Published Articles
Measuring spot variance spillovers when (co)variances are time-varying – the case of multivariate GARCH models
with Helmut Herwartz
Oxford Bulletin of Economics and Statistics, 80(1), 2018, 135–159.
[Link to Journal Version]
Abstract: In highly integrated markets, news spreads at a fast pace and bedevils risk monitoring and optimal asset allocation. We therefore propose global and disaggregated measures of variance transmission that allow one to assess spillovers locally in time. Key to our approach is the vector ARMA representation of the second-order dynamics of the popular BEKK model. In an empirical application to a four-dimensional system of US asset classes – equity, fixed income, foreign exchange and commodities – we illustrate the second-order transmissions at various levels of (dis)aggregation. Moreover, we demonstrate that the proposed spillover indices are informative on the value-at-risk violations of portfolios composed of the considered asset classes.
Managing risk with a realized copula parameter
with Ostap Okhrin
Computational Statistics & Data Analysis, 100, 2016, 131-152.
[Link to Journal Version]
Abstract: 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 Hoeding's 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.
Are classical option pricing models consistent with observed option second-order moments? Evidence from high-frequency data
with Francesco Audrino
Journal of Banking & Finance, 64, 2015, pages 46–63.
[Link to Journal Version]
Abstract: As a means of validating an option pricing model, we compare the ex-post intra-day realized variance of options with the realized variance of the associated underlying asset that would be implied using assumptions as in the Black and Scholes (BS) model, the Heston and the Bates model. Based on data for the S&P 500 index, we find that the BS model is strongly directionally biased due to the presence of stochastic volatility. The Heston model reduces the mismatch in realized variance between the two markets, but deviations are still significant. With the exception of short-dated options, we achieve best approximations after controlling for the presence of jumps in the underlying dynamics. Finally, we provide evidence that, although heavily biased, the realized variance based on the BS model contains relevant predictive information that can be exploited when option high-frequency data is not available.
A simple and general approach to fitting the discount curve under no-arbitrage constraints
with Lin Yee Hin
Finance Research Letters, 15, 2015, pages 78–84.
[Link to Journal Version]
Abstract: We suggest a simple and general approach to fitting the discount curve under no-arbitrage constraints based on a penalized shape-constrained B-spline. The approach accommodates B-splines of any order and fitting both under the L1 and the L2 loss functions. An application to US STRIPS data from 2001-2015 suggests that polynomial splines of order three and four are mandatory to obtain reasonable fits. The choice of the loss function appears to be less relevant.
Specification and structural break tests for additive models with applications to realized variance data
with Enno Mammen & Michael Vogt
Journal of Econometrics 188 (1), 2015, pages 196–218.
[Link to Journal Version]
Abstract: We study two types of testing problems in a nonparametric additive model setting: We develop methods to test (i) whether an additive component function has a given parametric form and (ii) whether an additive component has a structural break. We apply the theory to a nonparametric extension of the linear heterogeneous autoregressive model which is widely employed to describe realized variance data. We find that the linearity assumption is often rejected, but actual deviations from linearity are mild.
A variance spillover analysis without covariances: what do we miss?
with Katja Gisler
Journal of International Money and Finance 51, 2015, pages 174–195.
[Link to Journal Version]
Abstract: 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.
Semi-nonparametric estimation of the call-option price surface under strike and time-to-expiry no-arbitrage constraints
with Lin Yee Hin
Journal of Econometrics 184 (2), 2015, pages 242-261
[Link to Journal Version]
Abstract: We suggest a semi-nonparametric estimator for the call-option price surface. The estimator is a bivariate tensor-product B-spline. To enforce no-arbitrage constraints across strikes and expiry dates, we establish sufficient no-arbitrage conditions on the control net of the B-spline surface. The conditions are linear and therefore allow for an implementation of the estimator by means of standard quadratic programming techniques. The consistency of the estimator is proved. By means of simulations, we explore the statistical efficiency benefits that are associated with estimating option price surfaces and state-price densities under the full set of no-arbitrage constraints. We estimate a call-option price surface, families of first-order strike derivatives, and state-price densities for S&P 500 option data.
A dynamic copula approach to recovering the index implied volatility skew
with Helmut Herwartz & Christian Werner
Journal of Financial Econometrics 10 (3), 2012, pages 457-493.
[Link to Journal Version]
Abstract: 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.
Static hedges for reverse barrier options with robustness against skew risk : an empirical analysis
with Jan Maruhn & Morten Nalholm
Quantitative Finance 11 (5), 2011, pages 711-727.
[Link to Journal Version]
Abstract: 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.
Arbitrage-free smoothing of the implied volatility surface
Quantitative Finance 9 (4), 2009, pages 417-428.
[Link to Journal Version]
Abstract: 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.
