Reliance on linear correlation coefficients and joint normal distribution of returns in multi-asset trading strategies can be badly misleading. Such conventions often overestimate diversification benefits and underestimate drawdowns in times of market stress. Copulas can describe the joint distribution of multiple returns or price series more realistically. They separate the modelling of dependence structures from the marginal distributions of the individual returns. Copulas are particularly suitable for assessing joint tail distributions, such as the behaviour of portfolios in extreme market states. This is when risk management matters most. A critical choice is the appropriate marginal distributions and copula functions based on the stylized features of contract return data. Multivariate distributions based on these assumptions can be simulated in Python.

The below post is based on a range of articles and posts which are linked next to the respective quote.

This post ties in with this site’s summary on managing systemic risk, particularly the section on calibrating tail risk.

### What is a copula?

“Copulas are used to __model joint distribution of multiple underlying [data series]__. They permit a rich ‘correlation’ structure between [them]. They are used for pricing, for risk management, for pairs trading and so forth., and are especially popular in credit derivatives. [For example] a basket of stocks [may] 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.” [ebrary]

“A copula is…just a joint cumulative density function of multiple random variables with marginal distributions [of the type] uniform (0,1). A copula __allows us to separate modeling of dependence structure from modeling marginal distributions__.[That is] because we can use the probability integral transform [which] says that we can transform any random variable to uniform random variable.”[Pavlov]

“Copulas can be used to model extreme markets and asset interdependencies, i.e. __joint tail realizations__, a feature that is not obtained when using linear correlations.” [Rachev, Stein, and Sun]

“An n-dimensional copula is a multivariate distribution function C, with uniform distributed margins in [0, 1] (U(0, 1)) [that is] and the following properties:

- C:[0, 1]
^{n}-> [0, 1] - C is grounded and n-increasing
- C has margins Ci which satisfy Ci(u) = C(1, . . . , 1, u, 1, . . . , 1) =u for all u ∈ [0, 1].”

“So what is the point of all this? Why don’t we just use multivariate Gaussian? In multivariate Gaussian marginal distributions must also be Gaussian. When we use copula, we can use probability integral transform to convert uniform marginals to any distribution we want (while preserving dependency structure).” [Pavlov]

“For many financial applications, __the problem is not to use a given multivariate distribution but consists in finding a convenient distribution to describe some stylized facts__, for example the relationships between different asset returns…In most applications, the distribution is assumed to be a multivariate gaussian or a log-normal distribution for tractable calculus, even if the Gaussian assumption may not be appropriate…It is well known that __asset returns are fat-tailed__. In the portfolio analysis framework, the variance corresponds to the risk measure, but it implies that the world is Gaussian. Two difficulties…Gaussian assumption and joint distribution modelling…can be treated as a problem of copulas…The copula is in fact the dependence structure of the model…A copula is a function that links univariate marginals to their multivariate distribution.” [Bouye]

### How are copulas calculated and estimated?

“Generally, a copula is used to __separate the pure randomness of one variable [say] a financial asset [return] from the interdependencies between it and other variables. __By doing so, one can model each variable separately and, in addition, have a measure of the relations between those variables…__One can choose for each and any asset in a spectrum the most appropriate type of distribution, not influencing the interdependencies__ between those variables/assets. The interdependencies between those variables are represented by a multivariate probability distribution function, which is informative on the joint outcomes of the variables, and this multivariate distribution function is the copula.” [Rachev, Stein, and Sun]

“We do not need any technical representation to make clear why the interdependencies are modelled fully flexible by a copula: An N-dimensional multivariate distribution representing the copula of N financial variables (for example returns, rates, etc.) has a support on the N-dimensional cube and is a standardized measure being able to capture all possible relationships between the N-variables. To summarize: the use of copula __allows the separation of univariate randomness (defined by the individual probability distribution functions of financial random variables) and dependence structure defined by the copula__.” [Rachev, Stein, and Sun]

“The copulas allow us to model the dependence structure independently from the marginal distributions. In this way, we may construct a multivariate distribution with different margins and the dependence structure given from a particular type of copula function. Therefore, __a crucial step in this context is the choice and the calibration of the most adequate copula function from the real financial data__…the issue is selecting the type of copula function which fits better the empirical data.” [Di Clemente and Romano]

“Practical algorithms for generating Monte Carlo scenarios from a multivariate distribution with a fixed copula and different margins are easily implemented to simulate financial asset returns. The traditional models use the multinormal distribution to simulate asset log-returns. We can choose different marginal distributions for building more efficient algorithms also using a normal copula. The __choice of the margins seems to have a more significant impact than the choice of the type of the copula__ on the results of the simulation. In this paper, only the copula effects are taken into account.” [Di Clemente and Romano]

For the calculation and visualization of copulas in Python view post here and here.

