1. Lévy copulas: Basic ideas and a new estimation method J L van Velsen, EC Modelling, ABN Amro TopQuants, November 2013

2. 2 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

3. 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: processconstructionlimitation 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? 3

4. 4 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 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) VG1 Lévy copula bivariate Lévy process with VG1 and VG2 margins VG2 This work: Example bottom-up approach:

5. 5 Basics of Lévy copulas 2 Given univariate distribution functions F i and a copula C, the function is a joint distribution function with margins F i.. Example: standard normal and beta(2,2) coupled by Gumbel copula (right graph): F1 F2 CF Distributional copula: distribution function on unit hypercube with uniform margins Sklar’s theorem:

6. Basic ideas Lévy copula: observationconsequence Lévy jump process fully characterized by Lévy measureDefine 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

7. joint tail integral: Definition tail integral for bivariate positive jump process: marginal tail integrals: 7 Sklar’s theorem for Lévy 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

8. 8 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 C without divergence of tail integral: almost surely no jumps in C

9. 9 3 Examples of Lévy copulas independence copula comonotonic copula Archimedean copulas pure common shock copula independence copula: comonotonic copula:

10. 10 Archimedean Lévy copula: Clayton copula (example of Archimedean copula): dependence structure bivariate CPP: common shock frequency: common shock severities: Clayton survival copula with parameter copula density:

11. 11 pure common shock Lévy copula: dependence structure: common shock frequency: common shock severities: independent severities copula density:

12. 12 4 Operational risk modelling Typical 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)

13. 13 granularity problem: no CP random variable merge cells 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 cells CPP1 CPP2 nature of the model is invariant with respect to the level of granularity also: appealing interpretation of dependence in terms of common shocks

14. 14 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

15. 15 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

16. 16 # losses=k max(losses)=x CPP1 likelihood function per time bin: CPP2 # losses=l max(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:

17. 17 B A C example of an element of the likelihood function: survival function severity CPP1 survival function of severity of 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

18. 18 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:

19. 19 6 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. Selection of a Lévy copula of a compound Poisson process with unknown common shocks

20. 20 distribution function maximum losses conditional on counts: distribution maxima and frequencies: distribution function maximum loss of CPP2 conditional on counts 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

21. 21 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).

22. 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: 22 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