George A. Christodoulakis

Correlated ARCH (CorrARCH): Modelling the time-varying conditional correlation between financial asset returns

Abstract Although the time variation of the conditional correlations of asset returns is a well established stylized fact (and of crucial importance for efficient financial decisions) there is no explicit general model available for its estimation and forecasting. In this paper, we propose a bivariate GARCH covariance structure in which conditional variances can follow any GARCH-type process, while conditional correlation is generated by an explicit discrete-time stochastic process, the CorrARCH process. A high order CorrARCH can parsimoniously be represented by a CorGARCH process. The model successfully generates the reported stylized facts, establishes an autocorrelation structure for correlations and thus provides an explicit framework for out-of-sample forecasting. We provide empirical evidence from the G7 Stock Market Indexes.

Generating Composite Volatility Forecasts with Radom Factor Betas

Publisher Summary Out-of-sample volatility forecasts are of crucial importance in virtually all financial decisions such as portfolio selection, the pricing of primary and derivative assets, and risk management methodologies. This chapter presents a methodology for generating composite out-of-sample volatility forecasts when returns are driven by random beta multiple factor processes. Given time varying factor and idiosyncratic conditional variances, analysis focuses on two cases: conditionally autoregressive betas in which the factor and its beta share the same innovation and latent autoregressive betas in which the factor and its beta experience distinct but correlated innovations. The literature has proposed both direct and indirect parameterization methodologies to model the temporal characteristics of beta coefficients, with different implications for volatility forecasting. In direct parameterization, beta coefficients are modeled as random parameters in that beta consists of a constant plus a noise term. Indirect approaches to modeling consider the beta coefficient as the ratio of the asset and factor conditional covariance over the factor conditional variance and model the numerator and the denominator of the ratio separately.

Sharpe style analysis in the msci sector portfolios: a monte carlo integration approach

We examine a decision-theoretic Bayesian framework for the estimation of Sharpe Style portfolio weights of the MSCI sector returns. Following [van Dijk and Kloek (1980)] an appropriately defined prior density of style weights can incorporate non-negativity and other constraints. We use factor-mimicking portfolios as proxies to global style factors such as Value, Growth, Debt and Size. Our computational approach is based on Monte Carlo Integration (MCI) of [Kloek and van Dijk (1978)] for the estimation of the posterior moments and distribution of portfolio weights. MCI provides a number of advantages, such as a flexible choice of prior distributions, improved numerical accuracy of the estimated parameters, the use of inequality restrictions in prior distributions and exact inference procedures. Our empirical findings suggest that, contrary to existing evidence, style factors do explain the MSCI sector portfolio returns for the particular sample period. Further, non-negativity constraints on portfolio weights were found to be binding in all cases.

Common volatility and correlation clustering in asset returns

Abstract We present a new multivariate framework for the estimation and forecasting of the evolution of financial asset conditional correlations. Our approach assumes return innovations with time dependent covariances. A Cholesky decomposition of the asset covariance matrix, with elements written as sines and cosines of spherical coordinates allows for modelling conditional variances and correlations and guarantees its positive definiteness at each time t . As in Christodoulakis and Satchell [Christodoulakis, G.A., Satchell, S.E., 2002. Correlated ARCH (CorrARCH): Modelling the time-varying conditional correlation between financial asset returns. European Journal of Operational Research 139 (2), 350–369] correlation is generated by conditionally autoregressive processes, thus allowing for an autocorrelation structure for correlation. Our approach allows for explicit out-of-sample forecasting and is consistent with stylized facts as time-varying correlations and correlation clustering, co-movement between correlation coefficients, correlation and volatility as well as between volatility processes (co-volatility). The latter two are shown to depend on correlation and volatility persistence. Empirical evidence on a trivariate model using monthly data from Dow Jones Industrial, Nasdaq Composite and the 3-month US Treasury Bill yield supports our theoretical arguments.

Forecast Evaluation in the Presence of Unobserved Volatility

Abstract A number of volatility forecasting studies have led to the perception that the ARCH- and Stochastic Volatility-type models provide poor out-of-sample forecasts of volatility. This is primarily based on the use of traditional forecast evaluation criteria concerning the accuracy and the unbiasedness of forecasts. In this paper we provide an analytical assessment of volatility forecasting performance. We use the volatility and log volatility framework to prove how the inherent noise in the approximation of the true- and unobservable-volatility by the squared return, results in a misleading forecast evaluation, inflating the observed mean squared forecast error and invalidating the Diebold–Mariano statistic. We analytically characterize this noise and explicitly quantify its effects assuming normal errors. We extend our results using more general error structures such as the Compound Normal and the Gram–Charlier classes of distributions. We argue that evaluation problems are likely to be exacerbated by non-normality of the shocks and that non-linear and utility-based criteria can be more suitable for the evaluation of volatility forecasts.

