An analytical approximation to the option formula for the GARCH model

This article derives an analytical approximation to the option formula for a spot asset price whose conditional variance equation follows a nonlinear asymmetric GARCH (NGARCH) process. The approximate option formula, which is just a volatility adjustment in comparison to the Black-Scholes (BS) formula, is very simple and provides the volatility term structure of spot asset prices. Also, the formula shows that the most characteristic feature of an NGARCH model appears in the vega of a European option, which depends on both the spread between the long-run variance and the current one and a parameter reproduced from the stationary property of the conditional variance. This methodology can be easily extended to an option formula for the generalized GARCH process.     

REGIME-SWITCHING IN AUSTRALIAN SHORT-TERM INTEREST RATES

This paper applies a generalized regime-switching (GRS) model of the short-term interest rate to Australian data. The model allows the short rate to exhibit both mean reversion and conditional heteroscedasticity and nests the popular generalized autoregressive conditional heteroscedasticity (GARCH) and regime-switching specifications. It is shown that empirical estimates of many popular interest rate models provide curious results which imply that innovations to the short rate process are extremely persistent, and that the short rate is potentially non-stationary. The source of these curious results, which are also present in US and European interest rates, is identified in the context of the GRS model, which is shown, via specification and forecasting tests, to capture the features of Australian short-term interest rate data better than existing models. The stochastic process of short-term interest rates in Australia is compared with evidence from the US and Europe, highlighting a number of important differences.     

Positivity constraints on the conditional variances in the family of conditional correlation GARCH models

In this article, we derive a set of necessary and sufficient conditions for positivity of the vector conditional variance equation in multivariate GARCH models with explicit modelling of conditional correlation. These models include the constant conditional correlation GARCH model of Bollerslev [1990. Review of Economics and Statistics 72, 498–505] and its extensions. Under the new conditions, it is possible to introduce negative volatility spillovers in the model. An empirical example illustrates usefulness of having such conditions in practice.     

Are there nonlinearities in short‐term interest rates?

The present paper investigates the characteristics of short‐term interest rates in several countries. We examine the importance of nonlinearities in the mean reversion and volatility of short‐term interest rates. We examine various models that allow the conditional mean (drift) and conditional variance (diffusion) to be functions of the current short rate. We find that different markets require different models. In particular, we find evidence of nonlinear mean reversion in some of the countries that we examine, linear mean reversion in others and no mean reversion in some countries. For all countries we examine, there is strong evidence of the need for the volatility of interest rate changes to be highly sensitive to the level of the short‐term interest rate. Out‐of‐sample forecasting performance of one‐factor short rate models is poor, stemming from the inability of the models to accommodate jumps and discontinuities in the time series data.     

Modelling conditional heteroscedasticity and jumps in Australian short-term interest rates

The present paper explores a class of jump–diffusion models for the Australian short‐term interest rate. The proposed general model incorporates linear mean‐reverting drift, time‐varying volatility in the form of LEVELS (sensitivity of the volatility to the levels of the short‐rates) and generalized autoregressive conditional heteroscedasticity (GARCH), as well as jumps, to match the salient features of the short‐rate dynamics. Maximum likelihood estimation reveals that pure diffusion models that ignore the jump factor are mis‐specified in the sense that they imply a spuriously high speed of mean‐reversion in the level of short‐rate changes as well as a spuriously high degree of persistence in volatility. Once the jump factor is incorporated, the jump models that can also capture the GARCH‐induced volatility produce reasonable estimates of the speed of mean reversion. The introduction of the jump factor also yields reasonable estimates of the GARCH parameters. Overall, the LEVELS–GARCH–JUMP model fits the data best.     

Modeling the U.S. Short-Term Interest Rate by Mixture Autoregressive Processes

A new kind of mixture autoregressive model with GARCH errorsis introduced and applied to the U.S. short-term interest rate.According to the diagnostic tests developed in the article andfurther informal checks, the model is capable of capturing bothof the typical characteristics of the short-term interest rate:volatility persistence and the dependence of volatility on thelevel of the interest rate. The model also allows for regimeswitches whose presence has been a third central result emergingfrom the recent empirical literature on the U.S. short-terminterest rate. Realizations generated from the estimated modelseem stable and their properties resemble those of the observedseries closely. The drift and diffusion functions implied bythe new model are in accordance with the results in much ofthe literature on continuous-time diffusion models for the short-terminterest rate, and the term structure implications agree withhistorically observed patterns.     

