Approximating Correlation Matrices Using Stochastic Lie Group Methods

Abstract

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.

Higher strong order methods for linear Itô SDEs on matrix Lie groups

“…For checking the convergence order, we set G = SO(3) and g = so(3). In order to ensure the conditions of Theorem 2.1 we have used the set up of matrices K t and V t proposed by Muniz et al [22] for d = 1. Specifically, we chose the time-dependent functions…”

“…This problem can be solved by the stochastic correlation model presented in [22]. The main ideas of the approach are outlined below.…”

“…Q t ∈ SO (2). Following the approach in [22] the matrix Q t is now assumed to be driven by the SDE (2.3) which can be solved by using Algorithm 3.1. With the resulting matrices approximations of P t can be computed with (4.1), which can then be transformed to corresponding correlation matrices .…”

“…With the resulting matrices approximations of P t can be computed with (4.1), which can then be transformed to corresponding correlation matrices . At last, a density function is estimated from this correlation flow and the free parameters involved are calibrated such that the density function matches the density function from the historical data, see [22] for details.…”

“…We are confronted with geometric constraints, e.g. in the form of a positivity constraint on interest rates [14,25,30] or a symmetry and positivity constraint on covariance and correlation matrices [22], which are important in, for example, risk management and portfolio optimization.…”