Christian M. Hafner

Discrete time option pricing with flexible volatility estimation

Abstract. By extending the GARCH option pricing model of Duan (1995) to more flexible volatility estimation it is shown that the prices of out-of-the-money options strongly depend on volatility features such as asymmetry. Results are provided for the properties of the stationary pricing distribution in the case of a threshold GARCH model. For a stock index series with a pronounced leverage effect, simulated threshold GARCH option prices are substantially closer to observed market prices than the Black/Scholes and simulated GARCH prices.

On the estimation of dynamic conditional correlation models

The maximum likelihood estimator applied to the dynamic conditional correlation model is severely biased in high dimensions. This is, in particular, the case where the cross-section dimension is close to the sample size. It is argued that one of the reasons for the bias lies in an ill-conditioned sample covariance matrix, which is used in the so-called variance targeting technique to match sample and theoretical unconditional covariances. A reduction of the bias is proposed by using shrinkage to target methods for the sample covariance matrix. Alternatively, the identity matrix, a single factor model, and equicorrelation are used as targets. Since the shrinkage intensity decreases towards zero with increasing sample size, the estimator is asymptotically equivalent to the maximum likelihood estimator. The finite sample performance of the proposed estimator over alternative estimators is demonstrated through a Monte Carlo study. Finally, an illustrative application to financial time series compares alternative estimation methods by means of commonly used statistical and economic criteria.

A Generalized Dynamic Conditional Correlation Model: Simulation and Application to Many Assets

In this article, we put forward a generalization of the Dynamic Conditional Correlation (DCC) Model of Engle (2002). Our model allows for asset-specific correlation sensitivities, which is useful in particular if one aims to summarize a large number of asset returns. We propose two estimation methods, one based on a full likelihood maximization, the other on individual correlation estimates. The resultant generalized DCC (GDCC) model is considered for daily data on 39 U.K. stock returns in the FTSE. We find convincing evidence that the GDCC model improves on the DCC model and also on the CCC model of Bollerslev (1990).

Efficient estimation of a semiparametric dynamic copula model

A new semiparametric dynamic copula model is proposed where the marginals are specified as parametric GARCH-type processes, and the dependence parameter of the copula is allowed to change over time in a nonparametric way. A straightforward two-stage estimation method is given by local maximum likelihood for the dependence parameter, conditional on consistent first stage estimates of the marginals. First, the properties of the estimator are characterized in terms of bias and variance and the bandwidth selection problem is discussed. The proposed estimator attains the semiparametric efficiency bound and its superiority is demonstrated through simulations. Finally, the wide applicability of the model in financial time series is illustrated, and it is compared with traditional models based on conditional correlations.

Multivariate mixed normal conditional heteroskedasticity

A new multivariate volatility model where the conditional distribution of a vector time series is given by a mixture of multivariate normal distributions is proposed. Each of these distributions is allowed to have a time-varying covariance matrix. The process can be globally covariance stationary even though some components are not covariance stationary. Some theoretical properties of the model such as the unconditional covariance matrix and autocorrelations of squared returns are derived. The complexity of the model requires a powerful estimation algorithm. A simulation study compares estimation by maximum likelihood with the EM algorithm. Finally, the model is applied to daily US stock returns.

On asymptotic theory for multivariate GARCH models

The paper investigates the asymptotic theory for a multivariate GARCH model in its general vector specification proposed by Bollerslev, Engle and Wooldridge (1988) [4], known as the VEC model. This model includes as important special cases the so-called BEKK model and many versions of factor GARCH models, which are often used in practice. We provide sufficient conditions for strict stationarity and geometric ergodicity. The strong consistency of the quasi-maximum likelihood estimator (QMLE) is proved under mild regularity conditions which allow the process to be integrated. In order to obtain asymptotic normality, the existence of sixth-order moments of the process is assumed.

Semi-Parametric Modelling of Correlation Dynamics

In this paper we develop a new semi-parametric model for conditional correlations, which combines parametric univariate GARCH-type specifications for the individual conditional volatilities with nonparametric kernel regression for the conditional correlations. This approach not only avoids the proliferation of parameters as the number of assets becomes large, which typically happens in conventional multivariate conditional volatility models, but also the rigid structure imposed by more parsimonious models, such as the dynamic conditional correlation model. An empirical application to the 30 Dow Jones stocks demonstrates that the model is able to capture interesting asymmetries in correlations and that it is competitive with standard parametric models in terms of constructing minimum variance portfolios and minimum tracking error portfolios.

Nonlinear Time Series Analysis with Applications to Foreign Exchange Rate Volatility

The book deals with the econometric analysis of high frequency financial time series. It emphasizes a new nonparametric approach to volatility models and provides theoretical and empirical comparisons with conventional ARCH models, applied to foreign exchange rates. Nonparametric models are discussed that cope with asymmetry and long memory of volatility as well as heterogeneity of higher conditional moments.

Asymptotic theory for a factor GARCH model

This paper investigates the asymptotic theory for a factor GARCH model. Sufficient conditions for strict stationarity, existence of certain moments, geometric ergodicity and - mixing with exponential decay rates are established. These conditions allow for volatility spill-over and integrated GARCH. We then show the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) of the model parameters. The results are obtained under the finiteness of the fourth order moment of the innovations.

Estimating high-frequency foreign exchange rate volatility with nonparametric ARCH models

High-frequency foreign exchange rate (HFFX) series are analyzed on an operational time scale using models of the ARCH class. Comparison of the estimated conditional variances focuses on the asymmetry and persistence issue. Estimation results for parametric models confirm standard results for HFFX series, namely high persistence and no significance of the asymmetry coefficient in an EGARCH model. To find out whether these results are robust against alternative specifications, nonparametric models are estimated. Local linear estimation techniques are applied to a nonparametric ARCH model of order one (CHARN). The results show significant asymmetry of the volatility function. To allow for both flexibility and persistence, a higher-order multiplicative model is fitted. The results show important asymmetries in volatility. In contrast to the EGARCH specification, the news impact curves have different shapes for different lags and tend to increase slower at the boundaries.