Does hedging with implied volatility factors improve the hedging efficiency of barrier options?
with Szymon Borak & Wolfgang Härdle
The Journal of Risk Model Validation 3 (1), 2009, pages 73-92.
[Link to Journal Version]
Abstract: The price of a barrier option depends on the shape of the entire implied volatility surface which is a high-dimensional dynamic object. Barrier options are hence exposed to non-trivial volatility risk. We extract the key risk factors of implied volatility surface fluctuations by means of a semiparametric factor model. Based on the factors we define a practical hedging procedure within a local volatility framework. The hedging performance is evaluated using DAX index options.
Hedging under alternative stickiness assumptions: an empirical analysis for barrier options
with Bernd Engelmann & Peter Schwendner
Journal of Risk 12 (1), 2009, pages 53-77.
[Link to Journal Version]
Abstract: 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.
Variability and price dispersion in a stable monetary environment: evidence from Germany
with Joachim Winter
Managerial and Economic Decisions 28 (7), 2007, pages 789-801.
[Link to Journal Version]
Abstract: We investigate the relationship between inflation and price variation using highly disaggregated, weekly price data for consumption goods recorded in Germany during 1995, a low-inflation period. We find a significant positive correlation between the rates of price change and price dispersion, both at the level of individual products and product groups. However, we find no correlation between the rates of price change and price variability. Together with results from similar studies, Tommasi (1993. Optimal Pricing, Inflation, and the Cost of Price Adjustment. MIT Press: London, Cambridge, MA; 485–511) and Parsley (1996. J. Money Credit Banking28: 323–341), a remarkable pattern emerges: when aggregate nominal shocks are small, only price dispersion is correlated with price changes. As the rate of inflation rises, both variability and dispersion become affected. During hyper-inflation, systematic movements of price dispersion seem to disappear. We conclude that price dispersion is best explained by micro-economic frictions in price adjustment, whereas price variability appears to be related to costly price search and information problems.
On extracting information implied in options
with Michal Benko & Wolfgang Härdle
Computational Statistics 22 (4), 2007, pages 543-553.
[Link to Journal Version]
Abstract: Options are financial instruments with a payoff depending on future states of the underlying asset. Therefore option markets contain information about expectations of the market participants about market conditions, e.g., current uncertainty on the market and corresponding risk. A standard measure of risk calculated from plain vanilla options is the implied volatility (IV). IV can be understood as an estimate of the volatility of returns in future period. Another concept based on the option markets is the state-price density (SPD) that is a density of the future states of the underlying asset. From raw data we can recover the IV function by nonparametric smoothing methods. Smoothed IV estimated by standard techniques may lead to a non-positive SPD which violates no arbitrage criteria. In this paper, we combine the IV smoothing with SPD estimation in order to correct these problems. We propose to use the local polynomial smoothing technique. The elegance of this approach is that it yields all quantities needed to calculate the corresponding SPD. Our approach operates only on the IVs—a major improvement comparing to the earlier multi-step approaches moving through the Black–Scholes formula from the prices to IVs and vice-versa.
A semiparametric factor model for implied volatility surface dynamics
with Wolfgang Härdle & Enno Mammen
Journal of Financial Econometrics 5 (2), 2007, pages 189-218.
[Link to Journal Version]
Abstract: 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.
Static versus dynamic hedges: an empirical comparison for barrier options
with Bernd Engelmann & Morten Nalholm & Peter Schwendner
Review of Derivatives Research 9 (3), 2006, pages 239-264
[Link to Journal Version]
Abstract: We conduct an empirical comparison of static versus dynamic hedges of barrier options. Using more than five years of data, we compare a number of static hedges from the literature with dynamic hedges based on the local volatility model. The main result is that the variability of profit-and-loss distributions from certain static hedges is significantly smaller than that of dynamic hedges and robust to changing market scenarios. Furthermore, these static hedges are able to provide a robust tracking of barrier options’ sensitivities.
Quoting multiasset equity options in the presence of errors from estimating correlations
with Peter Schwendner
Journal of Derivatives 11 (4), 2004, pages 43-54
[Link to Journal Version]
Abstract: 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.
The dynamics of implied volatilities: A common principle components approach
with Wolfgang Härdle & Christophe Villa
Review of Derivatives Research 6 (3), 2003, pages 179-202
[Link to Journal Version]
Abstract: 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.
Common factors governing VDAX movements and the maximum loss
with Wolfgang Härdle & Peter Schmidt
Financial Markets and Portfolio Management 16 (1), 2002, pages 16-29
[Link to Journal Version]
Abstract: Based on daily VDAX data we analyse the factors governing the movements of implied volatilities of options on the German stock index DAX. We derive common factors representing shift and slope of the term structure of ATM implied volatilities. Further we present a risk management tool for option portfolios using the maximum loss concept and give empirical results.