Copulae as a new tool in financial modelling

The paper presents an overview of financial applications of copulas. Copulas permit to represent joint distribution functions by splitting the marginal behavior, embedded in the marginal distributions, from the dependence, captured by the copula itself. The splitting proves to be very helpful not only in the modelling phase, but also in the estimation or simulation one. Essentially, it provides a straighforward way to extend financial modelling from the usual joint normality assumption to more general joint distributions, even preserving the normality assumption on the marginals. The paper puts into evidence the advantages of the copula representation with respect to the joint distribution one, with special reference to applications in pricing and risk measurement. Copula functions have a long history in Probability theory, since they date back to [Sklar (1959)]. They have been studied under a number of different names, such as t-norms, dependence functions, doubly stochastic measures, Markov operators. Their application to Finance is very recent: the idea first appears in [Embrechts et al. (1999)], in connection with the limits of linear correlation as a measure of dependence or association. Copulas permit to represent joint distribution functions so as to split the marginal behavior, embedded in the marginal distributions, from the dependence, captured by the copula itself. The splitting proves to be very helpful not only in the modelling phase, but also in the estimation or simulation one. Essentially, it provides a straightforward way to extend financial modelling from the usual joint normality assumption to more general joint distributions, even preserving the normality assumption on the marginals. In what follows we will try to put into evidence the advantages of the copula representation with respect to the joint distribution one, with special reference to applications in Finance. The paper is organized as follows: in section 1 we recall the copula definition and its main mathematical and modelling properties, including the a priori relevance for finance. In section 2 we illustrate some pricing applications, in section 3 some market risk ones. We complete risk measurement applications in section 4, where we analyze credit risk. Section 5 summarizes and concludes. 1. Definition, basic properties and financial relevance Let us consider, for the sake of simplicity, the bivariate case (for the definition and properties in the n-dimensional case see for instance [Nelsen (1999)]). Informally, a copula is a joint distribution function defined on the unit square, with uniform marginals. Formally, define as I the unit interval, I=[0,1], and recall that a function C defined on I2=I×I is named 2-increasing if for every rectangle [v1,v2] × [z1,z2] whose vertices lie in I2, and such that v1 = v2, z1 = z2 0 ) ( ) ( ) ( ) ( 1 1 2 1 1 2 2 2 ≥ ,z v +C ,z v -C ,z v -C ,z v C (1) The lhs of (1) measures the mass or volume, according to function C, of the rectangle [v1,v2] × [z1 ,z2]. According to it, 2-increasing functions assign non-negative mass to every rectangle in their domain. With these preliminary notation, one can introduce the copula definition: Definition 1. A two -dimensional copula C(v,z) is a 2-increasing real function C:I2? I such that, for every v, z ∈ I i) C(0 ,z) = C(v,0) = 0 ii) C(v,1) = v, C(1,z) = z Example 1. The functions max(u+v-1,0), uv, min(u,v) can be easily checked to be copula functions. They are called respectively the minimum, product and maximum copula, and are denoted as C, C, C. Example 2. Consider the function )) ( ), ( ( ) , ( 1 1 XY z v z v C r Ga − − Φ Φ Φ = where XY r Φ is the joint distribution of a bi-dimensional standard normal, with linear correlation coefficient rXY, while F -1 is the inverse of the standard normal distribution F: ∫ ∞ −       − = Φ h dx x h 2 exp 2 1 ) ( 2 π One can easily verify that C is a copula, since it is 2-increasing and