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 (CAPM; a model recognised by a Nobel prize), 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, (y ≠ 0) 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 (as much as 70% of the true values). 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 Steven Heston 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 (1) and (2) where S is the underlying stock, is the expected growth rate of , and is the stochastic volatility of at time t.