v - INFN

Statistical modeling, financial data analysis and applications
Venice September2013
Financial data Complexity:
Dependency, Networks and Scaling
Tomaso Aste
Financial Computing & Analytics group
• 
• 
• 
• 
http://fincomp.cs.ucl.ac.uk/
Academic Members
Expertise:
•  Philip Treleaven
big data analytics
statistical modeling
•  Robert Smith
algorithmic trading
•  Antoaneta Serguieva
extreme market risk models
high frequency trading
•  Sebastian del Bano Rollin
price formation
•  Donald Lawrence
behavioral finance
•  Denise Gorse
market microstructure
systemic risk
•  Guido Germano
operation risk
•  Christopher D. Clack
credit derivatives
stochastic analysis
•  Tomaso Aste
portfolio optimization
External Members
market microstructure
•  Guillaume Bagnarosa
agent based simulations
information flow
•  Ariane Chapelle
news impact
•  Piotr Karasinski
network theory
financial mathematics
•  Jessica James
statistical physics
•  William Shaw
applied statistics
MSc Programmes
(….)
•  Financial Risk Management (~40 students)
•  Financial Computing (~30 students)
•  Financial Mathematics (~30 students)
•  (new) Quantitative Regulation / Business data Analytics
Doctoral training
•  PhD centre in financial computing (~ 70 students)
Information
We are witnessing interesting times rich of information, readily
available for us all. Using, understanding and filtering such
information has become one of the major tasks and a crucial
bottleneck for scientific and industrial endeavors
Information content and flow are often associated with large degrees of
redundancy both in time (repeating and scaling patterns) and across
different variables (dependency and causality).
Financial markets are information processing systems where news
propagate and impact on orders, transactions and -ultimately- prices.
Information impacts different operators and different products in different
ways resulting in complex changes that can be seen at different
temporal scales and across different financial products in a complex
network of dependency.
Financial Signals Complexity
•  Several time-scales of the processes
•  Large number of variables
•  Several kinds of interactions
Complexity of each
variable over its
time evolution
Complexity of the
collective
dynamics of all
variables
Scaling
properties
Dependency/
Causality
structure
Complexity of each variable over
its time evolution
Single variable Complexity
1sec-5min
AAPL 1 sec data 3 Dic 2012
AAPL 1 sec data 3 Dic 2012
0
0
10
10
1
10
30
60
300
−1
10
−1
1
10
30
60
300
rescaled
10
relative freq
relative freq
H(2) = 0.34633
−2
10
−3
−3
10
10
−4
−4
10
−2
−1.5
−1
−0.5
0
lor−return
0.5
1
1.5
10
−1.5
2
1min-1h
AAPL 1 min data Nov 2012
F>
relative freq
−2
10
−2
10
−3
−3
10
10
−4
10 −3
10
−2
10
−1
10
Log−returns
0
10
og-returns for the stock prices of Lehman Brothers (left panel) and
reen lines mark respectively one, three and ten standard deviations
l region with the power law function F> (r) ⇥ r
. The estimated
−6
−4
−2
0
log−return
Log-return
2
4
1
2
3
4
5
6
7
8
9
10
12
15
20
25
30
35
40
45
50
55
60
0.5
1
1.5
−3
x 10
AAPL 1 min data Nov 2012
10
rescaled
10
−2
10
−3
10
6
−3
x 10
H(2) = 0.47532
−1
relative freq
−1
−1
0
log−return
0
10
10
−0.5
Log-return/τH(2)
0
10
0
−1
−3
x 10
Log-return
10
−2
10
−6
−4
−2
0
log−return
2
Log-return/τH(2)
4
1
2
3
4
5
6
7
8
9
10
12
15
20
25
30
35
40
45
50
55
60
6
−3
x 10
Scaling and multiscaling: the observation of financial fluctuations at different scales
(seconds, minutes, hours, days, months, years) reveals similarities and differences
showing (non-trivial) connections between different time-scales
Basic Materials
Complexity across time-scales
Quantification of scaling and multi-scaling
K q (τ ) =
x(t + τ ) − x(t)
x(t)
q
q
~ g(q)τ qH (q)
We developed a numerical tool to study directly the scaling properties of the data via the qth-order moments
of the distribution of the increments:
The generalized Hurst exponent method
http://www.mathworks.com/matlabcentral/fileexchange/30076
When H(q)=H constant and independent of q the process is uniscaling or unifractal, H coincides with the Hurst
coefficient or the self-affine index.
When H(q) is not constant the process is multi-scaling (or multi-fractal) and different exponents characterize the
scaling of different q-moments of the distribution.
The nonlinearity of qH(q) is a solid argument against the Brownian, fractional Brownian, Levy, and fractional Levy
models, which are all additive models, therefore giving for qH(q) straight lines or portions of straight lines.
T Di Matteo
Raffaello Morales, T. Di Matteo, Ruggero Gramatica, TA, “Dynamical Hurst exponent as a tool to monitor unstable periods in financial time series”, Physica A, 391 (2012) 3180-3189.
Jozef Barunik, T A, Tiziana Di Matteo and Ruipeng Liu, “Understanding the source of multifractality in financial markets”, Physica A, 391 (2012) 4234–4251 M. Bartolozzi, C. Mellen, F. Chan, D. Oliver,
T. Di Matteo and T.A, "Applications of physical methods in high-frequency futures markets", Proc. SPIE, Vol. 6802, 680203 (Jan. 5, 2008).
Ruipeng Liu, TA and T. Di Matteo, "Multi-scaling Modelling in Financial Markets", Proc. SPIE Vol. 6802, 68021A (Jan. 5, 2008).
M. Bartolozzi, C. Mellen, T. Di Matteo, and T.A, Multi-scale correlations in different futures markets, Eur. Phys. J. B 58 (2007) 207-220.
T. Di Matteo, TA and M. M. Dacorogna, "Long term memories of developed and emerging markets: using the scaling analysis to characterize their stage of development", Journal of Banking & Finance
29/4 (2005) 827-851.
