What are Copulas and How are they Used in Quantitative Finance?

Short answer

Copulas are used to model joint distribution of multiple underlyings. They permit a rich 'correlation' structure between underlyings. They are used for pricing, for risk management, for pairs trading, etc., and are especially popular in credit derivatives.

Example

You have a basket of stocks which, during normal days, exhibit little relationship with each other. We might say that they are uncorrelated. But on days when the market moves dramatically they all move together. Such behaviour can be modelled by copulas.

Long answer

The technique now most often used for pricing credit derivatives when there are many underlyings is that of the copula. The copula^{[1]} function is a way of simplifying the default dependence structure between many underlyings in a relatively transparent manner. The clever trick is to separate the distribution for default for each individual name from the dependence structure between those names. So you can rather easily analyse names one at a time, for calibration purposes, for example, and then bring them all together in a multivariate distribution. Mathematically, the copula way of representing the dependence (one marginal distribution per underlying, and a dependence structure) is no different from specifying a multivariate density function. But it can simplify the analysis.

The copula approach in effect allows us to readily go from a single-default world to a multiple-default world almost seamlessly. And by choosing the nature of the dependence, the copula function, we can explore models with rich 'correlation structure. For example, having a higher degree of dependence during big market moves is quite straightforward.

Take N uniformly distributed random variables U1, U2,..., Un, each defined on [0,1]. The copula function is defined as

Clearly we have

That is the copula function. The way it links many univariate distributions with a single multivariate distribution is as follows.

Let xi, x2, xn be random variables with cumulative distribution functions (so-called marginal distributions) of Fi(xi), F2(x2), Fn(xn). Combine the Fs with the copula function,

and it s easy to show that this function F (x1, x2, ... , xN )isthe same as

In pricing basket credit derivatives we would use the copula approach by simulating default times of each of the constituent names in the basket. And then perform many such simulations in order to be able to analyse the statistics, the mean, standard deviation, distribution, etc., of the present value of resulting cashflows.

Here are some examples of bivariate copula functions. They are readily extended to the multivariate case.

Bivariate normal

where N2 is the bivariate Normal cumulative distribution function, and N--1 is the inverse of the univariate Normal cumulative distribution function.

Frank

Fréchet—Hoeffding upper bound

Gumbel—Hougaard

This copula is good for representing extreme value distributions.

Product

One of the simple properties to examine with each of these copulas, and which may help you decide which is best for your purposes, is the tail index. Examine

This is the probability that an event with probability less than u occurs in the first variable given that at the same time an event with probability less than u occurs in the second variable. Now look at the limit of this as u 0,

This tail index tells us about the probability of both extreme events happening together.

References and Further Reading

Li, D 2000 On Default Correlation: A Copula Function Approach. Risk-Metrics Working Paper

Nelsen, RB 1999 An Introduction to Copulas. Springer Verlag