Potential PCA interpretation problems for volatility smile dynamics - Dimitri Reiswich; Robert Tompkins (Frankfurt School of Finance & Management, Centre for Practical Quantitative Finance (CPQF), 2009)
Principal Component Analysis (PCA) is a common procedure for the analysis of financial market data, such as implied volatility smiles or interest rate curves. Recently, Pelsser and Lord [11] raised the question whether PCA results may not be 'facts but artefacts'. We extend this line of research by considering an alternative matrix structure which is consistent with foreign exchange option markets. For this matrix structure, PCA effects which are interpreted as shift, skew and curvature can be generated from unstructured random processes. Furthermore, we find that even if a structured system exists, PCA may not be able to distinguish between these three effects. The contribution of the factors explaining the variance in the original system are incorrect. Finally, for a special case, we provide an analytic correction that recovers correct factor variances from those incorrectly estimated by PCA.
Keywords: 330; Principal Component Analysis; PCA; Level; Slope; Curvature; Twist; Bisymmetric Matrices; Centro-symmetric Matrices; Finanzmarkt; Zeitreihenanalyse; Hauptkomponentenanalyse; Varianzanalyse; Korrelation; Devisenoptionsgeschäft; Wechselkurs; Volatilität; Theorie