T. Di Matteo, TA and M. M. Dacorogna, "Scaling behaviors in differently developed markets", Physica A 324 (2003) 183-188.
Financial signal are multi-scaling
log(K q (τ )) ~ qH (q)log(τ ) + log(g(q))
K q (τ ) =
H(q) depends on q
H(1) ≠ H(2) ≠ 0.5
multiscaling
T
O
N
.
….
x(t + τ ) − x(t)
x(t)
q
..
…
r
a
e
n
i
L
q
~ g(q)τ qH (q)
Deviations from pure Brownian motion are meaninghful
T. Di Matteo, TA and M. M. Dacorogna, "Scaling behaviors in differently developed markets", Physica A 324 (2003) 183-188.
0.2
1996
1998
2000
2002
t
2004
2006
0.2
1996
2008
1998
2000
2002
t
2004
2006
2008
Multiscaling, switching and market stress
Figure 3:
Weighted Generalized Hurst exponent H w (q = 1) as a function of time for American International Group (AIG).
Left panel:
t = 200 days time-window. Right panel:
t = 400 days time-window. The characteristic time is kept constant
at ✓ = 300 days in both plots. The points are reported in correspondence of the end of the time-window.
0.7
0.7
American International Group
0.5
0.7
0.6
Htw (1)
Htw (1)
0.6
Freddie Mac
0.4
0.7
Fannie Mae
0.6
0.3
2000
2002
2004
t
2006
2008
H tw (1)
H tw (1)
0.5
0.4
0.6
0.5
Lehman Brothers Holdings
0.4
0.3
0.5
2000
2002
2004
t
2006
2008
0.4
2000
H tw (1)
H tw (1)
Figure 4:
Left panel: weighted Generalized Hurst exponent as a function of time for American International Group (AIG).
Freddie
Mac
Fannie Mae
0.7
0.7
Right panel: weighted Generalized Hurst exponent as a function of time for Lehman Brothers Holdings (LBH). Note the abrupt
jump in the value of the GHE at the end of the time-period. The overlapping time-windows are
t = 750 days, with ✓ = 250
days.
The values are plotted in correspondence of the end of the time-window
(solid black line). The shaded areas around the
0.3
0.3
0.6 plot represent the sizes of the standard deviations.
0.6
tick-line
2002
2004
2006
2008
2000
2002
2004
2006
2008
t
t
0.5
0.5
interesting if we compare the two panels. According to [11, 12] these trends might suggest a transition from
The wGHE is
conveying important
information about the
stability of a company
and by tracking its
value in time one has
a further tool to
assess risk
a stable stage of the companies to an unstable one.
Other
same trend are
shown in Fig.6. Again the trend is increasing
0.4 bailed-out companies which show the w
0.4
Figure
5:
Weighted
Generalized
Hurst
exponent the
H (q
= of
1) as
function of time.
Left
panel:
Mac. Right panel:
and crossing
over the
value of
0.5 towards
end
thea time-period
when
the
crisisFreddie
fully unfolded.
Fannie
Mae.
The increasing
trend over
the
whole
period highlights
a transition
values of
H w (1)
< 0.5
tofinancial
values of
We
have
compared
these
results
with
those
obtained
by
looking
at otherfrom
companies
either
from
the
w (1)
0.3
0.3
H
>
0.5.
This
suggests
a
progressive
change
in
the
stability
of
the
companies
under
study.
sector or belonging to other market sectors to test the significance of these results. For example, in the Basic
Materials
sector,2002
we find2004
many companies
whose dynamical wGHE
in
time, thus
an
2000
2006
2008
2000decreases
2002
2004
2006exhibiting
2008
opposite behavior to that shown
by the bailed-out companies in the financial sector. tAn example is reported
t
in Fig.7 where the dynamical wGHE’s for two companies belonging to the sector of Basic Materials are
shown. We notice a very definite overall decreasing trend, as if the companies securities gained persistence
Motors
Washington
Mutual
Corp
0.75: General
0.7
Figure
Weighted
Generalized Hurst exponent H w (q = 1) as a function
of time. Left
panel:
Freddie Mac. Right panel:
5
Fannie Mae.
The increasing trend over the whole period highlights
a transition from values of H w (1) < 0.5 to values of
H w (1) > 0.5. This suggests a progressive change in the stability of the companies under study.
0.5
0.7
0.6
H tw (1)
H tw (1)
0.6
General Motors
0.4
0.7
Washington Mutual Corp
0.4
H tw (1)
0.6
H tw (1)
0.6
0.3
0.5
0.4
0.5
2000
2002
2004
t
2006
0.3
0.5
2008
0.4
2000
2002
2004
t
2006
2008
0.3
0.3
Noble
Energy
Inc
Occidental
Petroleum
Corp
0.7
0.7 of
Figure
6:
Weighed
Generalized
Hurst exponent H w (q = 1) as a function
time. Left
panel: General
Motors, a company
2000
2002
2004
2006
2008Its bankruptcy was classified
2000 as 2002
2004
2006
2008 Right
that went bankrupt
following
Chrysler
in
June 2009.
the fourth
largest in
U.S. history.
t
t
panel: Washington Mutual. The increasing trend over the whole period highlights a transition from values of H w (1) < 0.5 to
w
values of H (1) > 0.5. This suggests a progressive change in the stability of the companies under study.
0.6
0.6
H tw (1)
H tw (1)
Figure 6:
Weighed Generalized Hurst exponent H w (q = 1) as a function of time. Left panel: General Motors, a company
that went bankrupt fol lowing Chrysler in June 2009. Its bankruptcy was classified as the fourth largest in U.S. history. Right
w (1) < 0.5 to
panel:
Washington
The increasing
the
whole period
highlights
transition
from values of
in
going
through Mutual.
the period
of crisis. trend
This over
is in
agreement
with
what ahas
been considered
asH
the
boost of
values of H w (1) > 0.5. This suggests a progressive change in the stability of the companies under study.
