Author(s)

K. Sch¨ottle & R. Werner

Abstract

Jaeckel and Rebonato [1] develop two different methods of creating valid correlation matrices: construction by hypersphere decomposition and by singular value (i.e. spectral) decomposition. Although both methods yield satisfactory results in practice, from a mathematical point of view they both share some theoretical drawbacks. Using results from semidefinite programming (SDP) we give the most general problem formulation to compute valid correlation matrices. We present numerical results which prove that these SDPs are rather easily solvable with efficient solvers for SDP problems (e.g. PENNON, see [3]). In contrast to Higham [2] we do not find numerical difficulties in solving the stated SDP problems. We close the article with two more very important features which have been neglected in literature so far. First, regularity and second, control of the condition number of the resulting correlation matrix are easily guaranteed by linear SDPconstraints. Numerical experiments show that these additional constraints do not harm the efficient numerical solution. Keywords: correlation matrix, semidefinite programming, risk management. 1 Motivation In their article [1] Jaeckel and Rebonato introduce the problem of finding the best correlation matrix. In this context, a correlation matrix is given by the following Definition 1. Let Sn be the space of real symmetric n × n matrices. Further, let Sn + ⊂ Sn be the cone of positive semidefinite matrices. A matrix G ∈ Sn + (G  0) is called correlation matrix, if Gii = 1 for i = 1, . . . , n.

Keywords

correlation matrix, semidefinite programming, risk management.