Symmetric GARCH Models Research Articles

Stock Market Exuberance: Linkages between U.S. and the European Markets

This paper investigates the high frequency behaviour of US, British and German stock market exuberance using an index provided by standard portfolio arbitrage relationships. Symmetric and asymmetric multivariate GARCH models are implemented to quantify international volatility comovements. In the period from January 1992 to April 2000 a change in the pattern of volatility transmission is detected at the beginning of summer 1997. Empirical analysis suggests that equity markets volatility modelling with exuberance indexes is more accurate than modelling with stock returns. Furthermore, the estimated conditional covariances between exuberance indexes fluctuate over time and tend to rise whenever volatility increases.

Asymmetric covariance in spot‐futures markets

AbstractThis article studies how the spot‐futures conditional covariance matrix responds to positive and negative innovations. The main results of the article are achieved by obtaining the Volatility Impulse Response Function (VIRF) for asymmetric multivariate GARCH structures, extending Lin (1997) findings for symmetric GARCH models. This theoretical result is general and can be applied to analyze covariance dynamics in any financial system. After testing how multivariate GARCH models clean up volatility asymmetries, the Asymmetric VIRF is computed for the Spanish stock index IBEX‐35 and its futures contract. The empirical results indicate that the spot‐futures variance system is more sensitive to negative than positive shocks, and that spot volatility shocks have much more impact on futures volatility than vice versa. Additionally, evidence is obtained showing that optimal hedge ratios are insensitive to the well‐known asymmetric volatility behavior in stock markets. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:1019–1046, 2003

Tests for time reversibility: a complementarity analysis

Since time reversibility (TR) is a necessary condition for an independent and identically distributed (iid) sequence, several tests for TR have been suggested to be applied as tests for model misspecification. In this paper, we analyze possible complementarities among two well known TR tests (Ramsey and Rothman's test, and Chen et al.'s test) in two situations: (1) the fitted model is a linear ARMA model when the true data generating process is a nonlinear-in-mean model (either threshold autoregressive or bilinear), and (2) the fitted model is a symmetric GARCH model but the true process belongs to the asymmetric GARCH family (either EGARCH or GJR). The results suggest that there are not significant complementarities among both tests, although moment restrictions do not allow applying always the most powerful test.

Forecasting Volatility in European Stock Markets with Non-Linear GARCH Models

This paper investigates the forecasting performance of three popular variants of the non-linear GARCH models, namely VS-GARCH, GJR-GARCH and Q-GARCH, with the symmetric GARCH(1,1) model as a benchmark. The application involves ten European stock price indexes. Forecasts produced by each non-linear GARCH model and each index are evaluated using a common set of classical criteria, as well as forecast combination techniques with constant and non-constant weights. With respect to the standard GARCH specification, the non-linear models generally lead to better forecasts in terms of both smaller forecast errors and lower biases. In-sample forecast combination regressions are better than those from single Mincer-Zarnowitz regressions. The out-of-sample performance of combining forecasts is less satisfactory, irrespective of the type of weights adopted.

Measuring Oil Price Volatility

In this paper we try to measure oil price uncertainty. The measure of uncertainty is based on the conditional standard deviations which are derived from univariate (G)ARCH models. The measure of uncertainty we choose is the within-year high-low range of the conditional standard deviations. It is likely that the higher the uncertainty, the higher the high-low range within a year will be. We focus on volatility of the price of a barrel Brent crude, over the period 5 January, 1982 to 23 April, 2002 representing 5296 observations. The preferred model is a symmetric GARCH(1,3) model. Asymmetric leverage effects are not found. We also examine the volatility in monthly time series for the period January, 1970 to April, 2002. For this time span and frequency we prefer the GARCH(1,1) model.

