Functional Principal Component Analysis for Derivatives of Multivariate Curves

Abstract

We present two methods based on functional principal component analysis (FPCA) for the estimation of smooth derivatives of a sample of random functions, which are observed in a more than one-dimensional domain. We apply eigenvalue decomposition to a) the dual covariance matrix of the derivatives, and b) the dual covariance matrix of the observed curves. To handle noisy data from discrete observations, we rely on local polynomial regressions. If curves are contained in a finite-dimensional function space, the second method performs better asymptotically. We apply our methodology in a simulation and empirical study, in which we estimate state price density (SPD) surfaces from call option prices. We identify three main components, which can be interpreted as volatility, skewness and tail factors. We also find evidence for term structure variation.

A functional time series analysis of forward curves derived from commodity futures

“…For instance, it has been previously observed (see e.g. Grith et al, 2018) that under-smoothed representations of the sampled curves can be more effective when the goal is to estimate a principal component basis, although such representations may sacrifice accuracy in estimating each individual curve.…”