Semiparametric Modeling of Implied Volatility
Description
Reviews / Votes
From the reviews:
"This book brings together recent advances in the theory of implied volatility and refined semiparametric estimation strategies and dimension reduction methods for functional surfaces. The theory of implied and local volatility is presented. The smile-consistent modeling approaches are discussed in detail. . This book is for readers with a preknowledge of stochastic processes and interest in financial derivatives, as for example plain vanilla or exotic options." (Klaus Ehemann, Zentralblatt MATH, Vol. 1084, 2006)
"The parameter that measures volatility has long caused many problems in financial modeling. . Fengler has written a research monograph. . Concepts are presented in detail, elegantly connecting the past and current research, mathematical presentation, and numerical output (graphics). . The appendices serve primarily for presentation of proofs and some results from stochastic calculus. This book is suitable for researchers, graduate students, and finance professionals." (Ita Cirovic Donev, MathDL, March, 2006)
"This short book addresses one of the most . fundamental questions in financial mathematics and derivatives trading, namely, volatility modeling and management. . the author does a good job in presenting the local volatility models, their implementation, and the problem in using this approach for hedging. . It is an admirable attempt at the daunting task of modeling the dynamics of the IVS." (Andrew Carter and Jean-Pierre Fouque, SIAM Review, Vol. 49 (1), 2007)
More details
Other editions
Additional editions
Person
Content
5.1 Introduction
The IVS is a complex, high-dimensional random object. In building a model, it is thus desirable to have a low-dimensional representation of the IVS. This aim can be achieved by employing dimension reduction techniques. Generally it is found that two or three factors with appealing .nancial interpretations are su.cient to capture more than 90% of the IVS dynamics. This implies for instance for a scenario analysis in risk-management that only a parsimonious model needs to be implemented to study the vega-sensitivity of an option portfolio, Fengler et al. (2002b). This section will give a general overview on dimension reduction techniques in the context of IVS modeling. We will consider techniques from multivariate statistics and methods from functional data analysis. Sections 5.2 and 5.3 will provide an in-depth treatment of the CPC and the semiparametric factor model of the IVS together with an extensive empirical analysis of the German DAX index data.
In multivariate analysis, the most prominent technique for dimension reduction is principal component analysis (PCA). The idea is to seek linear combinations of the original observations, so called principal components (PCs) that inherit as much information as possible from the original data. In PCA, this means to look for standardized linear combinations with maximum variance. The approach appears to be sensible in an analysis of the IVS dynamics, since a large variance separates out systematic from idiosyncratic shocks that drive the surface. As a nice byproduct, the structure of the linear combinations reveals relationships among the variables that are not apparent in the original data. This helps understand the nature of the interdependence between di.erent regions in the IVS.
In .nance, PCA is a well-established tool in the analysis of the term structure of interest rates, see Gouri´eroux et al. (1997) or Rebonato (1998) for textbook treatments: PCA is applied to a multiple time series of interest rates (or forward rates) of various maturities that is recovered from the term structure of interest rates. Typically, a small number of factors is found to represent the dynamic variations of the term structure of interest rates. The studies of Bliss (1997), Golub and Tilman (1997), Ni.keer et al. (2000), and Molgedey and Galic (2001) are examples of this kind of literature.
This approach does not immediately carry over to the analysis of IVs due to the surface structure. Consequently, in analogy to the interest rate case, empirical work .rst analyzes the term structure of IVs of ATM options, only, Zhu and Avellaneda (1997) and Fengler et al. (2002b). Alternatively, one smile at one given maturity can be analyzed within the PCA framework, Alexander (2001b). Skiadopoulos et al. (1999) group IVs into maturity buckets, average the IVs of the options, whose maturities fall into them, and apply a PCA to each bucket covariance matrix separately. A good overview of these methods can be found in Alexander (2001a).