0.5
0.5 turning away from the financial sector.
the commodities
market during the crisis, where investors were
There are other sectors that have revealed instead no particular trend in the dynamical wGHE. We stress
in going
period sector
of crisis.
This
is increasing
in agreement
with
what for
hasthe
been
considered
as the boost
of
that
eventhrough
in the the
Financial
itself,
the
trend
found
bailed-out
companies
is not
0.4
0.4
the commodities
market
during the
crisis,
where investors
were turning
away
from
financial
sector.
common
to others;
for instance,
many
companies,
like American
Express
Co
andthe
Morgan
Stanley
show
There behaviors,
are other sectors
that have
revealed
instead
no particular
trend
the
dynamical
wGHE.
We stress
stable
with wGHE
values
steadily
fluctuating
about 0.5.
We in
will
see
in the next
paragraph
that
that
even in
the Financial
sector
itself,
thedynamical
increasing
trend are
found
theshowing
bailed-out
companies
the
sectors
exhibiting
a defining
trend
in the
wGHE
alsofor
those
extreme
values is
in not
the
0.3
0.3
common to others; for instance, many companies, like American Express Co and Morgan Stanley show
tail exponents of their distributions of returns. Although the increase or decrease of the wGHE is not simply
stable behaviors, with wGHE values steadily fluctuating about 0.5. We will see in the next paragraph that
related
with
the return
only,
behaviors
are
associated
with
the showing
fluctuations
of the
log-returns
the sectors
exhibiting
defining
trend
in the
dynamical
wGHE
are
also
those
values
2000
2002 a statistics
2004
2006 both
2008
2000
2002
2004 extreme
2006
2008in the
distributions.
tail exponents of their distributions
of returns. Although the increase or decrease of t
the wGHE is not simply
t
related with the return statistics only, both behaviors are associated with the fluctuations of the log-returns
R Morales
distributions.
6
Raffaello
Morales, T. Di Matteo, Ruggero Gramatica, TA, “Dynamical
Hurst exponent as a tool to monitor unstable periods in financial time series”, Physica
Figure 7: Weighed Generalized Hurst exponent H w (q = 1) as a function of time for: Left panel - Noble Energy Inc.; Right
6
A,panel
391 (2012)
3180-3189.
- Occidental Petroleum. The time-window is taken to be
t = 750 days and ✓ = 250 days.
Decomposing financial signals
3.48
3.46
3.44
3.42
3.4
3.38
3.36
3.34
0
50
100
150
200
250
300
350
400
Empirical Mode Decomposition
1.4
0
50
100
150
200
2
a (t) = Cτ (t)
1.2
250
2 H (t )
300
350
400
(Hilbert-Huang Transform)
1
0.8
0.6
0.4
0.2
0
0
Instantaneous Hurst exponent
50
100
150
200
250
300
350
N Nava
400
Anomalous instantaneous scaling
2
log a (t) − 2H log τ (t)
Log prices
Log prices
3.6
3.625
3.58
3.62
3.56
Log Price
Log Price
3.615
3.61
3.605
3.52
3.5
3.48
3.6
3.595
3.54
3.46
0
100
200
300
400
Time
500
600
700
800
0
100
200
Instantaneous var/period
400
Time
500
600
700
800
Instantaneous var/period
-17
-15
IMF1
IMF2
IMF3
IMF4
IMF5
IMF6
-18
-20
IMF1
IMF2
IMF3
IMF4
IMF5
IMF6
IMF7
IMF8
-16
-17
var/period
-19
var/period
300
-18
-19
-21
-20
-22
-23
-21
0
100
200
300
400
Time
500
600
700
800
-22
0
100
200
300
400
Time
500
600
700
800
Complexity of the collective
dynamics of all variables
Measuring dependency and causality
X
•  Granger causality
•  Transfer Entropy
•  Partial Correlations
Y
Z
Non-linearity and kernel measure
•  Kernelized Granger/Geweke’s causality
•  Hilbert-Schmidt Normalised Conditional Independence Criterion (HSNCIC)
8.2. Applying to financial data – Bitcoin
•  Kernelized Transfer Entropy
Chapter 8. Application to real data
Ensemble of methods
TE bitcoin EURJPY
TE EURJPY bitcoin
1.2
1.1
20
1
0.9
30
0.8
40
0.7
0.6
50
1.2
10
time window number, from 1 to 68
time window number, from 1 to 68
10
1.1
20
1
0.9
30
0.8
0.7
50
0.6
0.5
60
0.4
2
4
6
8
10
number of lags, between 0 and 11
SRE
PRG
40
0.5
60
INF
1.3
1.3
0.4
2
4
6
8
10
number of lags, between 0 and 11
SRE − real stock returns
INF − inflation rate
PRG − production growth
IRE − real itnerest rates
IRE
A Zaremba
Information Filtering
Collective Dynamics
pi (t)
401 firms on the US equity market from 01/01/96 to
01/01/2009
data over a window of 250 days with 20 days steps
stocks prices
Dependency structure
Ci, j = (x i − x i )(x j − x j )
x i (t) = log(pi (t + τ )) − log(pi (t))
ρ i, j =
Ci, j
σ iσ j
N(N −1)
= 80200
2
€
I consider ρi,j as a similarity measure
N=64
N=10
N=20
N=5
€
€
O(N2) relations with a lot of redundancy
Distribution of
eigenvalues
λmax = 1+
1
1
+
Q Q
λmin = 1+
1
1
−
Q Q
€
€
€
€
p( λ) =
Q=
T
N
Q (λmax − λ)(λ − λmin )
2π
λ
only order N are significant
We must reduce connectivity without
tresholding
Simplifying Dependency Structure
Information Filtering
By keeping the network of important links only
How can we build such a network?
By embedding on a surface
Why surfaces?
• locally planar
• natural hierarchy
• elementary moves
• computability
Any network can be embedded on a surface!