Bayesian option pricing using asymmetric GARCH models

This paper shows how one can compute option prices from a Bayesian inference viewpoint, using a GARCH model for the dynamics of the volatility of the underlying asset. The proposed evaluation of an option is the predictive expectation of its payoff function. The predictive distribution of this function provides a natural metric, provided it is neutralised with respect to risk, for gauging the predictive option price or other option evaluations. The proposed method is compared to the Black and Scholes evaluation, in which a marginal mean volatility is plugged, but which does not provide a natural metric. The methods are illustrated using symmetric, asymmetric and smooth transition GARCH models with data on a stock index in Brussels.

Can portmanteau nonlinearity tests serve as general mis-specification tests?: Evidence from symmetric and asymmetric GARCH models

A number of recent papers have employed the BDS test as a general test for mis-specification for linear and nonlinear models. We show that for a particular class of conditionally heteroscedastic models, the BDS test is unable to detect a common mis-specification. Our results also demonstrate that specific rather than portmanteau diagnostics are required to detect neglected asymmetry in volatility. However for both classes of tests reasonable power is only obtained using very large sample sizes.

Modelling conditional heteroskedasticity: Application to the "IBEX-35" stock-return index

This paper compares alternative time-varying volatility models for daily stock-returns using data from Spanish equity index IBEX-35. Specifically, we estimate a parametric family of models of generalized autoregressive heteroskedasticity (which nests the most popular symmetric and asymmetric GARCH models), a semiparametric GARCH model, the generalized quadratic ARCH model, the stochastic volatility model, the Poisson Jump Diffusion model and, finally, a nonparametric model. Those models which use conditional standard deviation (specifically, TGARCH and AGARCH models) produce better fits than all other GARCH models. We also compare the within sample predictive power of all models using a standard efficiency test. Our results show that the asymmetric behaviour of responses is a statistically significant characteristic of these data. Moreover, we observe that specifications with a distribution which allows for fatter tails than a normal distribution do not necessarily outperform specifications with a normal distribution.

Option pricing under stochastic volatility and stochastic interest rate in the Spanish case

Among the underlying assumptions of the Black-Scholes option pricing model, the largest empirical biases are caused by those with a fixed volatility of the underlying asset and a constant short-term riskless interest rate. Only recently has attention been paid to the simultaneous effects of the stochastic nature of both variables on the pricing of options. Using a discrete approach an attempt is made to estimate the effects of a stochastic volatility and a stochastic interest rate in the Spanish option market; symmetric and asymmetric GARCH models were tried. The presence of in-the-mean and seasonality effects was allowed. Also estimated were the stochastic processes for the MIBOR90, a Spanish short-term interest rate, from 19 March 1990 to 31 May 1994 and the stochastic processes for the volatility of the returns of the most important Spanish stock index (IBEX-35) from 1 October 1987 to 20 January 1994. These estimators were used on pricing call options on the stock index, from 30 November 1993 to 30 May 1994. Hull-White and Amin-Ng pricing formulas were used. These prices were compared with actual prices and with those derived from the Black-Scholes formula, trying to detect the biases reported previously in the literature. Whereas the conditional variance of the MIBOR90 interest rate seemed to be free of ARCH effects, an asymmetric GARCH with in-the-mean and seasonality effects and some evidence of persistence in variance, IEGARCH(1,2)-M-S, was found to be the model that best represent the behaviour of the stochastic volatility of the IBEX-35 stock returns. All the biases reported previously in the literature were found. All the formulas overpriced the options in near-the-money case and underpriced the options otherwise. Furthermore, in most option trading, Black-Scholes overpriced the options and, because of the time-to-maturity effect, implied volatility computed from the Black-Scholes formula underestimated the actual volatility.

All in the family Nesting symmetric and asymmetric GARCH models

This paper develops a parametric family of models of generalized autoregressive heteroskedasticity (GARCH). The family nests the most popular symmetric and asymmetric GARCH models, thereby highlighting the relation between the models and their treatment of asymmetry. Furthermore, the structure permits nested tests of different types of asymmetry and functional forms. Daily U.S. stock return data reject all standard GARCH models in favor of a model in which, roughly speaking, the conditional standard deviation depends on the shifted absolute value of the shocks raised to the power three halves and past standard deviations.