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Specifying time-dependent correlation matrices is a problem that occurs in several important areas of finance and risk management. The goal of this work is to tackle this problem by applying techniques of geometric integration in financial mathematics, i.e., to combine two fields of numerical mathematics that have not been studied yet jointly. Based on isospectral flows we create valid time-dependent correlation matrices, so called correlation flows, by solving a stochastic differential equation (SDE) that evolves in the special orthogonal group. Since the geometric structure of the special orthogonal group needs to be preserved we use stochastic Lie group integrators to solve this SDE. An application example is presented to illustrate this novel methodology.