Option data and modeling BSM implied volatility

This contribution to the Handbook of Computational Finance, Springer-Verlag, gives an overview on modeling implied volatility data. After introducing the concept of Black-Scholes-Merton implied volatility (IV), the empirical stylized facts of IV data are reviewed. We then discuss recent results on IV surface dynamics and the computational aspects of IV. The main focus is on various parametric, semi- and nonparametric modeling strategies for IV data, including ones which respect no-arbitrage bounds.Implied volatility

Arbitrage-Free Smoothing of the Implied Volatility Surface

The pricing accuracy and pricing performance of local volatility models crucially depends on absence of arbitrage in the implied volatility surface: an input implied volatility surface that is not arbitrage-free invariably results in negative transition probabilities and/ or negative local volatilities, and ultimately, into mispricings. The common smoothing algorithms of the implied volatility surface cannot guarantee the absence arbitrage. Here, we propose an approach for smoothing the implied volatility smile in an arbitrage-free way. Our methodology is simple to implement, computationally cheap and builds on the well-founded theory of natural smoothing splines under suitable shape constraints. Unlike other methods, our approach also works when input data are scarce and not arbitrage-free. Thus, it can easily be integrated into standard local volatility pricers.Arbitrage-Free Smoothing, Volatility, Implied Volatility Surface

A dynamic copula approach to recovering the index implied volatility skew

Equity index implied volatility functions are known to be excessively skewed in comparison with implied volatility at the single stock level. We study this stylized fact for the case of a major German stock index, the DAX, by recovering index implied volatility from simulating the 30 dimensional return system of all DAX constituents. Option prices are computed after risk neutralization of the multivariate process which is estimated under the physical probability measure. The multivariate models belong to the class of copula asymmetric dynamic conditional correlation models. We show that moderate tail-dependence coupled with asymmetric correlation response to negative news is essential to explain the index implied volatility skew. Standard dynamic correlation models with zero tail-dependence fail to generate a sufficiently steep implied volatility skew.Copula Dynamic Conditional Correlation, Basket Options, Multivariate GARCH Models, Change of Measure, Esscher Transform

DSFM fitting of Implied Volatility Surfaces

The implied volatility became one of the key issues in modern quantitative finance, since the plain vanilla option prices contain vital information for pricing and hedging of exotic and illiquid options. European plain vanilla options are nowadays widely traded, which results in a great amount of high-dimensional data especially on an intra day level. The data reveal a degenerated string structure. Dynamic Semiparametric Factor Models (DSFM) are tailored to handle complex, degenerated data and yield low dimensional representation of the implied volatility surface (IVS). We discuss estimation issues of the model and apply it to DAX option prices.dynamic semiparametric factor model, implied volatility, vanilla options, DAX option prices

A Dynamic Semiparametric Factor Model for Implied Volatility String Dynamics

A primary goal in modelling the implied volatility surface (IVS) for pricing and hedging aims at reducing complexity. For this purpose one fits the IVS each day and applies a principal component analysis using a functional norm. This approach, however, neglects the degenerated string structure of the implied volatility data and may result in a modelling bias. We propose a dynamic semiparametric factor model (DSFM), which approximates the IVS in a finite dimensional function space. The key feature is that we only fit in the local neighborhood of the design points. Our approach is a combination of methods from functional principal component analysis and backfitting techniques for additive models. The model is found to have an approximate 10% better performance than a sticky moneyness model. Finally, based on the DSFM, we devise a generalized vega-hedging strategy for exotic options that are priced in the local volatility framework. The generalized vega-hedging extends the usual approaches employed in the local volatility framework.Smile, local volatility, generalized additive model, backfitting, functional principal component analysis

Price-setting and price-adjustment behavior for fast-moving consumer goods

"In many economic models, it is assumed that prices adjust instantaneously to changes of economic conditions (e.g., to shocks in demand or production costs). Since the existence of price rigidities has been frequently documented, more realistic models require that infrequent and lumpy price adjustment have to be taken into account. There are still many unresolved issues in this area, both theoretically and empirically. In this paper, we show that the dynamics and dispersion of retail prices can be investigated using price data obtained from the GfK Consumer Panel for 1995. Our results document the importance of psychological pricing points for price setting, confirming results from many earlier studies. A new aspect of our analysis that has not been investigated in the literature is the relevance of psychological prices points for price adjustment and aggregation. We interpret our findings as suggestive evidence for the notion that rigidities are relevant for aggregate dynamics in Germany. However, we also confirm that a structural aggregation theory is necessary for a better understanding of the relevance of micro-level rigidities for aggregate dynamics. In such a more comprehensive model, price data obtained from the GfK Consumer Panel might also prove very helpful in the future. Among the three other areas of empirical research that could potentially be explored with price data from the GfK Consumer Panel, the analysis of the relationship between individual price dynamics, price dispersion and aggregate inflation proves particularly fruitful. Moreover, the very disaggregated, high-frequency data contained in this data-set are almost unique. In other research areas which require that prices changes (and not only distributions of prices) are observed over time, empirical tests unfortunately suffer from the fact that time series of individual prices can be constructed only under additional strong assumptions." (author's abstract

Spreading out: covid-19 and the changing geography of consumption

Lockdowns change where consumers spend their money; expenditures are more evenly spread in space, write Martin Brown, Matthias Fengler, Rafael Lalive, Robert Rohrkemper, and Thomas Spyche

Measuring spot variance spillovers when (co)variances are time-varying - the case of multivariate GARCH models

We propose global and disaggregated spillover indices that allow us to assess variance and covariance spillovers, locally in time and conditionally on time-t information. Key to our approach is the vector moving average representation of the half-vectorized `squared' multivariate GARCH process of the popular BEKK model. In an empirical application to a four-dimensional system of broad asset classes (equity, fixed income, foreign exchange and commodities), we illustrate the new spillover indices at various levels of (dis)aggregation. Moreover, we demonstrate that they are informative of the value-at-risk violations of portfolios composed of the considered asset classes