Trading, hedging and risk analysis of complex option portfolios depend on accurate pricing models. The modelling of implied volatilities (IV) plays an important role, since volatility is the crucial parameter in the Black-Scholes (BS) pricing formula. It is well known from empirical studies that the volatilities implied by observed market prices exhibit patterns known as volatility smiles or smirks that contradict the assumption of constant volatility in the BS pricing model. On the other hand, the IV is a function of two parameters: the strike price and the time to maturity and it is desirable in practice to reduce the dimension of this object and characterize the IV surface through a small number of factors. Clearly, a dimension reduced pricing-model that should reflect the dynamics of the IV surface needs to contain factors and factor loadings that characterize the IV surface itself and their movements across time.
A popular dimension reduction technique is the principal components analysis (PCA), employed for example by Fengler, Härdle, and Schmidt (2002) in the IV surface analysis. The discretization of the strike dimension and application of PCA yield suitable factors (weight vectors) in the multivariate framework. Noting that the IVs of fixed maturity could also be viewed as random functions, we propose to use the functional analogue of PCA. We utilize the truncated functional basis expansion described in Ramsay and Silverman (1997) to the IVs of the European options on the German stock index (DAX). The standard functional PCA, however, yields weight functions that are too rough, hence a smoothed version of functional PCA is proposed here.
Like Fengler, Härdle, and Villa (2003) we discover similarities of the resulting weight functions across maturity groups. Thus we propose an estimation procedure based on the Flury-Gautschi algorithm, Flury (1988), for the simultaneous estimation of the weight functions for two different maturities. This procedure yields common weight functions with the level, slope, and curvature interpretation known from the financial literature. The resulting common factors of the IV surface are the basic elements to be used in applications, such as simulation based pricing, and deliver a substantial dimension reduction.
The chapter is organized as follows. In Section 5.2 the basic financial framework is presented, while
in Section 5.3 we introduce the notation of the functional data analysis. In the following three
sections we analyze the IV functions using functional principal components, smoothed functional
principal components and common estimation of principal components, respectively.