Lévy copulas: Basic ideas and a new estimation method

J L van Velsen, EC Modelling, ABN Amro

TopQuants, November 2013

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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

1 Introduction & motivationMultivariate 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)

• 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

number of parameters grows exponentially with the number of dimensions

Question: More general way of constructing multivariate Lévy jump processes?

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Advantages Lévy copula:

•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

Answer: Yes, with a Lévy copula (Cont & Tankov, 2004)

Estimation and selection of a Lévy copula of a bivariate CPP with unknown common shocks with an application to operational risk modelling

Literature on applicationsof 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)

VG1

Lévy copulabivariate Lévy process

with VG1 and VG2 margins VG2

This work:

Example bottom-upapproach:

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Basics of Lévy copulas2Given univariate distribution functions Fi and a copula C, the function

))(),...,((),...,( 111 nnn xFxFCxxF

is a joint distribution function with margins Fi..

Example: standard normal and beta(2,2) coupled by Gumbel copula (right graph):

-3 -2 -1 0 1 2 3 40

0.2

0.4

0.6

0.8

1

x

y

-3 -2 -1 0 1 2 3 40

0.2

0.4

0.6

0.8

1

x

y

F1

F2

C F

Distributional copula: distribution function on unit hypercube with uniform margins

Sklar’s theorem:

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

B6

joint tail integral:

Definition tail integral for bivariate positive jump process:

marginal tailintegrals:

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Sklar’s theorem forLévy copula:

positive Lévy copula and Sklar’s theorem (Cont and Tankov, 2004):

Lévy copula:

A

Note: Lévy copulas are also defined for higher dimensions and non-positive processes

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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):

B

A

Cwithout divergence of tail integral: almost surely no jumps in C

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3 Examples of Lévy copulas

•independence copula•comonotonic copula •Archimedean copulas•pure common shock copula

independence copula: comonotonic copula:

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Archimedean Lévy copula:

Clayton copula (example of Archimedean copula):

0

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100 0

50

1000

0.01

0.02

0.03

0.04

0.05

tail integral CPP2tail integral CPP1

copu

la d

ensi

ty

dependence structure bivariate CPP:

• common shock frequency:

• common shock severities: Clayton survival copula with parameter

copula density:

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0

50

100

0

50

1000

0.5

1

1.5

2

x 10-3

copu

la d

ensi

ty

tail integral CPP1tail integral CPP2

pure common shock Lévy copula:

dependence structure:

• common shock frequency:

• common shock severities: independent severities

copula density:

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4 Operational risk modellingTypical structure AMA model for OpRisk:

BL

ET

: dependence introduced by distributional copula

• 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

important characteristics of the model:

•severity distributions (sub-exponential)•dependence structure (cell structure and distributional copula)

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granularity problem: no CP random variable

merge cellsCP random variable CP random variable

nature of the model is not invariant with respect to the level of granularity

solution: use Lévy copula

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

solution granularity problem: CPP

merge cellsCPP1 CPP2

nature of the model is invariant with respect to the level of granularityalso: appealing interpretation of dependence in terms of common shocks

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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:

CPP1 CPP2

unknown common shocks

How to estimate and select a Lévy copula with unknown common shocks?

•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)

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

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5 Estimation of a Lévy copula of a compound Poisson

CPP1 CPP2

unknown common shocks

•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)

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

available information:

process with unknown common shocks

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# losses=kmax(losses)=x

CPP1

likelihood function per time bin:

CPP2

# losses=lmax(losses)=y

likelihood function sample:

note on distribution maximum: for a sequence of k iid random variables

•marginal frequencies•parameters marginal severity distributions•parameters Levy copula

entries of likelihood function:

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).

limit behaviour:

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B

A

C

example of an element of the likelihood function:

survival function severity CPP1

survival function of severityof jumps in dim 1 only:

frequency CPP1 frequency CPP2

frequency of jumps in dim 1 only

multiply lhs and rhs by and observe that:

•lhs corresponds to C•first part rhs corresponds to A•second part rhs corresponds to B

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Two-step maximum likelihood estimation (similar to the inference function for margins [IFM] approach of distributional copulas):

1.estimate frequency and parameters of severity distribution for CPP1 and CPP2 separately 2.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:

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6Selection 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 severities3. 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 thenumber of losses) and use an ordinary copula goodness of fit test.

Selection of a Lévy copula of a compound Poisson process with unknown common shocks

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distribution function maximum losses conditional on counts:

distribution maxima and frequencies:

distribution function maximum loss of CPP2 conditional oncounts and maximum loss of CPP1:

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 •dependence significant Lévy copula probably not correct

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Example with Danish fire loss data (publicly availableon http://www.ma.hw.ac.uk/~mcneil):

0

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0

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0.5

1

1.5

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x 10-3

copu

la d

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tail integral CPP1tail integral CPP2

0

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100 0

50

1000

0.01

0.02

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0.04

0.05

tail integral CPP2tail integral CPP1

copu

la d

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test Clayton and pure common shock Lévy copula

Clayton: pure common shock:

result: pure common shock copula is rejected at 5%

Results is in line with the finding of Esmaeili and Kluppelberg (2010) thatthe Clayton Lévy copula provides a good fit to the Danish fire loss data (analysis based on known common shocks).

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:

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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