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Lévy copulas: Basic ideas and a new estimation method

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Lévy copulas: Basic ideas and a new estimation method

J L van Velsen, EC Modelling, ABN Amro

TopQuants, November 2013

- Introduction & motivation
- Basics of Lévy copulas
- Examples of Lévy copulas
- Operational risk modelling
- Estimation of a Lévy copula of a compound Poisson process with unknown common shocks
- Selection of a Lévy copula of a compound Poisson process with unknown common shocks
- Conclusions

2

Introduction & motivation

1

- 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:

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

3

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

- 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

Example bottom-up

approach:

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)

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

This work:

Basics of Lévy copulas

2

Distributional copula: distribution function on unit hypercube with uniform margins

Sklar’s theorem:

F1

C

F

Given univariate distribution functions Fi and a copula C, the function

F2

is a joint distribution function with margins Fi..

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

Basic ideas Lévy copula:

Lévy measure bivariate positive process:

marginal Lévy measure:

: expected # of jumps in A per unit time

A

B

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

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

Examples of Lévy copulas

3

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

comonotonic copula:

independence copula:

Archimedean Lévy copula:

Clayton copula (example of Archimedean copula):

copula density:

- dependence structure bivariate CPP:
- common shock frequency:
- common shock severities:
- Clayton survival copula with
- parameter

pure common shock Lévy copula:

copula density:

- dependence structure:
- common shock frequency:
- common shock severities:
- independent severities

Operational risk modelling

4

Typical structure AMA model for OpRisk:

BL

- 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

ET

: dependence introduced by distributional copula

- important characteristics of the model:
- severity distributions (sub-exponential)
- dependence structure (cell structure and distributional copula)

no CP random variable

CPP

CP random variable

CPP2

CP random variable

CPP1

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:

merge cells

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

solution: use Lévy copula

solution granularity

problem:

merge cells

nature of the model is invariant with respect to the level of granularity

also: appealing interpretation of dependence in terms of common shocks

CPP2

CPP2

sub-CPP2

CPP1

CPP1

sub-CPP2

Selecting and estimating a suitable Lévy copula requires knowledge of common

shocks. In operational risk modelling, however, this information is typically not available.

- 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)

available information:

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)

CPP2

CPP1

Estimation of a Lévy copula of a compound Poisson

5

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)

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

CPP1

# losses=k

max(losses)=x

likelihood function per time bin:

CPP2

# losses=l

max(losses)=y

likelihood function sample:

for a sequence of k iid random variables

note on distribution maximum:

- 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:

example of an element of the likelihood function:

frequency CPP2

frequency CPP1

survival function of severity

of jumps in dim 1 only:

survival function severity CPP1

frequency of jumps in dim 1 only

A

B

- multiply lhs and rhs by and observe that:
- lhs corresponds to C
- first part rhs corresponds to A
- second part rhs corresponds to B

C

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

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:

Selection of a Lévy copula of a compound Poisson

6

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.

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 Lévycopula:
- apply G to maxima CPP1 and H to maxima CPP2 pseudosample of probabilities
- determine dependence between columns of pseudosample
- dependence significant Lévycopula 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:

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).

Conclusions

7

- 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