Financial correlation


Financial correlations measure the relationship between the changes of two or more financial variables over time. For example, the prices of equity stocks and fixed interest bonds often move in opposite directions: when investors sell stocks, they often use the proceeds to buy bonds and vice versa. In this case, stock and bond prices are negatively correlated.
Financial correlations play a key role in modern finance. Under the capital asset pricing model, an increase in diversification increases the return/risk ratio. Measures of risk include value at risk, expected shortfall, and portfolio return variance.

Financial correlation and the Pearson product-moment correlation coefficient

There are several statistical measures of the degree of financial correlations. The Pearson product-moment correlation coefficient is sometimes applied to finance correlations. However, the limitations of Pearson correlation approach in finance are evident. First, linear dependencies as assessed by the Pearson correlation coefficient do not appear often in finance. Second, linear correlation measures are only natural dependence measures if the joint distribution of the variables is elliptical. However, only few financial distributions such as the multivariate normal distribution and the multivariate student-t distribution are special cases of elliptical distributions, for which the linear correlation measure can be meaningfully interpreted. Third, a zero Pearson product-moment correlation coefficient does not necessarily mean independence, because only the two first moments are considered. For example, will lead to Pearson correlation coefficient of zero, which is arguably misleading. Since the Pearson approach is unsatisfactory to model financial correlations, quantitative analysts have developed specific financial correlation measures. Accurately estimating correlations requires the modeling process of marginals to incorporate characteristics such as skewness and kurtosis. Not accounting for these attributes can lead to severe estimation error in the correlations and covariances that have negative biases. In a practical application in portfolio optimization, accurate estimation of the variance-covariance matrix is paramount. Thus, forecasting with Monte-Carlo simulation with the Gaussian copula and well-specified marginal distributions are effective.

Financial correlation measures

Correlation Brownian motions

applied a correlation approach to negatively correlate stochastic stock returns and stochastic volatility. The core equations of the original Heston model are the two stochastic differential equations, SDEs
and
where S is the underlying stock, is the expected growth rate of, and is the stochastic volatility of at time t. In equation, g is the mean reversion rate, which pulls the variance to its long term mean, and is the volatility of the volatility σ. dz is the standard Brownian motion, i.e., is i.i.d., in particular is a random drawing from a standardized normal distribution n~. In equation, 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 and is introduced by correlating the two Brownian motions and. The instantaneous correlation between the Brownian motions is
The definition can be conveniently modeled with the identity
where and are independent, and and are independent, t ≠ t’.

The binomial correlation coefficient

A further financial correlation measure, is the binomial correlation approach of Lucas. 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
By construction, equation can only model binomial events, for example default and no default. The binomial correlation approach of equation 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. Copulas were introduced to finance by Vasicek and Li.
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 into a unit-dimensional one:
More explicitly, let be a uniform random vector with and. Then there exists a copula function such that
where F is the joint cumulative distribution function and, i = 1,..., ni are the univariate marginal distributions. is the inverse of. If the marginal distributions are continuous, it follows that C is unique. For properties and proofs of equation, see Sklar and Nelsen.
Numerous types of copula functions exist. They can be broadly categorized in one-parameter copulas as the Gaussian copula, and the Archimedean copula, which comprise Gumbel, Clayton and Frank copulas. Often cited two-parameter copulas are student-t, Fréchet, and Marshall-Olkin. For an overview of these copulas, see Nelsen.
for example in a collateralized debt obligation, CDO. This was first done by Li in 2006. He defined the uniform margins as cumulative default probabilities Q for entity i at a fixed time t, :
Hence, from equations and we derive the Gaussian default time copula CGD,
In equation the terms map the cumulative default probabilities Q of asset i for time t,, percentile to percentile to standard normal. The mapped standard normal marginal distributions are then joined to a single n-variate distribution by applying the correlation structure of the multivariate normal distribution with correlation matrix R. The probability of n correlated defaults at time t is given by.

Copulae and the 2008 financial crisis

Numerous non-academic articles have been written demonizing the copula approach and blaming it for the 2008 financial crisis, see for example Salmon 2009, Jones 2009, and Lohr 2009..
Tail dependence
In a crisis, financial correlations typically increase, see studies by Das, Duffie, Kapadia, and Saita and Duffie, Eckner, Horel and Saita and references therein. Hence it would be desirable to apply a correlation model with high co-movements in the lower tail of the joint distribution. It can be mathematically shown that the Gaussian copula has relative low tail dependence, as seen in the following scatter plots.


Figure 1: Scatter plots of different copula models
As seen in Figure 1b, the student-t copula exhibits higher tail dependence and might be better suited to model financial correlations. Also, as seen in Figure 1, the Gumbel copula exhibits high tail dependence especially for negative co-movements. Assuming that correlations increase when asset prices decrease, the Gumbel copula might also be a good correlation approach for financial modeling.
Calibration
A further criticism of the Gaussian copula is the difficulty to calibrate it to market prices. In practice, typically a single correlation parameter is used to model the default correlation between any two entities in a collateralized debt obligation, CDO. Conceptually this correlation parameter should be the same for the entire CDO portfolio. However, traders randomly alter the correlation parameter for different tranches, in order to derive desired tranche spreads. Traders increase the correlation for ‘extreme’ tranches as the equity tranche or senior tranches, referred to as the correlation smile. This is similar to the often cited implied volatility smile in the Black–Scholes–Merton model. Here traders increase the implied volatility especially for out-of-the money puts, but also for out-of-the money calls to increase the option price..
In a mean-variance optimization framework, accurate estimation of the variance-covariance matrix is paramount. Thus, forecasting with Monte-Carlo simulation with the Gaussian copula and well-specified marginal distributions are effective. Allowing the modeling process to allow for empirical characteristics in stock returns such as auto-regression, asymmetric volatility, skewness, and kurtosis is important. Not accounting for these attributes lead to severe estimation error in the correlations and variances that have negative biases.
Risk management
A further criticism of the Copula approach is that the copula model is static and consequently allows only limited risk management, see Finger or Donnelly and Embrechts. The original copulas models of Vasicek and Li and several extensions of the model as Hull and White or Gregory and Laurent do have a one period time horizon, i.e. are static. In particular, there is no stochastic process for the critical underlying variables default intensity and default correlation. However, even in these early copula formulations, back testing and stress testing the variables for different time horizons can give valuable sensitivities, see Whetten and Adelson and Meissner, Hector, and. Rasmussen.
In addition, the copula variables can be made a function of time as in Hull, Predescu, and White. This still does not create a fully dynamic stochastic process with drift and noise, which allows flexible hedging and risk management. The best solutions are truly dynamic copula frameworks, see section ‘Dynamic Copulas’ below.