In this paper a novel method is developed for the problem of finding a low-rank correlation matrix nearest to a given correlation matrix. The method is based on majorization and therefore it is globally convergent. The method is computationally efficient, is straightforward to implement, and can handle arbitrary weights on the entries of the correlation matrix. A simulation study suggests that majorization compares favourably with competing approaches in terms of the quality of the solution within a fixed computational time. The problem of rank reduction of correlation matrices occurs when pricing a derivative dependent on a large number of assets, where the asset prices are modelled as correlated log-normal processes.

Geometric optimisation algorithms are developed that efficiently find the nearest low-rank correlation matrix. We show, in numerical tests, that our methods compare favourably to the existing methods in the literature. The connection with the Lagrange multiplier method is established, along with an identification of whether a local minimum is a global minimum. An additional benefit of the geometric approach is that any weighted norm can be applied. The problem of finding the nearest low-rank correlation matrix occurs as part of the calibration of multi-factor interest rate market models to correlation.

Abstract. Geometric optimisation algorithms are developed that efficiently find the nearest low-rank correlation matrix. We show, in numerical tests, that our methods compare favourably to the existing methods in the literature. The connection with the Lagrange multiplier method is established, along with an identification of whether a local minimum is a global minimum. An additional benefit of the geometric approach is that any weighted norm can be applied. The problem of finding the nearest low-rank correlation matrix occurs as part of the calibration of multi-factor interest rate market models to correlation.

A novel algorithm is developed for the problem of finding a low-rank correlation matrix nearest to a given correlation matrix. The algorithm is based on majorization and, therefore, it is globally convergent. The algorithm is computationally efficient, is straightforward to implement, and can handle arbitrary weights on the entries of the correlation matrix. A simulation study suggests that majorization compares favourably with competing approaches in terms of the quality of the solution within a fixed computational time. The problem of rank reduction of correlation matrices occurs when pricing a derivative dependent on a large number of assets, where the asset prices are modelled as correlated log-normal processes. Such an application mainly concerns interest rates.

This paper shows that the forward rates process discretized by a single time step together with a separability assumption on the volatility function allows for representation by a low-dimensional Markov process. This in turn leads to efficient pricing by for example finite differences. We then develop a discretization based on the Brownian bridge especially designed to have high accuracy for single time stepping. The scheme is proven to converge weakly with order 1. We compare the single time step method for pricing on a grid with multi step Monte Carlo simulation for a Bermudan swaption, reporting a computational speed increase of a factor 10, yet pricing sufficiently accurate.