(2005). Applied Multivariate Statistical Analysis. Technometrics: Vol. 47, No. 4, pp. 517-517.

Applied Multivariate Statistical Analysis

Econometric policy evaluation: A critique

I Descriptive Techniques: Comparison of Batches.- II Multivariate Random Variables: A Short Excursion into Matrix Algebra Moving to Higher Dimensions Multivariate Distributions Theory of the Multinormal Theory of Estimation Hypothesis Testing.- III Multivariate Techniques: Decomposition of Data Matrices by Factors Principal Components Analysis Factor Analysis Cluster Analysis Discriminant Analysis.- Correspondence Analysis.- Canonical Correlation Analysis.- Multidimensional Scaling.- Conjoint Measurement Analysis.- Application in Finance.- Computationally Intensive Techniques.- A: Symbols and Notations.- B: Data.- Bibliography.- Index.

The Bootstrap and Edgeworth Expansion

The prices of index options at a given date are usually represented via the corresponding implied volatility surface, presenting skew/smile features and term structure which several models have attempted to reproduce. However, the implied volatility surface also changes dynamically over time in a way that is not taken into account by current modelling approaches, giving rise to `Vega' risk in option portfolios. Using time series of option prices on the SP500 and FTSE indices, we study the deformation of this surface and show that it may be represented as a randomly fluctuating surface driven by a small number of orthogonal random factors. We identify and interpret the shape of each of these factors, study their dynamics and their correlation with the underlying index. Our approach is based on a Karhunen-Loeve decomposition of the daily variations of implied volatilities obtained from market data. A simple factor model compatible with the empirical observations is proposed. We illustrate how this ...

http://www.cmap.polytechnique.fr/%7Erama/papers/dynamics.pdf

Dynamics of implied volatility surfaces

The pricing accuracy and pricing performance of local volatility models depends on the absence of arbitrage in the implied volatility surface. An input implied volatility surface that is not arbitrage-free can result in negative transition probabilities and consequently mispricings and false greeks. We propose an approach for smoothing the implied volatility smile in an arbitrage-free way. The method is simple to implement, computationally cheap and builds on the well-founded theory of natural smoothing splines under suitable shape constraints.

/pdf/arbitrage-free-smoothing-of-the-implied-volatility-surface-32oiwsacla.pdf

Arbitrage-free smoothing of the implied volatility surface

We propose a semiparametric factor model, which approximates the implied volatility surface (IVS) in a finite dimensional function space. Unlike standard principal component approaches typically used to reduce complexity, our approach is tailored to the degenerated design of IVS data. In particular, we only fit in the local neighborhood of the design points by exploiting the expiry effect present in option data. Using DAX index option data, we estimate the nonparametric components and a low-dimensional time series of latent factors. The modeling approach is completed by studying vector autoregressive models fitted to the latent factors.

A semiparametric factor model for implied volatility surface dynamics

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

Semiparametric Modeling of Implied Volatility

It is common practice to identify the number and sources of shocks that move, e.g., ATM implied volatilities by principal components analysis. This approach, however, is likely to result in a loss of information, since the surface structure of implied volatilities is neglected. In this paper we analyze the implied volatility surface along maturity slices with a common principal components analysis (CPC), known from morphometrics. In CPC analysis, the space spanned by the eigenvectors is identical across groups, whereas variances associated with the common principal components vary. Our analysis shows that implied volatility surface dynamics can be traced back to a common eigenstructure in maturity slices. This empirical result is used to set up a factor model for implied volatility surface dynamics.

/pdf/the-dynamics-of-implied-volatilities-a-common-principal-100yw0zq81.pdf

The Dynamics of Implied Volatilities: A Common Principal Components Approach

It is common practice to identify the number and sources of shocks that move implied volatilities across space and time by applying Principal Components Analysis (PCA) to pooled covariance matrices of changes in implied volatilities. This approach, however, is likely to result in a loss of information, since the surface structure of implied volatilities in the maturities and moneyness dimension is neglected. In this paper we propose to estimate the implied volatility surface at each point in time nonparametrically and to analyze the implied volatility surface slice by slice with a common principal components analysis (CPCA). As opposed to traditional PCA, the basic assumption of CPCA is that the space spanned by the eigenvectors is identical across groups, whereas variances associated with the components are allowed to vary. This allows us to study a p variate random vector of k groups, say the volatility smile at p different grid points of moneyness for k maturities, simultaneously. Our evidence suggests that surface dynamics can indeed be traced back to a common eigenstructure between covariance matrices of the surface slices, which allow for the usual shift, slope, and twist interpretation of shocks to implied volatilities. This insight is a suitable starting point for VaR Monte Carlo Simulations of delta-gamma neutral, vega sensitive option portfolios.

Matthias R. Fengler

Papers

Arbitrage-free smoothing of the implied volatility surface

A semiparametric factor model for implied volatility surface dynamics

Semiparametric Modeling of Implied Volatility

The Dynamics of Implied Volatilities: A Common Principal Components Approach

The dynamics of implied volatilities: A common principal components approach