Copulas are an innovative tool in finance to separate marginal distributions, for example of single asset returns, from their dependence structure in modelling multivariate distributions. In principle this allows for the whole variety of univariate distributions that have been developed and introduced into finance in recent years, for example, heavy-tailed distributions, to be used as marginals. Merging the marginals and describing co-movements is left to the copula. Dependence structures expressed by copulas are not fully determined by linear correlation, as is the case with the multivariate normal distribution. Some classes of copulas additionally allow for capturing so-called tail dependence, describing, such as co-movements of asset returns conditional on one being (extremely) negative or positive. With this feature, copulas extend the notion of multivariate normals that is widely used for multi-dimensions, yet the complexity regarding parameters needed to describe certain copulas does not increase proportionately.
Copulas in finance are motivated by the application of multivariate distributions of returns arising in pricing, the aggregation of risk parameters as in VaR or credit default modeling, to mention just some. We will give examples showing the impact of using copulas below, identifying model risk.
Elaborate Example Of A Copula: The Bivariate Clayton Copula
For two random variables X and Y with marginal distributions FX (x) =Prob [X¾ x] and Fy =Prob [Y¾ y] a copula C defines a dependence structure by the joint distribution
FX,Y (x ,y)=Prob [X¾ x,Y¾ y]:=C (FX (x),F Y (y ))
The density of the joint distribution is derived by differentiation. As an example we define the bivariate Clayton copula
C clayton (u,v)= (u-*+v -*-1) -1/*
The Clayton copula is described by just one parameter, *, making it easy to handle. The parameter * is thus equivalent to the correlation in the bivariate normal case. It is directly related to Kendall's * - a measure of association - by *= . * also controls the notion of lower tail dependence, with formally given by
From Figure 1 we see what this means: increasing * corresponds to increasing lower tail dependence and the contour lines of the density plot extend more and more into the lower left corner. This leads to an increase of density at joint negative returns a feature that has been sought in modelling asset returns to describe market downturns and is fundamentally different from bivariate normal.
Other classes of copulas exist that are comparatively easy to handle such as Gauss, Frank and Gumbel copulas. They are all driven by one parameter similar to the Clayton copula yet exhibit different features, as can be seen from the examples in Figure 3. While the Clayton copula is preferred when modelling joint negative returns, the Gumbel copula should be used when emphasis is on joint positive returns. All of the copulas shown have the same marginals FX and FY and the same Kendall's *. From this we see that the concept of marginals and correlation does not suffice to define a multivariate distribution uniquely. Here the multivariate normal approach can be extended by the copula approach with the same marginals and Kendall's * such that different multivariate distributions result.
In practice, empirical data sets being available, the choice of copula can be made based on the tail dependence and the parameter of the copula can be estimated by sample data, as for the Clayton copula * can be estimated from sample Kendall's *.
Areas of Application: Examples
In all examples given below, Kendall's * and the marginals are the same, but the bivariate distribution is modelled by different copulas. We will use normal marginals.
The payout function of a European style minimum-of-two call on two assets S1 and S2 with strike K is given by
max(min(S1 (T),S2(T))-K,0)
We apply Monte Carlo Methods to generate option prices. Current asset prices are S1 (0) =S2(0) =100 and volatilities are 0.2 and 0.25, respectively. Time to maturity T is set to 0.5, strike price is K=98. Drift parameters and the risk free interest rate are set to 0. Figure 2 shows the computed prices of the option for different values of * and four classes of copulas. Bivariate densities in the cases of *=0.333 and *=-0.333 are given as contour plots in Figures 3 and 4. It is in the lower left quadrant of these density plots where the option payout is zero. Comparing the density plots in this quadrant explains most of the option price curves in Figure 2. Looking at *=0.5 we observe a difference in the option's price of nearly 20%. For a fixed * the differences are primarily based on the modelling of the dependence structure beyond linear correlation.
Most interesting is the observation that for each single * (-1,1) the corresponding sample linear correlations for the four copulas are about the same. Thus the price differences in Figure 2 translate directly into price differences where * is used instead of *. Using * is a common approach among modellers working with bivariate normals. Therefore a model builder who only considers Kendall's * (or an estimated linear correlation *) and marginals is not able to observe these structural differences in the data and the results.
A simple example showing the impact of using copulas on quantile numbers of portfolios for a portfolio consisting of a long S1 and a long S2 position is given in Figure 5. For *=0.333, compare Figure 3, samples have been drawn from the specified copulas and then the empirical quantiles for returns of S1+S 2 have been computed to generate the curves shown. At a 5% level q=0.05, for example, the range of quantile numbers/VaR attained moves from about 23 to 20. It is the Clayton copula, showing the lower tail dependence, which assumes the smallest value, as expected.
David X. Li (Working Paper Number 99-07, The RiskMetrics Group) shows how to apply the concept of copulas to the calculation of default correlation. He outlines that the current approach of default calculation in CreditMetrics™ is implicitly based on the assumption of a Gauss copula. The idea followed by Li is to model the 'time-until-default' by a positive continuous random variable (survival time of each defaultable asset). Then, the default correlation of two assets is defined as the correlation of their survival times. The marginal distributions of the survival times are obtained via the marketed credit curves. This is a typical setting for the use of copula functions that can easily be generalized by the use of alternative copulas.
Much work remains to be done in the copula approach in finance, ranging from developing test procedures for the detection of tail dependence of data sets to the set-up of stochastic differential equations, linked by copulas, up to the derivation of equivalent martingale measures in case of copula dependence. Nevertheless copulas provide a valuable tool when modelling dependence structures in multivariate data, especially when it comes to tail dependence. These dependence structures cannot be captured by the usual approach of using the multivariate normal distribution.
Figure 1: Clayton copula density contour plots with standard normal marginals.
Figure 2: Prices of a minimum-of-two call for varying * and different copulas. Also refer to Figures 3 and 4.
Figure 3: Copulas for *=0.333 with normal marginals contour plots of densities.
Figure 4: Copulas for *=-0.333 with normal marginals contour plots of densities.
Figure 5: Quantiles of return on S1 + S2. Single asset log-returns (r1 ,r2 ) are drawn from different copulas with normal marginals and *=0.333.
This week's Learning Curve was written by Volker Schmitz, PhD candidate in statistics at the Institute of Statisticsat the Aachen University of Technology, and Erwin Pier-Ribbert, risk methodology trading-head of research at Dresdner Bankin Frankfurt.