“__Simulations have an important role in statistical inference.__ They especially help to investigate properties of estimators. Moreover, they are necessary to understand the underlying multivariate distribution…You have to perform simulations to get an idea of the shape of the distribution.” [Bouye]

“In many applications, __parametric models are useful in order to study the properties of the underlying statistical model__. The idea to decompose the complex problems of multivariate modelling into two more simplified statistical problems is justified…The parameters of the model should be in a form most suitable for easy interpretation (e.g., a parameter is interpreted as either a dependence parameter or a univariate parameter but not some mixture).” [Bouye]

### How can copulas be used for trading strategies?

“__Copula functions can be utilized in financial applications to determine the dependence structure of the financial asset returns in the portfolio__…In particular, the copula functions can model the dependence structure between risk factors (for example, equity returns, interest rate returns, and foreign exchange returns) in a more reliable way…Empirical evidence has proved the inadequacy of the multi-normal distribution, traditionally adopted to model the financial asset returns distribution. Copula functions can be employed in a flexible way for building efficient algorithms and to simulate a more adequate distribution of the financial assets.” [Di Clemente and Romano]

“The usual __linear correlation is not a satisfactory measure of the dependence among different securities__ for several reasons. First…when extreme events are frequently observed, the linear correlation between these securities is undefined…Second, [linear] correlation is a measure for linear dependence only. Third, linear correlation is not invariant under nonlinear strictly increasing transformations…Fourth, linear correlation only measures the degree of dependence but does not clearly discover the structure of dependence.” [Rachev, Stein, and Sun]

“Copulas are also a powerful tool for finance because the modelling problem can be split into two steps: the first step deals with the identification of the marginal distributions…the second step consists in defining the appropriate copula in order to represent the dependence structure…Building multivariate distributions with copulas becomes very easy…__Financial applications [of copulas include] asset returns modelling [for] portfolio aggregation [and] time series modelling [as well as] risk measurement__… Before 1999, copulas have not been used in finance.” [Bouye]

“Empirical evidence has widely proved that the multinormal distribution is inadequate to model portfolio’s financial asset returns distribution…

- The empirical marginal distributions are skewed and fat-tailed.
- The normal distribution does not consider the possibility of extreme joint co-movements for financial asset returns.

In other words, __the real dependence structure of the financial assets is different from the Gaussian one and especially under situations of market stress__…In particular, the credit assets clearly show a non-normal return distribution and the phenomena of asymmetry, leptokurtosis, and tail dependence.” [Di Clemente and Romano]

“While it is an obvious mathematical fact that the multivariate Gaussian distribution is not capable of tail dependencies, it was used in growing numbers and frequency in financial markets…__The use of Gaussian distributions in financial market applications is widely accepted as being flawed__ due to the fact that this distribution type attributes too low probabilities to extreme observations…It is generally no real improvement to the linear correlation coefficient concerning extreme dependencies” [Rachev, Stein, and Sun]

“[The figure below] shows the__ usefulness of the right type of copula and marginal distribution modelling and the drawbacks of using Gaussian copulas in financial markets__, here with an example for two stock market indices. Comparing Figures 1a and 1d, it is obvious that a combination of Gaussian copula and Gaussian marginal distribution does a poor job when being fitted to the two stock markets’ returns. In addition, it can be seen from Figure 1c that even when modelling the single markets adequately using stable distributions for the univariate randomness of one variable, the occurrences of joint extreme observations is not reflected appropriately when using a Gaussian copula. Figure 1b, in contrast, shows __a very good fit to the historical returns for the skewed t copula and marginal distribution combination__.” [Rachev, Stein, and Sun]

“Correlation trading denotes the trading activity aimed at exploiting changes in correlation or more generally in the dependence structure of assets or risk factors. Likewise, __correlation risk is defined as the exposure to losses triggered by changes in correlation__. The copula function technique, which enables analyzing the dependence structure of a joint distribution independently from the marginal distributions, is the ideal tool to assess the impact of changes in market co-movements on the prices of assets and the amount of risk in a financial position.” [Cherubini]