Approximation of eigenvalues of spot cross volatility matrix with a view toward principal component analysis

TL;DR

This paper compares two methods for estimating eigenvalues of the spot cross volatility matrix in interest rate markets, using limit theorems and a Heston-type model to evaluate their effectiveness.

Contribution

It introduces a new estimation scheme based on Quadratic Variation and compares it with the existing Fourier Series-based method.

Findings

The Quadratic Variation scheme is effective in estimating eigenvalues.

Limit theorems are established for both estimation schemes.

Comparison using a Heston-type model shows relative performance differences.

Abstract

In order to study the geometry of interest rates market dynamics, Malliavin, Mancino and Recchioni [A non-parametric calibration of the HJM geometry: an application of It\^o calculus to financial statistics, {\it Japanese Journal of Mathematics}, 2, pp.55--77, 2007] introduced a scheme, which is based on the Fourier Series method, to estimate eigenvalues of a spot cross volatility matrix. In this paper, we present another estimation scheme based on the Quadratic Variation method. We first establish limit theorems for each scheme and then we use a stochastic volatility model of Heston's type to compare the effectiveness of these two schemes.