Fengler Semiparametric Modeling of Implied Volatility

2005
ISBN: 978-3-540-30591-0
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 224 Seiten, eBook

Reihe: Springer Finance Lecture Notes

Yet that weakness is also its greatest strength. People like the model because they can easily understand its assumptions. The model is often good as a ?rst approximation, and if you can see the holes in the assumptions you can use the model in more sophisticated ways. Black (1992) Expected volatility as a measure of risk involved in economic decision making isakeyingredientinmodern?nancialtheory:therational,risk-averseinvestor will seek to balance the tradeo? between the risk he bears and the return he expects. The more volatile the asset is, i.e. the more it is prone to exc- sive price ?uctuations, the higher will be the expected premium he demands. Markowitz (1959), followed by Sharpe (1964) and Lintner (1965), were among the ?rst to quantify the idea of the simple equation ‘more risk means higher return’ in terms of equilibrium models. Since then, the analysis of volatility and price ?uctuations has sparked a vast literature in theoretical and quan- tative ?nance that re?nes and extends these early models. As the most recent climax of this story, one may see the Nobel prize in Economics granted to Robert Engle in 2003 for his path-breaking work on modeling time-dependent volatility.

Zielgruppe

Research

Autoren/Hrsg.

Fengler, Matthias R.

Inhaltsverzeichnis

The Implied Volatility Surface.- Smile Consistent Volatility Models.- Smoothing Techniques.- Dimension-Reduced Modeling.- Conclusion and Outlook.

Leseproben

5 Dimension-Reduced Modeling (p.125)

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).

Über Autor(innen)

Matthias Fengler took his PhD in Finance at the Humboldt-Universität zu Berlin and is now a quantitative analyst at Sal. Oppenheim, Frankfurt.