Clustering CDS: algorithms, distances, stability and convergence rates

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Introduction
The standard methodology
Exploring dependence between returns
Copula-based dependence coeﬃcients (clustering distances)
Empirical convergence rates
Beyond dependence: a (copula,margins) representation
Clustering CDS: algorithms, distances,
stability and convergence rates
CMStatistics 2016, University of Seville, Spain
Gautier Marti, Frank Nielsen, Philippe Donnat
HELLEBORECAPITAL
December 9, 2016
Gautier Marti Clustering CDS: algorithms, distances, stability and convergence r

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Introduction
1 Introduction
2 The standard methodology
3 Exploring dependence between returns
4 Copula-based dependence coeﬃcients (clustering distances)
5 Empirical convergence rates
6 Beyond dependence: a (copula,margins) representation

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Introduction
Introduction
Goal: Finding groups of ’homogeneous’ assets that can help to:
• build alternative measures of risk,
• elaborate trading strategies. . .
But, we need a high confidence in these clusters (networks).
So, we need appropriate AND fast converging methodologies [8]:
to be consistent yet efficient (bias–variance tradeoff),
to avoid non-stationarity of the time series (too large sample).
A good model selection criterion:
Minimum sample size to reach a given ’accuracy’.

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Introduction
The standard methodology - description
The methodology widely adopted in empirical studies: [7].
Let N be the number of assets.
Let Pi (t) be the price at time t of asset i, 1 ≤ i ≤ N.
Let ri (t) be the log-return at time t of asset i:
ri (t) = log Pi (t) − log Pi (t − 1).
For each pair i, j of assets, compute their correlation:
ρij =
ri rj − ri rj
( r2
i − ri
2) r2
j − rj
2
.
Convert the correlation coeﬃcients ρij into distances:
dij = 2(1 − ρij ).

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Introduction
The standard methodology - description
From all the distances dij , compute a minimum spanning tree:
Figure: A minimum spanning tree of stocks (from [1]); stocks from the
same industry (represented by color) tend to cluster together

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Introduction
The standard methodology - limitations
• MST clustering equivalent to Single Linkage clustering:
• chaining phenomenon
• not stable to noise / small perturbations [11]
• Use of the Pearson correlation:
• can take value 0 whereas variables are strongly dependent
• not invariant to variable monotone transformations
• not robust to outliers
Is it still useful for ﬁnancial time series? stocks? CDS??!

HELLEBORECAPITAL
Introduction
The standard methodology - limitations
• MST clustering equivalent to Single Linkage clustering:
• chaining phenomenon
• not stable to noise / small perturbations [11]
• Use of the Pearson correlation:
• can take value 0 whereas variables are strongly dependent
• not invariant to variables monotone transformations
• not robust to outliers
Is it still useful for ﬁnancial time series? stocks? CDS??!

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Introduction
Copulas
Sklar’s Theorem [13]
For (Xi , Xj ) having continuous marginal cdfs FXi
, FXj
, its joint cumulative
distribution F is uniquely expressed as
F(Xi , Xj ) = C(FXi
(Xi ), FXj
(Xj )),
where C is known as the copula of (Xi , Xj ).
Copula’s uniform marginals jointly encode all the dependence.

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Introduction
From ranks to empirical copula
ri , rj are the rank statistics of Xi , Xj respectively, i.e. rt
i is the rank
of Xt
i in {X1
i , . . . , XT
i }: rt
i = T
k=1 1{Xk
i ≤ Xt
i }.
Deheuvels’ empirical copula [3]
Any copula ˆC deﬁned on the lattice L = {( ti
T ,
tj
T ) : ti , tj = 0, . . . , T} by
ˆC( ti
T ,
tj
T ) = 1
T
T
t=1 1{rt
i ≤ ti , rt
j ≤ tj } is an empirical copula.
ˆC is a consistent estimator of C with uniform convergence [4].

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Introduction
Clustering of bivariate empirical copulas
Generate the N
2 bivariate empirical copulas
Find clusters of copulas using optimal transport [10, 9]
Compute and display the clusters’ centroids [2]
Some code available at www.datagrapple.com/Tech.

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Introduction
Copula-centers for stocks (CAC 40)
Figure: Stocks: More mass in the bottom-left corner, i.e. lower tail
dependence. Stock prices tend to plummet together.

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Introduction
Copula-centers for Credit Default Swaps (XO index)
Figure: Credit default swaps: More mass in the top-right corner, i.e.
upper tail dependence. Insurance cost against entities’ default tends to
soar in stressed market.

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Introduction
Dependence as relative distances between copulas
C copula of (Xi , Xj ),
|u − v|/
√
2 distance between (u, v) to the diagonal
Spearman’s ρS :
ρS (Xi , Xj ) = 12
1
0
1
0
(C(u, v) − uv)dudv
= 1 − 6
1
0
1
0
(u − v)2
dC(u, v)
Many correlation coefficients can be expressed as distances to the
Fréchet–Hoeffding bounds or the independence [6]. Some are explicitely
built this way (e.g. [12, 5, 9]).

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Introduction
A metric space for copulas: Optimal Transport

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Introduction
The Target/Forget Dependence Coefficient (TFDC)
Now, we can define our bespoke dependence coefficient:
Build the forget-dependence copulas {CF
l }l
Build the target-dependence copulas {CT
k }k
Compute the empirical copula Cij from xi , xj
TFDC(Cij ) =
minl D(CF
l , Cij )
minl D(CF
l , Cij ) + mink D(Cij , CT
k )

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Introduction
Spearman vs. TFDC
0.0 0.2 0.4 0.6 0.8 1.0
discontinuity position a
0.0
0.2
0.4
0.6
0.8
1.0
Estimatedpositivedependence
Spearman & TFDC values as a function of a
TFDC
Spearman
Figure: Empirical copulas for (X, Y ) where
X = Z1{Z < a} + X 1{Z > a},
Y = Z1{Z < a + 0.25} + Y 1{Z > a + 0.25}, a = 0, 0.05, . . . , 0.95, 1,
and where Z is uniform on [0, 1] and X , Y are independent noises (left).
TFDC and Spearman coeﬃcients estimated between X and Y as a
function of a (right).
For a = 0.75, Spearman coeﬃcient yields a negative value, yet X = Y
over [0, a].

