Multivariate Probability Distributions: Applications and Risk Models

Abstract

Multiple, simultaneous and dependent sources of risks are common to risk modeling. To model their manifestations, we use multivariate probability distributions and models to express their dependence and their interactions. The purpose of this chapter is to summarize a number of approaches to multivariate probability modeling and measurement of dependence. These include statistical and functional models, Bayesian techniques, families of multivariate probability distributions and copula. Both short term and long-run memory and fractal models are relegated to Chap. 5.

Additional Sources and References

Risk and dependence models are necessarily multivariate and are therefore more difficult to analyze and define their underlying properties. Multivariate probability distributions may be found in books such as Joe 1997 on Multivariate Models and Dependence Concepts (see also Joe and Hu 1996), Aas et al. 2009; Bairamov and Gultekin 2010; Rodriguez 1977 and in Statistical Handbooks.

Malvergne and Sornette (2002) have demonstrated the transformational relationship of distributions to the normal probability distribution.

Bisawa and Wang (2002) have considered in particular the bivariate binomial distribution. Chib and Greenberg 1998, analyze multivariate probit models, Denuit and Lamber (2005) provide constraints on concordance measures in bivariate discrete data, while John Peter (1962) provides a tolerance region for multivariate distributions applicable to multivariate risk models. Fang et al. (1987) book on Symmetric Multivariate and Related Distributions provide an exhaustive list of such distributions and their properties. In 1988 Marshall and Olkin characterize families of multivariate distributions while Oakes 1994 defines families of multivariate survival distributions.

Specific applications to financial risk models have been pointed out by Stein 1973 and to Stochastic Programming problems by Tintner and Sengupta 1975.

Nelsen 1999 and an extensive research conducted at the Federal Polytechnic University in Zurich by Embrechts et al. has contributed immensely to the mathematical development of Copulas (Sklar 1973) as a modeling approach to multivariate distributions (Embrechts et al. 2001; Embrechts 2000a, b; Embrechts et al. 1997, 2001) Additional references include Cherubini et al. (2004), Frey and McNeil (2001), Everitt and Hand (1981), Anderson and Sidenius (2004) extending the normal copula, Li (1999, 2000a, b) who has been used profusely on default correlation to price credit derivatives and Lindskog (2000) on modeling copulas, Aas et al. (2009) construct a Pair copula of multiple dependence with application to insurance, Nikoloupoulos and Karlis (2008) study Multivariate logit copula model with application to dental data, that can also be applied to financial default models, Patton (2006) models asymmetric exchange rate dependence.