On a strong metric on the space of copulas and its induced dependence measure

Using the one-to-one correspondence between copulas and Markov operators on L¹([0,1]) and expressing the Markov operators in terms of regular conditional distributions (Markov kernels) allows to define a metric D₁ on the space of copulas C that is a metrization of the strong operator topology of the corresponding Markov operators. It is shown that the resulting metric space (C,D₁) is complete and separable and that the induced dependence measure ζ₁, defined as a scalar times the D₁-distance to the product copula Π, has various good properties. In particular the class of copulas that have maximum D₁-distance to the product copula is exactly the class of completely dependent copulas, i.e. copulas induced by Lebesgue-measure preserving transformations on [0,1]. Hence, in contrast to the uniform distance d_∞, Π cannot be approximated arbitrarily well by completely dependent copulas with respect to D₁. The interrelation between D₁ and the so-called ∂-convergence by Mikusinski and Taylor as well as the interrelation between ζ₁ and the mutual dependence measure ω by Siburg and Stoimenov is analyzed. ζ₁ is calculated for some well-known parametric families of copulas and an application to singular copulas induced by certain Iterated Functions Systems is given.     

Generalized Shuffles Related to Nijenhuis and TD-Algebras

Shuffle type products are well known in mathematics and physics. They are intimately related to Loday's dendriform algebras and were extensively used to give explicit constructions of free Rota–Baxter algebras. In the literature there exist at least two other Rota–Baxter type algebras, namely, the Nijenhuis algebra and the so-called TD-algebra. The explicit construction of the free unital commutative Nijenhuis algebra uses a modified quasi-shuffle product, called the right-shift shuffle. We show that another modification of the quasi-shuffle, the so-called left-shift shuffle, can be used to give an explicit construction of the free unital commutative TD-algebra. We explore some basic properties of TD-operators. Our construction is related to Loday's unital commutative tridendriform algebra, including the involutive case. The concept of Rota–Baxter, Nijenhuis and TD-bialgebras is introduced at the end, and we show that any commutative bialgebra provides such objects.     

A scalar product for copulas

We introduce a scalar product for n-dimensional copulas, based on the Sobolev scalar product for W1,2-functions. The corresponding norm has quite remarkable properties and provides a new, geometric framework for copulas. We show that, in the bivariate case, it measures invertibility properties of copulas with respect to the ∗-operation introduced by Darsow et al. (1992). The unique copula of minimal norm is the null element for the ∗-operation, whereas the copulas of maximal norm are precisely the invertible elements.     

Bivariate option pricing with copulas

The adoption of copula functions is suggested in order to price bivariate contingent claims. Copulas enable the marginal distributions extracted from vertical spreads in the options markets to be imbedded in a multivariate pricing kernel. It is proved that such a kernel is a copula function, and that its super-replication strategy is represented by the Fréchet bounds. Applications provided include prices for binary digital options, options on the minimum and options to exchange one asset for another. For each of these products, no-arbitrage pricing bounds, as well as values consistent with the independence of the underlying assets are provided. As a final reference value, a copula function calibrated on historical data is used.     

Archimedean copulas derived from utility functions

The inverse of the (additive) generator of an Archimedean copula is a strictly decreasing and convex function, while utility functions (applying to risk averse decision makers) are nondecreasing and concave. This provides a basis for deriving an inverse generator of an Archimedean copula from a utility function. If we derive the inverse of the generator from the utility function, there is a link between the magnitude of measures of risk attitude (like the very common Arrow–Pratt coefficient of absolute risk aversion) and the strength of dependence featured by the corresponding Archimedean copula. Some new copula families are derived, and their properties are discussed. A numerical example about modeling dependence of coupled lives is included.     

Best-possible bounds on the set of copulas with given degree of non-exchangeability

We establish best-possible bounds on the set of quasi-copulas with given degree of non-exchangeability. These bounds are shown to be best-possible bounds as well for the set of copulas with given degree of non-exchangeability, and, consequently, also on the set of bivariate distribution functions of continuous random variables with given margins and given degree of non-exchangeability. Non-exchangeability of a (quasi-)copula is measured in the sense of Nelsen, i.e.?proportional to the maximal absolute difference between this (quasi-)copula and its transpose.     