The embedding of KN is possible on
an orientable surface Sg of genus
(N − 3)(N − 4) 
g ≥ g =


12
∗
any ΓN is a sub-graph of KN and can be embedded on Sg
G. Ringel, Map Color Theorem, Springer-Verlag, Berlin, (1974) cap. 4
P. J. Gilbin, Graphs, Surfaces and Homology, Chapman and Hall, 2nd edition (1981)
G. Ringel and J. W. T. Youngs, Proc. Nat. Acad. Sci. USA 60 (1968) 438-445.
The surface constraints the complexity of the network
€
(the degree of interwoveness)
Planar Maximally Filtered Graph
max(∑ ai, j I i, j )
Sort similarities form the largest to the smallest
Connect the first two nodes on the top line of the list
i, j
Delete the top line from the list
no
Is the resulting graph planar?
yes
no
Keep the edge
Discard the edge
Have we reached the maximum
Results number of edges?
Guido Previde Massara, University
College London
yes
[email protected] [email protected]
Tomaso Aste, University College London
Tumminello,
TA, T. [email protected]
Matteo, R.N. Mantegna, “A tool for filtering
Tiziana Di Matteo, King’s College London
tiziana.di [email protected]
http://www.mathworks.com/matlabcentral/fileexchange/27360
M.
information in complex systems”, PNAS 102 (2005) 10421-10426.
Tetrahedron Maximally Planar Graph
5th September 2013
1
Results and discussion
1
1
0.98
0.96
0.8
0.94
0.92
0.6
0.9
0.4
0.88
0.86
0.2
0.84
0
0.82
(a)
PMFG
(b)
(c)
(d)
Deltahedron heuristic
BMPG
(e)
BMPG+T1
(a)
PMFG
(b)
(c)
Deltahedron heuristic
(d)
(e)
BMPG
BMPG+T1
Algorithm 3: BMP algorithm
input : W — a correlation matrix
output: BMP — a filtered version of W respecting the planarity
constraint
/* Initialise a triangle T1 e.g. by using the highest
correlated vertices
*/
1 T1
Three vertices with highest correlation ;
2 VertexList
List of vertices of W not belonging to T1 ;
3 Calculate Gains(VertexList, T1 ) ;
4 n
number of vertices in VertexList ;
5 i
0 ;
6 while i  n do
7
(Vi , Tabc ) = argmax Gains(Vk , Txyz ) ;
8
Gains(Vi , :) = 0 ;
9
Gains(:, Tabc ) = 0 ;
10
Ta1 , Ta2 , Ta3
triangles created by the insertion of Vi ;
11
Calculate Gains(i + 1 to n, Ta1 ), Gains(i + 1 to n, Ta2 ), and
Gains(i + 1 to n, Ta3 ) ;
12
Evaluate Gain by implementing T1 over Ta1 , Ta2 , Ta3 and execute
13 end
14 return BMP ;
Figure 1: Execution times relative to PMFG of Deltahedron heuristics, DMPG,Figure 2: Sum of weights in the extracted planar graph relative to the sum
and DMPG+T1 on a number of matrices: (a) Real correlation matrix 395x395,of the highest 3(N
2) edge weights of Deltahedron heuristics, DMPG, and
(b) Real correlation matrix 395x395, (c) Real correlation matrix 395x395, (d)DMPG+T1 on a number of matrices: (a) Real correlation matrix 395x395, (b)
Real correlation matrix 395x395 (e) Symmetric random uniform matrix 395x395Real correlation matrix 395x395, (c) Real correlation matrix 395x395, (d) Real
correlation matrix 395x395 (e) Symmetric random uniform matrix 395x395
4 Results and
discussion
The novelty of the method is that we do not longer rely on any particular
ordering
but at
4.1 Analysis of the performance of the new algorithm with
every stage we calculate the gain that would be obtained by adding
any of
the
weighted
measures
remaining vertices inside any triangle, complexity is O(n2) and results improve PMFG
G Previde Massara
1
We have tested the new algorithm for a number of matrices of various sizes and
di↵erent composition. For every type of matrix we compare the time perfor-
market. For each market day in t 2 [ t + 1, T
t + 1] we
peripheral stocks (with the largest values of X + Y ) and
investigate the behavior of a selection of N = 300 stocks with
built a portfolio with either uniform weights or Markowitz
high capitalization and that have performed well over the preweights [4], with or without short-selling (in the present
vious year ( t = 250 market days, see details in Materials and
study this corresponds to a total of 7071 portfolios, with t 2
Methods). Specifically, we computed correlations over a win[ t + 1, ..., T
t + 1]). For each portfolio we observe the redow of six months, reducing the excessive influence of remote
turns, defined as rt (⌧ ) = [P rice(t + ⌧ )
P rice(t)] /P rice(t),
market shocks on present correlations by using exponential
over a year following the investment date (⌧ 2 [1, 250]). The
smoothing [5] (which assigns higher weights to more recent
performance of each investment strategy is measured by comevents and incrementally reduces weights to past events). We
puting the average r̄(⌧ ) and the standard deviation, s(⌧ ) of
then improved the estimator by computing the average correthe returns over the 7071 investment dates. We then used the
)
lation matrix with shrinkage [6] over a period of six months
‘signal-to-noise ratio’, r̄(⌧
, as proxy for performance: good
s(⌧ )
obtaining in this way a robust estimation of the correlations
over the year preceding the investment day t (see details in
Materials and Methods). Such a matrix shows a remarkable
F Pozzi
persistence, with autocorrelation
values ranging around 50%
1.0
1.0
P
m= 5
m= 10
P
even after one year1 . This high persistence
is a very impor0.8
0.8
tant fact implying that measurements from the past are likely
to forecast the future and the ordering of the correlations is
P
0.6
0.6
expected to remain rather stable.