Hashing GARCH: a reassessment of volatility forecasting performance

Publisher Summary The autoregressive conditional heteroscedasticity (ARCH) and generalized ARCH (GARCH) models are widely used in economics and finance. There is a perception of poor out-of-sample performance on the basis of relatively high forecast error statistics as well as limited explanatory power of the ARCH forecasts for the “true” volatility as quantified by the squared or absolute returns over the relevant forecast horizons. This chapter investigates the noise inherent in such a forecast methodology. From the traditional statistical point of view, the chapter picks up the forecast that incorporates the maximum possible information. By contrast, there is a growing body of literature examining the extent to which more natural criteria such as expected utility, profit, or likelihood maximization can provide adequate forecast evaluation as opposed to the traditional statistical criteria. The problem of the economic value of forecasts, connecting forecast evaluation to the behavior of the economic agent, is addressed. Using profit measures, it is shown that a forecast can be of low value according to forecast error statistics, but at the same time it can be very profitable. It is shown that the inherent noise in the approximation of the actual and unobservable volatility by the squared return results in a misleading forecast evaluation.

Structural credit loss distributions under non-normality

In the context of Merton [1974] and Vasicek [1987, 2002] Gaussian single-factor credit risk models, the authors examine the impact of neglected non-normality of the underlying asset return process on the shape of the derived credit loss distribution and the resulting Basel capital requirements. They relax the Gaussian assumption and specify skew-normal and skew-student t densities to model the underlying asset return process, thus generalizing the credit loss distribution, and develop a maximum-likelihood estimator for the structural parameters. They apply their approach to aggregate charge-off rates published by the Federal Reserve Board for 10 U.S. sectors. The authors show that the non-gaussian modeling of the common factor provides a better characterization of data than its Gaussian counterpart and that it has a significant impact on the capital requirements, depending on the sign and magnitude of the skew-related coefficient. On the other hand, the non-gaussian modeling of the idiosyncratic factor does not provide a significantly better characterization than the Gaussian base case. The latter could stem from the fact that the sector portfolios are large, so the idiosyncratic component has been diversified away.

On the Evolution of Global Style Factors in the Morgan Stanley Capital International Universe of Assets

We explore the effects of global style factors on the Morgan Stanley Capital International (MSCI) universe of stocks from 1988 to 1998. An initial Bayesian analysis by Hall, Hwang and Satchell (2001) shows very low posterior probabilities for the significance of the (constant) beta coefficients in a linear factor model. When we allow for a flexible structure, in which betas follow conditionally autoregressive discrete time random processes, introduced by Christodoulakis and Satchell (2000), this result is reversed in nearly half of the cases. Style–factor betas are often found to fluctuate significantly around zero, exhibiting serial correlation, persistence and thus predictability. We report results for such style factors as value, growth, debt, and size on all the individual stocks of the MSCI universe as well as on capitalization–weighted and equally weighted aggregate sector returns.

The simulation of option prices with application to LIFFE options on futures

We build a framework for modelling the deviation of observed option prices from the Black & Scholes prices. We use a flexible model for a density, a two sided switching Weibull, to capture the implied volatility. The model can be used to generate prices, it can take into account no-arbitrage bounds for option prices and is capable of generating such stylised facts as the smile effect. We apply this methodology to LIFFE options on German government bond futures.

The Robustness of Estimators in Structural Credit Loss Distributions

This paper provides Monte Carlo results for the performance of the method of moments (MM), maximum likelihood (ML) and ordinary least squares (OLS) estimators of the credit loss distribution implied by the Merton (1974) and Vasicek (1987, 2002) framework when the common or idiosyncratic asset-return factor is non-Gaussian and, thus, the true credit loss distribution deviates from the theoretical one. We find that OLS and ML outperform MM in small samples when the true data-generating process comprises a non-Gaussian common factor. This result intensifies as the sample size increases and holds in all cases. We also find that all three estimators present a large bias and variance when the true data-generating process comprises a non-Gaussian idiosyncratic factor. This last result holds independently of the sample size, across different asset correlation levels, and it intensifies for positive shape parameter values.