Multivariate dependence concepts through copulas

In this paper, multivariate dependence concepts such as affiliation, association and positive lower orthant dependent are studied in terms of copulas. Relationships among these dependent concepts are obtained. An affiliation is a notion of dependence among the elements of a random vector. It has been shown that the affiliation property is preserved using linear interpolation of subcopula. Also our results are applied to the multivariate skew-normal copula. As an application, the dependence concepts used in auction with affiliated signals are discussed. Several examples are given for illustration of the main results.     

Extreme value properties of multivariate t copulas

The extremal dependence behavior of t copulas is examined and their extreme value limiting copulas, called the t-EV copulas, are derived explicitly using tail dependence functions. As two special cases, the Hüsler–Reiss and the Marshall–Olkin distributions emerge as limits of the t-EV copula as the degrees of freedom go to infinity and zero respectively. The t copula and its extremal variants attain a wide range in the set of bivariate tail dependence parameters. Supported by NSERC Discovery Grant.     

违约风险的交换期权的定价模型与解法

含交易对手违约风险的交换期权采用混合模型定价,借助公司价值模型中的补偿率,同时采用以强度为基础的违约函数来确定违约的发生.假定违约强度遵从均值回复的重随机Poisson过程：且违约强度过程与标的资产,企业价值都相关.利用等价鞅测度变换方法导出含有违约风险的交换期权的价格闭解.     

Doubly stochastic matrices whose powers eventually stop

In this note we characterize doubly stochastic matrices A whose powers A,A²,A³,… eventually stop, i.e., A^p=A^p+1= for some positive integer p. The characterization enables us to determine the set of all such matrices.     

An order of asymmetry in copulas,and implications for risk management

We study symmetry properties of bivariate copulas. For this, we introduce an order of asymmetry, as well as measures of asymmetry which are monotone in that order. In an empirical study, we illustrate that asymmetric dependence structures do indeed occur in financial market data and discuss its relevance for financial risk management.     

Construction of asymmetric copulas and its application in two-dimensional reliability modelling

Copulas offer a useful tool in modelling the dependence among random variables. In the literature, most of the existing copulas are symmetric while data collected from the real world may exhibit asymmetric nature. This necessitates developing asymmetric copulas that can model such data. In the meantime, existing methods of modelling two-dimensional reliability data are not able to capture the tail dependence that exists between the pair of age and usage, which are the two dimensions designated to describe product life. This paper proposes a new method of constructing asymmetric copulas, discusses the properties of the new copulas, and applies the method to fit two-dimensional reliability data that are collected from the real world.     

Model selection of copulas: AIC versus a cross validation copula information criterion

Akaike Information Criterion (AIC) is frequently employed in the semiparametric setting of selection of copula models, even though as a model selection tool it was developed in a parametric setting. Recently a Copula Information Criterion (CIC) has been especially designed for copula model selection. In this paper we examine the two approaches and present a simulation study where the performance of a cross-validated version of CIC is compared with the AIC criterion. Only minor differences are observed.     

Distorted Mix Method for constructing copulas with tail dependence

This paper introduces a method for constructing copula functions by combining the ideas of distortion and convex sum, named Distorted Mix Method. The method mixes different copulas with distorted margins to construct new copula functions, and it enables us to model the dependence structure of risks by handling the central and tail parts separately. By applying the method we can modify the tail dependence of a given copula to any desired level measured by tail dependence function and tail dependence coefficients of marginal distributions. As an application, a tight bound for asymptotic Value-at-Risk of order statistics is obtained by using the method. An empirical study shows that copulas constructed by this method fit the empirical data of SPX 500 Index and FTSE 100 Index very well in both central and tail parts.     

Joint distributions of random sets and their relation to copulas

Random sets are set-valued random variables. They have been applied in various fields like stochastic geometry, statistics, economics, engineering or computer science, and are often used for modeling uncertainty. In an earlier paper the author has defined joint capacity and joint containment functionals which are multivariate set functions describing the joint distribution of random sets. This paper is concerned with the question how copulas can be used to describe or model the dependence of random sets. It is demonstrated that a joint containment functional can be related to its margins by a family of copulas. Furthermore, the paper provides a first insight how copulas can be used to define joint containment functionals.     

Doubly stochastic quadratic operators and Birkhoff’s problem

In the present we introduce a concept of doubly stochastic quadratic operator. We study necessary and sufficient conditions for doubly stochasticity of operator. Moreover, we study analogue of Birkhoff’s theorem for the class of doubly stochastic operators.