We then used these average
r̄
r̄
weighted correlations with shrinkage
s
s
C to construct the financial
0.4
0.4
filtered networks: M ST and P M F G [2, 3, 7]. An example of
C
C
P M F G is shown in Figure 1.
C
0.2
0.2
We now discuss how an efficient investment strategy can
benefit from the knowledge of such market dependency struc0.0
0.0
0
42
84
126
168
210
252
0
42
84
126
168
210
252
Market days
Market days
P
ture. In particular
we built portfolios from a set of stocks
Portfolio
performance
C
1.0
1.0
selected from the peripheral regions of the financial filtered
m= 20
m= 30
networks and we compared the performance
of these portfoC
0.8
0.8
C
lios with
the
performance of Cportfolios built from Ca selection
C
C
C
of central stocks andC other portfolios
made with randomly se0.6
0.6
C or built by using other traditional methods.
lected stocks
To
C
r̄
r̄
C
s
s
C
C
this purpose, we first must distinguish between
stocks lying in
0.4
0.4
the networks’ central
regions from those lying in the periphC
C
C
eries. Numerous
centrality/peripherality measures have been
0.2
0.2
C
P
proposed in the literature [9, 10, 11, 12, 13]; they reflect difP
P happen that a vertex results central
0.0
0.0
ferent criteria and it can
0
42
84
126
168
210
252
0
42
84
126
168
210
252
Market days
Market days
C
for one measure
and peripheral for another. We have thereP
fore adopted an ‘agnostic’ perspective by looking at some of
Fig. 2. Comparison between the performance of di↵erent portfolios with uniform
6
6
6
the most common centrality/peripherality measures
(namely
P
weights (u) composed of m = 5, 10, 20, 30 stocks. The symbol ⇤ indicate portDegree (D), Betweenness Centrality (BC ), Eccentricity (E),
folios made with m most peripheral
stocks (i.e. with largest
X + Y ). O indicate
85
85
85
C
portfolios made with the m most
central stocks (i.e. with80smallest X + Y ). These
Closeness (C ) and Eigenvector Centrality (EC ) [13] and P
by
80
80
performances are compared with:
(/) portfolios made of m
randomly chosen stocks;
combining them in order to better identify central and pe75
75
75
(.) portfolios made with the 70
m stocks that have achieved70the best performance over
70
ripheral stocks in the financial filtered networks. Specifically,
C
P
the period preceding the investment date. The (tick line) is a ’market portfolio’ made
we constructed two hybrid centrality indices, X and Y , which
40 is the same across the four
40
MKT it
MKT
MKT
by40 taking
all 300 stocksP−pand
figures.
P−p
P−p
30
30
30
group together the rankings of the previous measures (see deM−p
M−p
M−p
20
20
20
PM−p
PM−p
PM−p
10
10
10
P−c
P−c
P−c
tails in Materials and Methods). In terms of these hybrid
5
5
5
M−c
M−c
M−c
PM−c
PM−c
PM−c
C
1.0
1.0
measures, Csmall values of (X + Y ) are associated with central
Probability
of
negative
returns
m=
10
m=
20
Pvertices whereas large values are associated with peripheral
P
12
12
12
0.8
0.8
vertices. From
the study
of the variation with time
P of these
C
C
85
85
85
centrality indices we observed that central vertices are sta0.6
0.6
80
80
80
C
P
ble with a large likelihood to be persistentlyP observed
in the
r̄
r̄
75
75
75
s
s
P
center over time.
Whereas peripheral vertices tend to be less
0.4
0.4
70
70
70
stable with a larger variability. [un po’ di piu’ e un po’ piu’
2
40
40
40
MKT
MKT
MKT
0.2
quatitativo ] We observe that, in terms of industrial sectors ,
P−p
P−p
P−p
30 0.2
30
30
M−p
M−p
M−p
20
20
20
PM−p
PM−p
PM−p
10
10
10
the peripheries are mainly populated
by companies belonging
P
P−c
P−c
P−c
5
5
5
M−c
M−c
M−c
0.0
PM−c 84
PM−c 0.00
PM−c 210
0
42
126
168
210
252
42
84
126
168
252
to “Electric, Gas, and Sanitary Services” (representing 20%
Market days
Market days
of peripheral companies vs. 11% of all companies), “Oil
and
Fig. 5. Probability of non negative returns (expressed in per-cent values) after six months from the date when the investment was made (upper panel)
F. Pozzi,
T. Di Matteo, and
TA , “Spread
of risk “Petroleum
across financialRefining
markets:
the
peripheries”,
Reports
1665.
Fig. in3.
Comparison
between
performance
di↵erent
portfolios
obtained (ns
by) and wi
Gas Extraction”
(7.0%
vs. 4.8%),
and
from the better
date
whento
theinvest
investment
was
made
(lower panel).
TheScientific
cases the
with uniform
weights (3
uof
),(2013)
Markowitz
solutions
with no short-selling
Efficient diversification / risk hedging
Correlation networks can be used for efficient portfolio differentiation
by selecting stocks from the periphery of the PMFG
P
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SA
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GR
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CTL ALL
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LIZ
ALL
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SA
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0.4
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0.2
MSFT
TAP
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PHA
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D
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BSC
DE
BNI
ARCINF
MOT
NC
DLX
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ABT
HTMXQ
DOW
GT
EQT
2004
2006
2008
JEC
AM
BSETHRS
AGC
T. Aste, W. Shaw and T. Di Matteo “Correlation structure and dynamics in volatile markets”, New Journal of Physics 12 (2010) 085009 1-21.
T. Di Matteo, F. Pozzi, T. Aste, "The use of dynamical networks to detect the hierarchical organization of financial market sectors", Eur. Phys. J. B 73 (2010) 3–11.
Complexity reduction: DBHT
We extract clusters and hierarchies form the PMFG in 5 main
steps built around the properties of 3-cliques in maximal planar
v4!
graphs
v4!
Won-Min Song, T. Di Matteo, TA, Nested
hierarchies in planar graphs, Discrete
Applied Mathematics 159 (2011)
2135-2146.
k2!
