Lvy copulas Basic ideas and a new estimation

Lévy copulas: Basic ideas and a new estimation method J L van Velsen, EC Modelling, ABN Amro Top. Quants, 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) • 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

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 VG 1 Lévy copula VG 2 bivariate Lévy process with VG 1 and VG 2 margins 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

2 Basics of Lévy copulas Distributional copula: distribution function on unit hypercube with uniform margins Sklar’s theorem: Given univariate distribution functions Fi and a copula C, the function is a joint distribution function with margins Fi. . F 1 C F F 2 Example: standard normal and beta(2, 2) coupled by Gumbel copula (right graph): 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 comonotonic copula: independence copula: 9

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 10

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

4 Operational risk modelling Typical structure AMA model for Op. Risk: 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 merge cells no CP random variable nature of the model is not invariant with respect to the level of granular solution: use Lévy copula solution granularity problem: CPP 1 CPP 2 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 shoc 13

Selecting and estimating a suitable Lévy copula requires knowledge of common shocks. In operational risk modelling, however, this information is typically not 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) CPP 1 CPP 2 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) CPP 1 CPP 2 sub-CPP 2 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) Basic idea: CPP 1 CPP 2 unknown common shocks • 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: CPP 1 CPP 2 # losses=k max(losses)=x # losses=l max(losses)=y likelihood function sample: note on distribution maximum: 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). for a sequence of k iid random variables 16

example of an element of the likelihood function: frequency CPP 1 frequency CPP 2 survival function of severity of jumps in dim 1 only: survival function severity CPP 1 frequency of jumps in dim 1 only multiply lhs and rhs by and observe that: 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 CPP 1 and CPP 2 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 CPP 2 conditional on counts and maximum loss of CPP 1: distribution maxima and frequencies: Goodness of fit test Lévy copula: • • • apply G to maxima CPP 1 and H to maxima CPP 2 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: 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, ar. Xiv: 1212. 0092 22