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LÃ©vy copulas: Basic ideas and a new estimation method

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LÃ©vy copulas: Basic ideas and a new estimation method

J L van Velsen, EC Modelling, ABN Amro

TopQuants, November 2013

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

2

Introduction & motivation

1

- Multivariate LÃ©vy 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 LÃ©vy jump processes:

Question: More general way of constructing multivariate LÃ©vy jump processes?

3

Answer: Yes, with a LÃ©vy copula (Cont & Tankov, 2004)

- Advantages LÃ©vy copula:
- bottom-up approach of modelling multivariate LÃ©vy jump processes
- all kinds of marginal LÃ©vy 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

LÃ©vy copula

bivariate LÃ©vy process

with VG1 and VG2 margins

VG2

Literature on applications

of the LÃ©vy copula (a selection)

- option pricing with a bivariate LÃ©vy process with VG margins (Tankov, 2006)
- estimation of a LÃ©vy 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 LÃ©vy copula of a bivariate CPP with unknown common shocks with an application to operational risk modelling

This work:

Basics of LÃ©vy copulas

2

Distributional copula: distribution function on unit hypercube with uniform margins

Sklarâ€™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 LÃ©vy copula:

LÃ©vy measure bivariate positive process:

marginal LÃ©vy 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 LÃ©vy copula and Sklarâ€™s theorem (Cont and Tankov, 2004):

LÃ©vy copula:

Sklarâ€™s theorem for

LÃ©vy copula:

Note: LÃ©vy 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 LÃ©vy copulas

3

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

comonotonic copula:

independence copula:

Archimedean LÃ©vy 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 LÃ©vy copula:

copula density:

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

Operational risk modelling

4

Typical structure AMA model for OpRisk:

BL ïƒ

- 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 ïƒ

: 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 LÃ©vy 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 LÃ©vy 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 LÃ©vy 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 LÃ©vy 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 LÃ©vy copula of a compound Poisson

6

process with unknown common shocks

Selection of a LÃ©vy copula in case of known common shocks:

1. select candidate LÃ©vy copulas based on the scatter plot of common

shock severities and the number of common shocks

2. estimate the parameters of the LÃ©vy copula based on common shock

severities

3. estimate the parameters of the LÃ©vy 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 LÃ©vycopula:
- apply G to maxima CPP1 and H to maxima CPP2 ïƒ pseudosample of probabilities
- determine dependence between columns of pseudosample
- dependence significant ïƒ LÃ©vycopula 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 LÃ©vy 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 LÃ©vy 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 LÃ©vy 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 LÃ©vy 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