{v2,v4,v5}!
v4!
k1!
{v2,v3,v4}!
v2!
v7! v5!
v2!
v9!
3-cliques
v8!
v6!
v5!
v2!
v3!
v4!
k3!
{v3,v4,v6}!
v6!
v3!
v3!
Some cliques contain inside others providing a natural hierarchy
Example of PMFG for Eurodollar rates
3-cliques on Maximal
Planar Graphs have a
unique property:
They contain other
cliques inside or/and
they are contained
inside the other
cliques.
Eurodollars
≥ 2 years
16 interest rates
with maturity dates
between 3 months
and 4 years.
T. Di Matteo and T. Aste,
"How does the Eurodollar
Interest Rate behave?",
Journal of Theoretical and
Applied Finance, 5 (2002)
122-127. (arXiv:cond-mat/
0101009, 2001).
< 2 years
DBHT
The clique structure provides automatically a classification
into communities organized into a nested hierarchy
v5!
v4!
v4!
v7! v5!
v2!
v6!
v7! v5!
v8!
b3!
v7!
v2!b3! v
4! k
2!
v6!
b2!
v8!
b1!
v3! v2!
v3!
v1!
v2!
v4!
v2!
v4!
k1!
v3!
k3!
!
α
v
v3!v5!6!
v9!
k3!
v4!
v6!
b4!
v8!
v3!
!
β
b
2!
bα= b1!
k1!
k2!
bβ = b4!
Directed Bubble Hierarchical Tree
v1
!
v2
bα!
!
v3
v5
b2!
!
v4
v6
bβ!
!
v8
v7!
b3!
!
We capture both local clustering and global hierarchical organization
without introducing any characteristic scale
Won-Min Song, T. Di Matteo, Tomaso Aste, Nested hierarchies in planar graphs, Discrete Applied Mathematics 159 (2011) 2135-2146.
W.M. Song, T. Di Matteo and T. Aste, “Hierarchical information clustering by means of topologically embedded graphs”, PLoS ONE, 7 (2012) e31929
WM Song
Clustering and Hierarchy - DBHT
PHA
PMFG-DBHT
16 clusters
CBE
NL
CN
BNL
MS
BW HLT
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KO
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DOW
CVX
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Kruskal-Wallis test p-values
0.000000
Energy
0.000000
Financial
0.000000
Technology
0.550930
Conglomerates
0.000001
Consumer Cyclical
12
oil industry
energy exploration & services
mining
energy production & distribution
rail roads
retailers & consumer products
car & transportation
pharmaceutical health beauty
beauty
food
semiconductors electronics
computers telecommunications
banks
retailers
14
constructions
16
finance & banking
meaningfulness
4
2
0
Two main factors contributing to financial
data complexity:
Complexity of
each variable
over its time
evolution
Complexity of
the collective
dynamics of
all variables
Are them related?
3
Are dependency Hierarchy and Multifractality related?
he hierarchical order is defined as the cardinality of i , ni = card( i ). In the example reproduced in
e stock
labeled by i,T.we
2, 4,
5, 8, 10}, i = {a
, a8 , aScaling
set of nodesof Financial Time Series Are Related
i = {1,
1 , a2 , a4 , a5and
10 }, i.e. the
R Morales,
Dihave
Matteo,
TA,
Dependency
Structure
Properties
red dots and ni = 6.
q-fin arXiv:1309.2411
Hierarchical order
e of hierarchical structure. The path highlighted in red is
Multifractality
ΔH=H(2)-H(1)
i,
while the thick red bullets are the nodes
R Morales
H(1, 2) = H(1) H(2),
(1)
0.8
r q = 1, 2 is the generalised Hurst exponent computed from the linear scaling of the empirical16q-moments
Section for more details). We have removed from the analysis all stocks whose multifractality
cannot
14
y distinguished from zero, which correspond to weak multifractal behaviour and hence would not be
12Methods).
s context. The benchmark value for multifractal stocks has been set to H(1, 2) > 0.015 (see
0.6
0.4
!ρ" t
N c ,t
Dependency Hierarchy and Multifractality are related!
10
0.2
6
8
Financial Sector
Industrial Sector
0
6
0.05
0.04
2006
t
2008
2010
−0.2
2012
0.035
0.025
8
10
12
14
n
16
2004
8
0.035
Cluster no. 2
14
2006
t
2008
2010
2012
!n" t
0.025
0.04
0.015
0.035
0.02
0.03
8
10
12
14
16
18
n
14
16
14
16
Cluster no. 3
0.02
0.01
120.035
0
0.03
110.025 −0.01
0.025
0.02 −0.02
0.015
0.02
10
0.015
0.015
6
12
0.03
0.04
0.045
∆H(1, 2)
!∆H (1, 2)" t
∆H(1,
2)
13
342 most capitalised stocks continuously
traded in the NYSE in the period 2-01-1997 to
31-12-2012.
10
0.045
0.03
0.05
0.02
!∆H (1, 2)"
6
4
2002
0.03
0.025
FIG. 4. Coalescence
of the hierarchy in time and dynamical correlation.
(Left) The number of cluster Nc,t as a
0.03
function of time is plotted in time in blue empty circles. The dashed red line is the best fit over the entire time period
0.025
2-01-1997
to 31-12-2012, while the magenta line is the fit over the shorter
period
preceding
the 2007/2008 financial crisis
0.02
0.05reported as too small to be visible. (Right)
September0.02
2002 through November 2007. Standard errors on the circles are not
Dynamical evolution of the average correlation (thick dots) in the same time period. The dashed coloured lines represent the
0.015
0.015
2.5%, 25%,
75%, 97.5%-quantiles, taken from the distribution of all the observed
correlation coefficients.
