A Workout in Computational Finance

14 Copulas and the Pitfalls of Correlation

The problem of finding and quantifying dependencies between time series of observables frequently occurs in quantitative finance and econometrics.¹ Here, by “dependence” we refer to any situation in which random variables are not probabilistically independent. Even though the term “correlation” is often used in the same sense, we use it in a technically more strict sense in this work to refer to particular types of dependencies, as detailed below. In addition to the correlation between two or more variables, a signal can also show cross-correlation with itself. This so-called autocorrelation describes, loosely speaking, the similarity between observations as a function of the time Δt separating them. Autocorrelation is also linked to the mean reversion property many of the models discussed in this book display, for example, the Hull-White models and the Black-Karasinski model.

Many financial models for dependent risk are based on the assumption of normally distributed factors (multivariate normality), and linear correlation is usually employed as a measure for dependence. However, as pointed out in Chapter 12, financial data are rarely normally distributed, but tend to follow distributions with heavier tails. Furthermore, synchronized extreme downturns, which are observed in reality, cannot be modeled using multivariate normal distributions. To correctly reproduce these properties, other elliptical distributions² are used. Estimators ...