000.
Boguña et al. PRE, 68 151011 (2003).
•3.7 WORLD TRADE WEB
Fitness for WTW
The fitness is
given by the
normalized Gross
Domestic Product
of the country.
Garlaschelli et al. PRL 93 188701 (2004).
•CONCLUSIONS
Financial Networks can help
1. In distinguishing behaviour of different markets
2. In visualizing important features as the chain of control
3. In testing the validity of market models
They might be an example of scale-free networks even more
general than those described by growth and preferential
attachment.
•CONCLUSIONS
•CONCLUSIONS
+
=
Random Graphs defined on Pareto distributed vertices
Naturally form scale-free networks.
(Scale-free networks arise also from Gaussian in specific
case)