0.04
0.03
!∆H (1, 2)"
2004
∆H(1, 2)
NYSE
0.035
4 0.045
2002
∆H(1, 2)
0.035
20
0.01
8
10
12
14
16
−0.03
8
18
10
12
5 200610
20
n2008 15
2006n 2008 2010 2012
2002 2004
2010 2012
t
t
FIG. 3. Correlation
between
multifractality
and
hierarchical order
inin
single
sectors
and clusters.
We plot in
black
FIG. 3. Correlation
between
multifractality
and
hierarchical
order
single
sectors
and clusters.
We
plot i
dots
H (1, 2) against the hierarchical order n for stocks in the Financial (top left) and Industrial (top right) sectors and cluster
dots
H (1,22)
against
the
hierarchical
orderright).
n for In
stocks
in the
Financial
(top
left)
and
Industrial
(top
right)
sectors
(bottom
left)
and
cluster 3 (bottom
all plots
the blue
line is the
best
fit of
thediamonds
dots, while
the
red
squares
are theand
0.04
FIG.
5.
Average
multifractality
and
hierarchical
order
in
time.
(Left)
The
blue
are
values
of
the
average
0.12
5. Average
multifractality
hierarchical
order
in time.
(Left)
The
blue
arethe
values
of the
0.07
2 (bottom
left)
and
3 for
(bottom
right).
In all plots
the blue
line is the
best
fit of
thediamonds
dots, while
red squares
averages
ofcluster
H (1, 2)
each and
fixed
order.
0.05 FIG.
over
50 overlapping
time
windows.
The
dashed green
green line
the
best
fit over
the entire
period
onstration that multifractality and hierarchical order are positively
correlated.
(Color
We
averages
ofmultifractality
H
(1,online)
2)
each
fixed time
order.
multifractality
over
50for
overlapping
windows.
The
dashed
lineis is
the
best
fit over
the time
entire
time2-01-1997
period 2to 31-12-2012, while the blue line is the fit over the shorter period preceding the 2007/2008 financial crisis, September 2002
cles with error bars the multifractal indicator h H(1, 2)i averaged over
sharing through
the same
hierarchical
31-12-2012,
while
the blue
line
is the
fitred
over
theareshorter
preceding
the order
2007/2008
financial crisis,
Septemb
0.04 thetostocks
November
2007.
(Right)
circles
values ofperiod
the average
hierarchical
over 50 overlapping
time windows.
0.035The
0.06
C.
A Multivariate Dynamical Hierarchical Model
through
2007.
The
circles
values
of the2-01-1997
average to
hierarchical
orderthe
over
overlapping
the hierarchical
order n. The blue solid line is the linear fit over the averages,
while the November
orange
horizontal
dashed
0.1
The
dashed
green
line is the
bestred
fit over
the are
entire
time period
31-12-2012,
while
blue50
line
is the fit overtime
the w
p (Right)
dashed
greenFigure
line
ispreceding
the
best
fit2007/2008
over the
entire
time
period
2-01-1997
to 31-12-2012,
while
the blue
lineerror
is the
fit o
period
the
financial
crisis,
September
2002
through
November
2007.
In hierarchical
both
plots the
bars
limit up to which the increasing trend is observed. The error bars 0.03
are theThe
standard
errorsshorter
computed
as
s/
N,
from
LSE
(see
3
in
SM).
The
two
Asian
markets
show
a
much
wider
range
of
order
due
p
In order to explain the mechanism underlying the observed link between multifractality and correlation hierarchy we
are
the
standard
mean
error
on0.03
thefinancial
mean
s/ crisis,
N , withSeptember
swhose
the standard
deviation.
shorter
period
preceding
the
2007/2008
2002
through
November
2007.
In bothThis
plots
thebia
err
appearance
of
one
very
large
hub
in
the
correlation
network,
which
includes
most
the stocks.
hub
standard
deviation over the stocks having same hierarchical order.
introduce
a dynamical
hierarchical
model
(DHM),
main
novelty
with respect
to of
standard
multivariate
models
0.05
!∆H (1, 2)"
!∆H (1, 2)"
n
TSE detrended
TSE
LSE
!∆H (1, 2)"
2004
HKSE
p
!∆H (1, 2)"
n order
Hierarchical
2002
[5]
in
theerror
introduction
of a order.
perturbation
term
on
correlation
matrix
associated
to can
its hierarchical
structure.
evaluation
of lies
the
proper
hierarchical
bythe
detrending
the
time
series we
remove the
large hub
the standard
mean
on the mean
s/ NHowever,
, with
s the
standard
deviation.
0.02 are
0.08
model
returns
of stock
as
forWe
TSE
data
and only
fori the
at least
small hierarchical orders, retrieve a similar behaviour to that observed in
p
volatility
0.025
Figure
5 in factor
Supplementary
Material). As explained
in detail
in the SM though, detrending the series
se
igure 20.04
the mean value h H(1, 2)i with standard error s/ N0.01
(with2s (see
the standard
deviation
on theas
ri,t = ✏Y
(2)
i,t i,t ,
scaling
properties of the data and the overall trend
H (1, 2)i vs n is fundamental
K in the plots h
ch observed hierarchical order on all stocks analysed (see Methods fora↵ects
details). the
We observe
a positive
=
x
e
,
(3)
C.
A
Multivariate
Dynamical
Hierarchical
Model
i,t
t
0.06
Nonetheless
we mention
report inmultivariate
Figure 4 in
SM) that
without
detrending
series
positive
depe
where
✏i,t = (✏t )i(and
is a stationary
Gaussian
random
variable,
i.e. ✏t ⇠ Nthe
(0, ⌃)
with the
⌃ the
covariance
m2
etween0.03
the two variables up to n = 14, followed by some noisier0 flatbetween
trend. The positive
correlation
0.02
matrix
and
is is
a volatility
factor.
from
usual
multivariate
models,
H (1, 2)
and
observed
(for Di↵erently
small n) in
both
TSE
and HKSE
data.
i,t n
i,t is not common to all stocks but
depends
explicitly
on
the
hierarchical
structure
of
the
market.