Modeling asset market volatility in a small market:: Accounting for non-synchronous trading effects

The use of GARCH modeling in empirical finance has so far to a great extent been restricted to larger asset markets. This paper considers whether the GARCH framework can be used on a smaller, less liquid market. In particular, selected stocks on the Vancouver Stock Exchange, a smaller market in Canada, are examined. Modeling return volatility in the standard GARCH framework and returns as autoregressive fails to remove significant serial correlation in the mean. The results indicate that once the parameters are adjusted for non-synchronous trading effects, GARCH can also be successful in modeling stochastic volatility on smaller markets. Persistence in both the mean and variance are eliminated with these adjustments. In addition, for some stocks, volumes add explanatory power for explaining return volatility.     

On the robustness of short–term interest rate models

This paper investigates the robustness of a range of short–term interest rate models. We examine the robustness of these models over different data sets, time periods, sampling frequencies, and estimation techniques. We examine a range of popular one–factor models that allow the conditional mean (drift) and conditional variance (diffusion) to be functions of the current short rate. We find that parameter estimates are highly sensitive to all of these factors in the eight countries that we examine. Since parameter estimates are not robust, these models should be used with caution in practice.     

Modeling Conditional Skewness in Stock Returns

Abstract

In this paper, we propose a new GARCH-in-Mean (GARCH-M) model allowing for conditional skewness. The model is based on the so-called z distribution capable of modeling skewness and kurtosis of the size typically encountered in stock return series. The need to allow for skewness can also be readily tested. The model is consistent with the volatility feedback effect in that conditional skewness is dependent on conditional variance. Compared to previously presented GARCH models allowing for conditional skewness, the model is analytically tractable, parsimonious and facilitates straightforward interpretation.Our empirical results indicate the presence of conditional skewness in the monthly postwar US stock returns. Small positive news is also found to have a smaller impact on conditional variance than no news at all. Moreover, the symmetric GARCH-M model not allowing for conditional skewness is found to systematically overpredict conditional variance and average excess returns.     

Sensitivity of the bank stock returns distribution to changes in the level and volatility of interest rate: A GARCH-M model

The objective of this paper is to employ the generalized autoregressive conditionally heteroskedastic in the mean (GARCH-M) methodology to investigate the effect of interest rate and its volatility on the bank stock return generation process. This framework discards the restrictive assumptions of linearity, independence, and constant conditional variance in modeling bank stock returns. The model presented here allows for shifts in the volatility equation in response to the changes in monetary policy regime in 1979 and 1982 to be estimated. ARCH, GARCH, and volatility feed back effects are found to be significant. Interest rate and interest rate volatility are found to directly impact the first and the second moments of the bank stock returns distribution, respectively. The latter also affects the risk premia indirectly. The degree of persistence in shocks is substantial for all the three bank portfolios and sensitive to the nature of the bank portfolio and the prevailing monetary policy regime.     

Smooth Transition ARCH Models: Estimation and Testing

In this paper, we suggest an extension of the ARCH model, the smooth-transition autoregressive conditional heteroskedasticity (STARCH) model. STARCH models endogenously allow for time-varying shifts in the parameters of the conditional variance equation. The most general form of the model that we consider is a double smooth-transition model, the STAR-STARCH model, which permits not only the conditional variance, but also the mean, to be a function of a smooth-transition term. The threshold ARCH model, the Markov-ARCH model and the standard ARCH model are special cases of our STARCH model. We also develop Lagrange multiplier tests of the hypothesis that the smooth-transition term in the conditional variance is zero. We apply our STARCH model to excess Treasury bill returns. We find some evidence of a smooth transition in excess returns, but in contrast to previous studies, we find almost no evidence of volatility persistence once we allow for smooth transitions in the conditional variance. Thus, the apparent persistence in the conditional variance reported by many researchers could be a mere statistical artifact. We conduct in-sample tests comparing STARCH models to nested competitors; these suggest that STARCH models hold promise for improved predictions. Finally, we describe further extensions of the STARCH model and suggest issues in finance to which they might profitably be applied.     

