Lévy copulas: Basic ideas and a new
estimation method
J L van Velsen, EC Modelling, ABN Amro
TopQuants, November 2013
Contents
1.
Introduction & motivation
2.
Basics of Lévy copulas
3.
Examples of Lévy copulas
4.
Operational risk modelling
5.
Estimation of a Lévy copula of a compound Poisson process with unknown
common shocks
6.
Selection of a Lévy copula of a compound Poisson process with unknown
common shocks
7.
Conclusions
2
1
Introduction & motivation
Multivariate Lévy jump processes are widely used in pricing and risk models. Examples:
•
•
•
pricing of multi-asset options
credit portfolio risk models and CDO pricing
insurance claim models and operational risk models
Examples of multivariate Lévy jump processes:
process
construction
limitation
multivariate variance gamma (VG)
multivariate Brownian motion with
common gamma subordinator
limited range of dependence
(independence not included)
multivariate compound Poisson
process (CPP)
•
number of parameters grows
exponentially with the number of
dimensions
•
specify the frequencies of all
kinds of jumps (jump only in first
dimension, only in the second
dimension, in both dimensions
and so on)
specify distribution functions of
all kinds of jumps
Question: More general way of constructing multivariate Lévy jump processes?
3
Answer: Yes, with a Lévy copula (Cont & Tankov, 2004)
Advantages Lévy copula:
•
•
•
•
Example bottom-up
approach:
bottom-up approach of modelling multivariate Lévy jump processes
all kinds of marginal Lévy jump processes are possible (example: combination of VG
and CPP)
full range of dependence
parsimonious construction of a multivariate CPP
VG1
Lévy copula
bivariate Lévy process
with VG1 and VG2 margins
VG2
Literature on applications
of the Lévy copula (a
selection)
• option pricing with a bivariate Lévy process with VG margins (Tankov, 2006)
• estimation of a Lévy copula of a bivariate CPP with known common shocks with an application to
insurance claim modelling (Esmaeli and Kluppelberg, 2010)
This work:
Estimation and selection of a Lévy copula of a bivariate CPP with unknown common
shocks with an application to operational risk modelling
4
Distributional copula: distribution function on unit hypercube with uniform margins
Sklar’s theorem:
F1
Given univariate distribution functions Fi and a copula C, the function
C
F ( x1 ,...,xn )  C( F1 ( x1 ),...,Fn ( xn ))
F
F2
is a joint distribution function with margins F i..
Example: standard normal and beta(2,2) coupled by Gumbel copula (right graph):
1
1
0.8
0.8
0.6
0.6
y
y
2
Basics of Lévy copulas
0.4
0.4
0.2
0.2
0
-3
-2
-1
0
1
x
2
3
4
0
-3
-2
-1
0
1
2
3
4
x
5
Basic ideas Lévy copula:
observation
consequence
Lévy jump process fully characterized by Lévy measure
Define Lévy copula with respect to Lévy measure
marginal Lévy measure similar to marginal probability
measure
Use same approach as for distributional copulas
(Sklar’s theorem)
Lévy measure may diverge at the origin (infinite activity
process)
Define tail integrals and use marginal tail integrals as
entries of the Lévy copula
Lévy measure bivariate positive process:
marginal Lévy measure:
: expected # of jumps in A per unit time
A
B
6
Definition tail integral for bivariate positive jump process:
A
joint tail integral:
marginal tail
integrals:
positive Lévy copula and Sklar’s theorem (Cont and Tankov, 2004):
Lévy copula:
Sklar’s theorem for
Lévy copula:
Note: Lévy copulas are also defined for higher dimensions and non-positive processes
7
Technical note about infinity of tail integral (by definition) for compound Poisson process:
marginal tail integral (A):
common jumps (B):
jumps in dim 1 only (C):
A
B
C
without divergence of tail integral: almost surely no jumps in C
8
3
Examples of Lévy copulas
•
•
•
•
independence copula
comonotonic copula
Archimedean copulas
pure common shock copula
independence copula:
comonotonic copula:
9
Archimedean Lévy copula:
Clayton copula (example of Archimedean copula):
copula density:
dependence structure bivariate CPP:
0.05
• common shock frequency:
copula density
0.04
0.03
• common shock severities:
Clayton survival copula with
parameter
0.02
0.01
0
0
100
50
50
100 0
tail integral CPP1
tail integral CPP2
10
pure common shock Lévy copula:
copula density:
-3
x 10
copula density
2
dependence structure:
1.5
• common shock frequency:
1
0.5
• common shock severities:
independent severities
0
100
100
50
tail integral CPP2
50
0
0
tail integral CPP1
11
4
Operational risk modelling
Typical structure AMA model for OpRisk:
BL 
ET 
• Separate CPPs within each combination of
business line (BL) and event type (ET).