We
suppose
there
is
a
volatility
factor
x
common
to
(1, 2)i and n is also confirmed by performing a t-test on the
correlation
validated
p-the
We
have
detected
samerandom
positive
dependence
betweenlink
hierarchical
order
and
multifractality
on specific
where
Kmwith
are
Bernoulli
variables
with
pm between
associated
to the
nodes
am and
along
thet hierarchical
In coefficient
orderalso
to
the
mechanism
underlying
the probabilities
observed
−0.01 sectors
all explain
stocks
and
then
a latent hierarchical
structure
of risks
am with
m
= 1,Q
. . .multifractality
,N
1, associated
withcorrelation
the nodes of hierar
the
(i)
(i)
and
on
clusters
found
through
DBHT
clustering
algorithm.
We
show
in
the
top
of
Figure
3
plots
K
0.04
3. All0.02
stocks with hierarchical order in the range [5, 14] (which accounts
for 90% ofatree
all
stocks)
exhibit
therefore
take
the form
= ✏whose
xtmain
, wherenovelty
Ystochastic
e
.autocorrelated
It istoworth
remarking
that the
the
dendrogram.
The
process x
chosen
as rai,tstationary
process
in time,
with
i . Returns
i,t Yn,t log-normal
t is
introduce
dynamical
hierarchical
model
(DHM),
respect
standard
multivariate
n,t = with
m2
0.015
multifractality
indicator
versus
the
hierarchical
order
computed
on 1stocks
to Financial
and Ind
autocovariance
as a power law,
i.e. Cov(x
, in line belonging
with a sweeping
amount of studies
(i)
t xt+h ) ⇠ h
properties increasing along with their depth in the hierarchy−0.02
of correlations.
OnThe
the
other
hand,
the function
hierarchical
term
Y
introduces
a single
richer
structure
ofthe
dependence,
where
the topology
of{a
the
risk
organisation
plays
[5]
lies in
the
introduction
of
a decaying
perturbation
term
on
correlation
matrix
associated
to
its
hierarchical
str
sectors.
black
dots
are
values
stocks
whereas
the hierarchical
red
squares
arei the
average
multifractal
ind
n,t
on multi
fractals
in
finance
[21,for
22].
For the
arbitrary
stock
i with
path
=
,
m
2
}
we
define
the
m
i
uration observed for orders larger than 14 suggests that the hierarchical
ofreturns
cross-correlations
forstructure
each order.
We
plot
We
model
of also
stock
i asthe best fit on the dots (thick blue line). Both sets of stocks show a very well d
0.02
0.01
between the hierarchical
order n and the multifractality
H (1, 2), which is eviden
nsible 0.01
for the multifractal
properties
of the stocks
only up to−0.03
a certainpositive
order.
Letcorrelation
us
in particular
5
10
15
2 indicator
4
6
8
10
5
10
15
20
5 positive
10 note
15
20
trend recovered in both examples. Again,
is found to be significant
with p-value p =
ri,tthe
= ✏dependence
n order n > 14 is too small to allowthe
n
n
n
i,t i,t ,
ber of stocks found with hierarchical
any
robust
statistical
conclusion,
In Figure 3 we also report the trends observed on two of the largest clusters found through the DBHT. Th
the reason why standard errors in Figure 2 are very large for n > 14. positive correlation between multifractality and hierarchical order is observed on the clusters best identifiable w
m
i
m
i
3
Model
he hierarchical order is defined as the cardinality of
e stock labeled by i, we have i = {1, 2, 4, 5, 8, 10},
red dots and ni = 6.
i,
i
ni = card( i ). In the example reproduced in
= {a1 , a2 , a4 , a5 , a8 , a10 }, i.e. the set of nodes
ri,t = εi,tσ i,t
σ i,t = xt ∏ e
Km
m∈γ i
0.25
!∆H (1, 2)"
e of hierarchical structure. The path highlighted in red is i , while the thick red bullets are the nodes
ch correspond to the risks stock i is exposed to. Note that the risk a1 is common to all stocks, while other
s of stocks only. The dashed branches of the dendrogram indicate an arbitrary hierarchy.
e conjecture that the hierarchical order ni can be actually viewed as a measure of the riskiness of
of its positive dependence with multifractality. This entails that the cross-correlation properties of
dataset are somehow entangled with the stylised facts displayed by the univariate time series. In
erarchical structure of correlation provides likewise a snapshot of the hierarchy of risks in the market.
ur knowledge, up to now no attempt to uncover this kind of underlying structure has been made in
II.
A.
0.2
0.15
RESULTS
Cluster detection and sectors
idered daily stock prices comprising the 342 most capitalised stocks continuously traded in the NYSE
01-1997 to 31-12-2012. Data have been provided by Bloomberg. The dataset includes stocks from
et sectors, according to the Bloomberg classification. The taxonomy of the stocks in the respective
in the Supplementary Material (SM), where we also report details on the clusters detected through
g algorithm. We have chosen to perform the clustering through DBHT after having verified that it
an other methods including the Single Linkage Cluster Analysis (SLCA) [33] in recovering a wellrchical structure: SLCA in fact tends to produce very large hubs of stocks that bias the correlation
er.
0.1
0.05
2
3
4
5
n
6
7
8
9
Conclusions and Perspectives
Complexity in the evolution in time of a variable
Complexity in the dynamics of all variables
Quantification of multi-scale temporal complexity
- Empirical mode decomposition
Noemi Nava
Quantification of hierarchical dependency and causality
- Kernel methods
Anna Zaremba
Information filtering through networks
- Embedded graphs
Guido Previde Massara
Modeling risk propagation
- Statistical mechanics approach
Annika Wipprecht
- Simulation and empirical study
Satjaporn Tungsong
Modeling financial processes with large fluctuations and non-linear dependency
- Scaling laws
Peter Divós
Thank YOU!
Si l’ordre satisfait la raison, le désordre fait les délices de l’imagination
Paul Claudel