In equation (2), g is the mean reversion rate (gravity), which pulls the variance to its long term mean , and is the volatility of the volatility σ(t). dz(t) is the standard Brownian motion, i.e. , is i.i.d., in particular is a random drawing from a standardized normal distribution n~(0,1). In equation (1), the underlying follows the standard geometric Brownian motion, which is also applied in Black–Scholes–Merton model, which however assumes constant volatility. The correlation between the stochastic processes (1) and (2) is introduced by correlating the two Brownian motions and . The instantaneous correlation between the Brownian motions is (3). The definition (3) can be conveniently modeled with the identity (4) where and are independent, and and are independent, t ≠ t’. The Cointelation SDE connects the SDE's above to the concept of mean reversion and drift which are usually concepts that are misunderstood by practitioners. The binomial correlation coefficient A further financial correlation measure, is the binomial correlation approach of Lucas (1995). We define the binomial events and where is the default time of entity and is the default time of entity . Hence if entity defaults before or at time , the random indicator variable will take the value in 1, and 0 otherwise. The same applies to . Furthermore, and is the default probability of and respectively, and is the joint probability of default. The standard deviation of a one-trial binomial event is , where P is the probability of outcome X. Hence, we derive the joint default dependence coefficient of the binomial events and as (5). By construction, equation (5) can only model binomial events, for example default and no default. The binomial correlation approach of equation (5) is a limiting case of the Pearson correlation approach discussed in section 1. As a consequence, the significant shortcomings of the Pearson correlation approach for financial modeling apply also to the binomial correlation model. Copula correlations A fairly recent, famous as well as infamous correlation approach applied in finance is the copula approach. Copulas go back to Sklar (1959). Copulas were introduced to finance by Vasicek (1987) and Li (2000). Copulas simplify statistical problems. They allow the joining of multiple univariate distributions to a single multivariate distribution. Formally, a copula function C transforms an n-dimensional function on the interval [0,1] into a unit-dimensional one: (6). More explicitly, let be a uniform random vector with and . Then