## What Is a Copula?

Copula is a probability model that represents a multivariate uniform distribution, which examines the association or dependence between many variables. Put differently, a copula helps isolate the joint or marginal probabilities of a pair of variables that are enmeshed in a more complex multivariate system. The copula is then the unique index or set of instructions for describing how those pairs fit together in the more complex system. This method is useful as it can help identify spurious correlations observed in the data. It is also useful in fine-tuning derivatives pricing models where the price of one security depends on the price of some underlying security (e.g., an options contract or CDO).

Although the statistical calculation of a copula was developed in 1959, it was not applied to financial markets and finance until the late 1990s.

### Key Takeaways

- A copula is a statistical method for understanding the joint probabilities of a multivariate distribution.
- The word copula comes from the Latin for "link" or "tie" together, where the term is used in linguistics to describe such linking words or phrases.
- Today, copulas are employed in advanced financial analysis to better understand outcomes that involve fat tails and skewness.

## Understanding Copulas

Latin for "link" or "tie," copulas are a set of mathematical tools used in finance to help identify capital adequacy, market risk, credit risk, and operational risk. Copulas rely on the interdependence of returns of two or more assets, and would usually be calculated using the correlation coefficient. However, correlation works best with normal distributions, while distributions in financial markets are most often non-normal in nature. The copula, therefore, has been applied to areas of finance such as options pricing and portfolio value-at-risk (VaR) to deal with skewed or asymmetric distributions.

Copulas are quite complex mathematical functions and require sophisticated algorithms and computing power to be practical in real-world applications.

Copulas were first developed by mathematician Abe Sklar in 1959. Sklar's theorem states that any multivariate joint distribution can be simplified and expressed in terms of univariate marginal distribution functions along with a unique copula that contains the information on how those distributions fit together.

## Copulas and Options Pricing

Options theory, particularly options pricing is a highly specialized area of finance. Multivariate options are widely used where there is a need to hedge against a number of risks simultaneously; such as when there is an exposure to several currencies. The pricing of a basket of options is not a simple task. Advancements in Monte Carlo simulation methods and copula functions offer an enhancement to the pricing of bivariate contingent claims, such as derivatives with embedded options.