Matthias R. Fengler

Arbitrage-free smoothing of the implied volatility surface

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.

Implied volatility string dynamics

A primary goal in modelling the dynamics of implied volatility surfaces (IVS) 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 severe modelling bias. We propose a dynamic semiparametric factor model, 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 the typical naive trader models. The model can be a backbone in risk management serving for value at risk computations and scenario analysis.

A variance spillover analysis without covariances: what do we miss?

We evaluate the relevance of covariances in the transmission mechanism of variance spillovers across the US stock, US bond and gold markets from July 2003 to December 2012. For that purpose, we perform a comparative spillover analysis between a model that considers covariances and a model that considers only variances. Our results emphasise the importance of covariances. Including covariances leads to an overall increase of the spillover level and detects the beginnings of the financial crisis and of the US debt ceiling crisis earlier than the spillover measure that considers only variances. Even for the low-dimensional system that we consider, one misses important variance spillover channels when covariances are excluded.

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.

The Analysis of Implied Volatilities

The analysis of volatility in financial markets has become a first rank issue in modern financial theory and practice: Whether in risk management, portfolio hedging, or option pricing, we need to have a precise notion of the markets expectation of volatility. Much research has been done on the analysis of realized historic volatilities, Roll (1977) and references therein. However, since it seems unsettling to draw conclusions from past to expected market behavior, the focus shifted to implied volatilities, Dumas, Fleming and Whaley (1998). To derive implied volatilities the Black and Scholes (BS) formula is solved for the constant volatility parameter a using observed option prices. This is a more natural approach as the option value is decisively determined by the markets assessment of current and future volatility. Hence implied volatility may be used as an indicator for market expectations over the remaining lifetime of the option. It is well known that the volatilities implied by observed market prices exhibit a pattern that is far different from the flat constant one used in the BS formula. Instead of finding a constant volatility across strikes, implied volatility appears to be non flat, a stylized fact which has been called smile effect. In this chapter we illustrate how implied volatilities can be analyzed. We focus first on a static and visual investigation of implied volatilities, then we concentrate on a dynamic analysis with two variants of principal components and interpret the results in the context of risk management.

Quoting multiasset equity options in the presence of errors from estimating correlations

Anyone using an option valuation model needs estimates of returns volatility. But volatility has been found to be hard to forecast accurately, partly because it varies randomly over time. Fortunately, the market provides a directly observable volatility estimate in the form of the implied volatility that can be backed out from an option’s market price. But for several increasingly popular types of options, the payoff distribution depends both on stock volatilities and also on the correlations among them. For such options, including basket options and options on the max or on the min, there are no dependable sources of implied correlations to use in pricing them. Correlations must be estimated from historical data, which leads to substantial estimation risk. In this article, Fengler and Schwendner describe a block bootstrap procedure that allows an investor to evaluate a multiasset option’s exposure to parameter risk from imperfectly estimated correlations. The results are translated into minimum bid-ask spreads that are required to account for this additional source of risk.

Managing risk with a realized copula parameter

A dynamic copula model is introduced, in which the copula structure is inferred from the realized covariance matrix estimated from within-day high-frequency data. The estimation is carried out in a method-of-moments fashion using Hoeffdings lemma. Applying this procedure day by day gives rise to a time series of daily copula parameters which can be approximated by an autoregressive time series model. This allows one to capture time-varying dependence. In an application to portfolio risk-management, it is found that this time-varying realized copula model exhibits very good forecasting properties for the one-day ahead value at risk.

DSFM fitting of implied volatility surfaces

Implied volatility is one of the key issues in modern quantitative finance, since 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 representations of the implied volatility surface (IVS). We discuss estimation issues of the model and apply it to DAX option prices.

Hedging under alternative stickiness assumptions: an empirical analysis for barrier options

In this study, we empirically analyze dynamic hedges of barrier options in the local volatility model using more than five years of data on the DAX, a major German equity index. The emphasis is on the comparison of the hedge performance of different hedging strategies under alternative stickiness assumptions on the dynamics of the implied volatility surface. We compare sticky-strike, sticky-moneyness and local volatility-implied (model-consistent) hedges for barrier options with a maturity of one and two years. We find that sticky-strike performs best, with the choice of the hedging strategy being a much more important factor for successful risk management than the stickiness assumption.

Static hedges for reverse barrier options with robustness against skew risk : an empirical analysis

We conduct an empirical evaluation of a static super-replicating hedge of barrier options. The hedge is robust to uncertainty about the future skew. Using almost seven years of current data on the DAX, we evaluate the performance of the hedge and compare it with those of both a dynamic and a static replicating hedge. The main result is that the robustness of the static super-replicating portfolio is also empirically confirmed in practice such that the hedge sets an upper bound for the price of skew risk for barrier options.