Levy Copula Research Articles

Special Issue on Recent Trends in Stochastic Models, Statistics and Their Applications

This special issue was organized on the occasion of the 13th Workshop on Stochastic Models, Statistics and Their Applications (SMSA 2017) held at Humboldt University of Berlin on February 20-24, 2017. Key areas of the workshop focused on models and methods for problems arising in industry, finance, natural sciences, engineering, and technology. In these areas, solutions are needed, which are able to deal with complex and possibly high-dimensional data, make use of innovative methodologies, and have proven applicability to real-world data. In this special issue, we have 13 papers. The papers consider stochastic models for diverse data sets and streams such as bridge vehicle loads, watermark identification data, text documents, images of digitized handwritten digits, financial asset returns, option prices, and equity data. The methods and algorithms required to handle come from machine learning, stochastic processes, sequential (on-line) change-point detection, regression analysis, image analysis, and robust divergence measures. The work of Barbulescu analyzes the properties of Generalized Regression Neural Network (GRNN) model applied to the prediction of the selected Romanian stock market prices. In the work of Luo et al, a nonlinear PDE model to estimate the American call options and put options under financial crisis scenario is implemented with additional penalty introduced to Black-Scholes formula. The work of Liu et al proposes a compound Poisson bivariate process model of vehicle load with Levy copula function and validates it in a reliability analysis of bridges in China. Migalska proposes in her paper a density-free symmetry verification test for arbitrary symmetries in images. In their work, Darkhovsky et al propose a novel method based on the notion of ϵ-complexity of continuous vector functions for an off-line detection of multiple change-points in multidimensional time series. The work of Rafajlowicz et al proposes a method of spike detection and suppression (SDS method), which extends the jump detector proposed by the Authors in their earlier work and applies it for a laser power control in a 3D additive manufacturing process. Sliwinski et al propose two algorithms for nonlinearity recovery in Hammerstein systems, one based on orthogonal series expansions and the other using aggregation and validate them in numerical simulations. The work of Sousa et al provides an insight into the behavior of GARCH process describing financial markets affected by rumours and other random factors through an analysis of process mean and variance by estimating run length. Kisslinger et al describe a new toolkit based on scaled Bregman divergences for detecting distributional changes in random data, including streams and clouds. The work of Sarnowski et al derives an optimal decision function for a sudden change detection in political business cycle model based on the Markovian process. The work of Walkowiak et al proposes a two-step procedure for open-set classification of text documents and validates it on simulated data and on the collection of Wikipedia documents. The work by Xie et al combines two-tier stochastic frontier model with the traditional Richardson Model and applies it to the Chinese stock market analysis. Finally, the work of Yuan et al derives the Hamilton-Jacobi-Bellman equation for an optimal consumption problem of maximizing the expected discounted value of utility in an economic growth model with random technological shocks. We thank numerous anonymous reviewers for helping in evaluating all submitted manuscripts.

Nonlinear filtering with correlated Lévy noise characterized by copulas

The objective in stochastic filtering is to reconstruct information about an unobserved (random) process, called the signal process, given the current available observations of a certain noisy transformation of that process. Usually X and Y are modeled by stochastic differential equations driven by a Brownian motion or a jump (or Levy) process. We are interested in the situation where both the state process X and the observation process Y are perturbed by coupled Levy processes. More precisely, L=(L_1,L_2) is a 2--dimensional Levy process in which the structure of dependence is described by a Levy copula. We derive the associated Zakai equation for the density process and establish sufficient conditions depending on the copula and $L$ for the solvability of the corresponding solution to the Zakai equation. In particular, we give conditions of existence and uniqueness of the density process, if one is interested to estimate quantities like P( X(t)>a), where a is a threshold.

Information Flow Dependence in Financial Markets

In response to empirical evidence, we propose a continuous-time model for multivariate asset returns with a two-layered dependence structure. The price process is subject to multivariate information arrivals driving the market activity modeled by non-decreasing pure-jump Levy processes. A Levy copula determines the jump dependence and allows for a generic multivariate information flow with a flexible structure. Conditional on the information flow, asset returns are jointly normal. Within this setup, we provide an estimation framework based on maximum simulated likelihood. We apply novel multivariate models to equity data and obtain estimates which meet an economic intuition with respect to the two-layered dependence structure.

Series Representations for Multivariate Time-Changed Lévy Models

In this paper, we analyze a Levy model based on two popular concepts - subordination and Levy copulas. More precisely, we consider a two-dimensional Levy process such that each component is a time-changed (subordinated) Brownian motion and the dependence between subordinators is described via some Levy copula. The main result of this paper is the series representation for our model, which can be efficiently used for simulation purposes.