Example BL: Retail Banking
Example ET: External Fraud
• The CPPs are connected at discrete times (months or
quarters) by a distributional copula
: dependence introduced by distributional copula
important characteristics of the model:
•
•
severity distributions (sub-exponential)
dependence structure (cell structure and distributional copula)
12
Note: connecting separate CPPs at t=1 with a distributional copula gives rise to a random
vector S with characteristic function that is not necessarily of the form
granularity problem:
CP random variable
CP random variable
merge cells
no CP random variable
nature of the model is not invariant with respect to the level of granularity
solution: use Lévy copula
solution granularity
problem:
CPP1
CPP2
merge cells
CPP
nature of the model is invariant with respect to the level of granularity
also: appealing interpretation of dependence in terms of common shocks
13
Selecting and estimating a suitable Lévy copula requires knowledge of common
shocks. In operational risk modelling, however, this information is typically not available.
available information:
•
•
•
severities of all losses within the cells
no common shock flags between cells
timing information typically assumed accurate on
monthly or quarterly basis (no continuous observation)
CPP1
CPP2
unknown common shocks
How to estimate and select a Lévy copula with unknown common shocks?
Note on granularity and common shocks:
Banks are required to flag and aggregate common shocks within cells. This means that each cell may
consist of many sub-CPPs connected by a Levy copula (these sub-CPPs and the Levy copula are not
estimated)
CPP1
CPP2
sub-CPP2
sub-CPP2
14
5
Estimation of a Lévy copula of a compound Poisson
process with unknown common shocks
available information:
•
•
•
severities of all losses within the cells
no common shock flags between cells
timing information typically assumed accurate on a
monthly or quarterly basis (no continuous observation)
CPP1
CPP2
unknown common shocks
Basic idea:
• make time bins (months or quarters) and determine the
number of losses and the maximum loss for each time bin
• maximize likelihood function for the sample of the previous step over
the parameters of the multivariate CPP (marginal frequencies, marginal
severity distributions and Levy copula)
Why sample based on maximum loss? Answer: With the maximum loss we
are able to determine an analytical expression for the likelihood function
15
likelihood function per time bin:
CPP1
CPP2
# losses=k
max(losses)=x
# losses=l
max(losses)=y
likelihood function sample:
note on distribution maximum:
for a sequence of k iid random variables
entries of likelihood function:
• marginal frequencies
• parameters marginal severity distributions
• parameters Levy copula
limit behaviour:
In the limit of infinitesimally small time bins (continuous observation), the likelihood
function collapses to the likelihood function known in the literature (Esmaeili and
Kluppelberg, 2010).
16
example of an element of the likelihood function:
frequency CPP1
frequency CPP2
survival function of severity
of jumps in dim 1 only:
survival function severity CPP1
frequency of jumps in dim 1 only
multiply lhs and rhs by
that:
and observe
A
B
• lhs corresponds to C
• first part rhs corresponds to A
• second part rhs corresponds to B
C
17
Two-step maximum likelihood estimation (similar to the inference function for margins
[IFM] approach of distributional copulas):
1.
2.
estimate frequency and parameters of severity distribution for CPP1 and CPP2
separately
substitute estimated parameters of step 2 in likelihood function and maximize the
resulting concentrated likelihood function wrt to the parameters of the Levy copula
Note: IFM method is particularly useful here because it makes use of all losses (not just
the maxima) in step 1.
results simulation study:
18
6
Selection of a Lévy copula of a compound Poisson
process with unknown common shocks
Selection of a Lévy copula in case of known common shocks:
1. select candidate Lévy copulas based on the scatter plot of common
shock severities and the number of common shocks
2. estimate the parameters of the Lévy copula based on common shock
severities
3. estimate the parameters of the Lévy copula based on the number of
common shocks
4. similar estimates in step 2 and 3?
Proposed selection method in case of unknown common shocks:
Determine the distributional copula for the maximum losses (given the
number of losses) and use an ordinary copula goodness of fit test.
19
distribution function maximum
losses conditional on counts:
distribution function maximum
loss of CPP2 conditional on
counts and maximum loss of
CPP1:
distribution maxima
and frequencies:
Goodness of fit test Lévy copula:
•
•
•
apply G to maxima CPP1 and H to maxima CPP2  pseudosample of probabilities
determine dependence between columns of pseudosample
20
dependence significant  Lévy copula probably not correct
Example with Danish fire loss data (publicly available
on http://www.ma.hw.ac.uk/~mcneil):
test Clayton and pure common shock Lévy copula
Clayton:
pure common shock:
-3
0.05
2
0.04
1.5
copula density
copula density
x 10
0.03
0.02
0.01
1
0.5
0
100
0
0
100
50
50
100 0
tail integral CPP1
tail integral CPP2
100
50
tail integral CPP2
50
0
0
tail integral CPP1
result: pure common shock copula is rejected at 5%
Results is in line with the finding of Esmaeili and Kluppelberg (2010) that
the Clayton Lévy copula provides a good fit to the Danish fire loss data
(analysis based on known common shocks).
21
7
Conclusions
• A method is developed to estimate and select a Lévy copula of a
discretely observed bivariate CPP with unknown common jumps
• The method is tested in a simulation study
• The method has been applied to a real data set and a goodness of fit test is developed
• With the new method, the Lévy copula becomes a realistic tool of the advanced
measurement approach of operational risk
For details see:
J. L. van Velsen, Parameter estimation of a Levy copula of a discretely observed bivariate compound Poisson
process with an application to operational risk modelling, arXiv:1212.0092
22