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Introduction
Process: Recovering a simulated ground-truth [8]
A simulation & benchmark process that needs to be reﬁned:
Extract (using a large sample) a ﬁltered correlation matrix R

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Introduction
Generate samples of size T = 10, . . . , 20, . . . from a relevant
distribution (parameterized by R)

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Introduction
Compute the ratio of the number of correct clustering
obtained over the number of trials as a function of T
100 200 300 400 500
Sample size
0.0
0.2
0.4
0.6
0.8
1.0
Score
Empirical rates of convergence for Single Linkage
Gaussian - Pearson
Gaussian - Spearman
Student - Pearson
Student - Spearman
100 200 300 400 500
Sample size
0.0
0.2
0.4
0.6
0.8
1.0
Score
Empirical rates of convergence for Average Linkage
Gaussian - Pearson
Gaussian - Spearman
Student - Pearson
Student - Spearman
100 200 300 400 500
Sample size
0.0
0.2
0.4
0.6
0.8
1.0
Score
Empirical rates of convergence for Ward
Gaussian - Pearson
Gaussian - Spearman
Student - Pearson
Student - Spearman
A full comparative study will be posted online at www.datagrapple.com/Tech.

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Introduction
ON CLUSTERING FINANCIAL TIME SERIES
GAUTIER MARTI, PHILIPPE DONNAT AND FRANK NIELSEN
NOISY CORRELATION MATRICES
Let X be the matrix storing the standardized re-
turns of N = 560 assets (credit default swaps)
over a period of T = 2500 trading days.
Then, the empirical correlation matrix of the re-
turns is
C =
1
T
XX .
We can compute the empirical density of its
eigenvalues
ρ(λ) =
1
N
dn(λ)
dλ
,
where n(λ) counts the number of eigenvalues of
C less than λ.
From random matrix theory, the Marchenko-
Pastur distribution gives the limit distribution as
N → ∞, T → ∞ and T/N fixed. It reads:
ρ(λ) =
T/N
2π
(λmax − λ)(λ − λmin)
λ
,
where λmax
min = 1 + N/T ± 2 N/T, and λ ∈
[λmin, λmax].
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
λ
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
ρ(λ)
Figure 1: Marchenko-Pastur density vs. empirical den-
sity of the correlation matrix eigenvalues
Notice that the Marchenko-Pastur density fits
well the empirical density meaning that most of
the information contained in the empirical corre-
lation matrix amounts to noise: only 26 eigenval-
ues are greater than λmax.
The highest eigenvalue corresponds to the ‘mar-
ket’, the 25 others can be associated to ‘industrial
sectors’.
CLUSTERING TIME SERIES
Given a correlation matrix of the returns,
0 100 200 300 400 500
0
100
200
300
400
500
Figure 2: An empirical and noisy correlation matrix
one can re-order assets using a hierarchical clus-
tering algorithm to make the hierarchical correla-
tion pattern blatant,
0 100 200 300 400 500
0
100
200
300
400
500
Figure 3: The same noisy correlation matrix re-ordered
by a hierarchical clustering algorithm
and finally filter the noise according to the corre-
lation pattern:
0 100 200 300 400 500
0
100
200
300
400
500
Figure 4: The resulting filtered correlation matrix
BEYOND CORRELATION
Sklar’s Theorem. For any random vector X = (X1, . . . , XN ) having continuous marginal cumulative
distribution functions Fi, its joint cumulative distribution F is uniquely expressed as
F(X1, . . . , XN ) = C(F1(X1), . . . , FN (XN )),
where C, the multivariate distribution of uniform marginals, is known as the copula of X.
Figure 5: ArcelorMittal and Société générale prices are projected on dependence ⊕ distribution space; notice their
heavy-tailed exponential distribution.
Let θ ∈ [0, 1]. Let (X, Y ) ∈ V2
. Let G = (GX, GY ), where GX and GY are respectively X and Y marginal
cdf. We define the following distance
d2
θ(X, Y ) = θd2
1(GX(X), GY (Y )) + (1 − θ)d2
0(GX, GY ),
where d2
1(GX(X), GY (Y )) = 3E[|GX(X) − GY (Y )|2
], and d2
0(GX, GY ) = 1
2 R
dGX
dλ − dGY
dλ
2
dλ.
CLUSTERING RESULTS & STABILITY
0 5 10 15 20 25 30
Standard Deviation in basis points
0
5
10
15
20
25
30
35
Numberofoccurrences
Standard Deviations Histogram
Figure 6: (Top) The returns correlation structure ap-
pears more clearly using rank correlation; (Bottom)
Clusters of returns distributions can be partly described
by the returns volatility
Figure 7: Stability test on Odd/Even trading days sub-
sampling: our approach (GNPR) yields more stable
clusters with respect to this perturbation than standard
approaches (using Pearson correlation or L2 distances).

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Introduction
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Introduction
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Introduction
Gautier Marti, Sébastien Andler, Frank Nielsen, and Philippe
Donnat.
Clustering financial time series: How long is enough?
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Conference on Artificial Intelligence, IJCAI 2016, New York,
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Introduction
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Introduction
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