Modeling Correlation Structure for Collateralized Debt Obligations

Pricing complex financial derivatives such as collateralized debt obligations (CDO) is considered as the main reason triggering the 2008 financial crisis. The correlation structure related to the credit risks involved in a portfolio for pricing issues have been tried to overcome via a Gaussian copula framework first introduced by David Li (2000). This approach regards the correlation among the credit risks as normally distributed (tied with a Gaussian copula framework), enabling us to derive analytical solutions. However, despite its simplicity, this approach is far from reality, which caused mispricing of the tranches of CDOs. This phenomenon is called the correlation smile. This paper takes the correlation smile issue by considering a Levy copula framework. When this is introduced to pricing equations, one can see that the correlation smile is “corrected”. Thus, a more accurate model of pricing the above-mentioned tranches is introduced.

Capturing Non-Exchangeable Dependence in Multivariate Loss Processes with Nested Archimedean LLvy Copulas

The class of spectrally positive Levy processes is a frequent choice for modelling loss processes in areas such as insurance or operational risk. Dependence between such processes (for example, between different lines of business) can be modelled with Levy copulas. This approach is a parsimonious, efficient, and flexible method which provides many of the advantages akin to distributional copulas for random variables. Literature on Levy copulas seems to have primarily focused on bivariate processes. When multivariate settings are considered, these usually exhibit an exchangeable dependence structure (whereby all subset of the processes have an identical marginal Levy copula). In reality, losses are not always associated in an identical way, and models allowing for non-exchangeable dependence patterns are needed. In this paper, we present an approach which enables the development of such models. Inspired by ideas and techniques from the distributional copula literature we investigate the procedure of nesting Archimedean Levy copulas. We provide a detailed analysis of this construction, and derive conditions under which valid multivariate (nested) Levy copulas are obtained. Our results are discussed and illustrated, notably with an example of model fitting to data.

Jump tail dependence in Lévy copula models

This paper investigates the dependence of extreme jumps in multivariate Levy processes. We introduce a measure called jump tail dependence, defined as the probability of observing a large jump in one component of a process given a concurrent large jump in another component. We show that this measure is determined by the Levy copula alone and that it is independent of marginal Levy processes. We derive a consistent nonparametric estimator for jump tail dependence and establish its asymptotic distribution. Regarding the economic relevance of the measure, a simulation study illustrates that jump tail dependence has a substantial impact on financial portfolio distributions and optimal portfolio weights.

Modelling Dependence in Insurance Claims Processes with Lévy Copulas

In this paper we investigate the potential of Levy copulas as a tool for modelling dependence between compound Poisson processes and their applications in insurance. We analyse characteristics regarding the dependence in frequency and dependence in severity allowed by various Levy copula models. Through the introduction of new Levy copulas and comparison with the Clayton Levy copula, we show that Levy copulas allow for a great range of dependence structures.Procedures for analysing the fit of Levy copula models are illustrated by fitting a number of Levy copulas to a set of real data from Swiss workers compensation insurance. How to assess the fit of these models with respect to the dependence structure exhibited by the dataset is also discussed.Finally, we provide a decomposition of the trivariate compound Poisson process and discuss how trivariate Levy copulas model dependence in this multivariate setting.

Dependence properties and comparison results for Lévy processes

In this paper we investigate dependence properties and comparison results for multidimensional Levy processes. In particular we address the questions, whether or not dependence properties and orderings of the copulas of the distributions of a Levy process can be characterized by corresponding properties of the Levy copula, a concept which has been introduced recently in Cont and Tankov (Financial modelling with jump processes. Chapman & Hall/CRC, Boca Raton, 2004) and Kallsen and Tankov (J Multivariate Anal 97:1551–1572, 2006). It turns out that association, positive orthant dependence and positive supermodular dependence of Levy processes can be characterized in terms of the Levy measure as well as in terms of the Levy copula. As far as comparisons of Levy processes are concerned we consider the supermodular and the concordance order and characterize them by orders of the Levy measures and by orders of the Levy copulas, respectively. An example is given that the Levy copula does not determine dependence concepts like multivariate total positivity of order 2 or conditionally increasing in sequence. Besides these general results we specialize our findings for subfamilies of Levy processes. The last section contains some applications in finance and insurance like comparison statements for ruin times, ruin probabilities and option prices which extends the current literature.

Anisotropic Stable Levy Copula Processes - Analysis and Numerical Pricing Methods

We consider the valuation of derivative contracts on baskets where prices of single assets are Levy-like Feller processes of tempered stable type. The dependence among the marginals' jump structure is parametrized by a Levy copula. For marginals of regular, exponential Levy type in the sense of Boyarchenko and Levendorskii we show that the infinitesimal generator ${\cal A}$ of the resulting Levy copula process is a pseudo-differential operator whose principal symbol is a distribution of anisotropic homogeneity. We analyze the jump measure of the corresponding Levy copula processes. We prove the domains of their infinitesimal generators ${\cal A}$ are certain anisotropic Sobolev spaces. In these spaces and for a large class of Levy copulas, we prove a G\aa rding inequality for ${\cal A}$. We design a wavelet-based dimension-independent tensor product discretization for the efficient numerical solution of the parabolic Kolmogoroff equation $u_t+{\cal A}u = 0$ arising in valuation of derivative contracts under possibly stopped Levy copula processes. We show that diagonal preconditioning yields bounded condition number of the resulting matrices.