Common Functional Principal Components

Functional principal component analysis (FPCA) based on the Karhunen-Loève decomposition has been successfully applied in many applications, mainly for one sample problems. In this paper we consider common functional principal components for two sample problems. Our research is motivated not only by the theoretical challenge of this data situation but also by the actual question of dynamics of implied volatility (IV) functions. For different maturities the logreturns of IVs are samples of (smooth) random functions and the methods proposed here study the similarities of their stochastic behavior. Firstly we present a new method for estimation of functional principal components from discrete noisy data. Next we present the two sample inference for FPCA and develop two sample theory. We propose bootstrap tests for testing the equality of eigenvalues, eigenfunctions, and mean functions of two functional samples, illustrate the test-properties by simulation study and apply the method to the IV analysis.

[1]  Enno Mammen,et al.  A Dynamic Semiparametric Factor Model for Implied Volatility String Dynamics , 2005 .

[2]  P. Hall,et al.  Asymptotically optimal difference-based estimation of variance in nonparametric regression , 1990 .

[3]  K. J. Utikal,et al.  Inference for Density Families Using Functional Principal Component Analysis , 2001 .

[4]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[5]  Henry W. Altland,et al.  Applied Functional Data Analysis , 2003, Technometrics.

[6]  J. Dauxois,et al.  Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference , 1982 .

[7]  Wolfgang K. Härdle,et al.  The Dynamics of Implied Volatilities: A Common Principal Components Approach , 2003 .

[8]  W. Härdle,et al.  Statistical Tools for Finance and Insurance , 2003 .

[9]  Matthias R. Fengler.  Semiparametric Modeling of Implied Volatility , 2005 .

[10]  D. Aldous.  The Central Limit Theorem for Real and Banach Valued Random Variables , 1981 .

[11]  C. R. Rao,et al.  Some statistical methods for comparison of growth curves. , 1958 .

[12]  Li-Shan Huang,et al.  Nonparametric estimation of quadratic regression functionals , 1999 .

[13]  N. Kahalé.  An Arbitrage-free Interpolation of Volatilities , 2022 .

[14]  I. I. Gikhman,et al.  The Theory of Stochastic Processes II , 1975 .

[15]  B. Silverman,et al.  Estimating the mean and covariance structure nonparametrically when the data are curves , 1991 .