ANALYSIS OF FEDERAL FUNDS RATE CHANGES AND VARIANCE PATTERNS

We analyze Fed funds rate changes in GARCH‐in‐mean (GARCH‐M) models and find that daily rate change and variance patterns differ with the timing of the rate observation, but that all patterns are generally consistent with optimal reserve account management. We also find that Fed funds daily and intraday variances exhibit trends and persistence, and that daily variance effects differ when using marginal rates versus daily weighted average rates. Furthermore, we find that conditional variances do not provide information about daily or intraday rate changes. Our results provide support for the use of GARCH models for studies on other financial assets. JEL classification: G21, G28     

Uncovering a positive risk-return relation: the role of implied volatility index

We report empirical evidence suggesting a strong and positive risk-return relation for the daily S&P 100 market index if the implied volatility index is included as an exogenous variable in the conditional variance equation. This result holds for alternative GARCH specifications and conditional distributions. Monte Carlo evidence suggests that if implied volatility is not included, whilst is should be, the risk-return relation is more likely to be negative or weak.     

Multivariate term structure models with level and heteroskedasticity effects

The paper introduces and estimates a multivariate level-GARCH model for the long rate and the term-structure spread where the conditional volatility is proportional to the γth power of the variable itself (level effects) and the conditional covariance matrix evolves according to a multivariate GARCH process (heteroskedasticity effects). The long-rate variance exhibits heteroskedasticity effects and level effects in accordance with the square-root model. The spread variance exhibits heteroskedasticity effects but no level effects. The level-GARCH model is preferred above the GARCH model and the level model. GARCH effects are more important than level effects. The results are robust to the maturity of the interest rates.     

Capturing volatility persistence: a dynamically complete realized EGARCH-MIDAS model

We introduce extensions of the Realized Exponential GARCH model (REGARCH) that capture the evident high persistence typically observed in measures of financial market volatility in a tractable fashion. The extensions decompose conditional variance into a short-term and a long-term component. The latter utilizes mixed-data sampling or a heterogeneous autoregressive structure, avoiding parameter proliferation otherwise incurred by using the classical ARMA structures embedded in the REGARCH. The proposed models are dynamically complete, facilitating multi-period forecasting. A thorough empirical investigation with an exchange-traded fund that tracks the S&P500 Index and 20 individual stocks shows that our models better capture the dependency structure of volatility. This leads to substantial improvements in empirical fit and predictive ability at both short and long horizons relative to the original REGARCH. A volatility-timing trading strategy shows that capturing volatility persistence yields substantial utility gains for a mean–variance investor at longer investment horizons.     

Forecasting Commodity Market Returns Volatility: A Hybrid Ensemble Learning GARCH-LSTM based Approach

This study investigates the advantage of combining the forecasting abilities of multiple generalized autoregressive conditional heteroscedasticity (GARCH)-type models, such as the standard GARCH (GARCH), exponential GARCH (eGARCH), and threshold GARCH (tGARCH) models with advanced deep learning methods to predict the volatility of five important metals (nickel, copper, tin, lead, and gold) in the Indian commodity market. This paper proposes integrating the forecasts of one to three GARCH-type models into an ensemble learning-based hybrid long short-term memory (LSTM) model to forecast commodity price volatility. We further evaluate the forecasting performance of these models for standalone LSTM and GARCH-type models using the root mean squared error, mean absolute error, and mean fundamental percentage error. The results highlight that combining the information from the forecasts of multiple GARCH types into a hybrid LSTM model leads to superior volatility forecasting capability. The SET-LSTM, which represents the model that combines forecasts of the GARCH, eGARCH, and tGARCH into the LSTM hybrid, has shown the best overall results for all metals, barring a few exceptions. Moreover, the equivalence of forecasting accuracy is tested using the Diebold–Mariano and Wilcoxon signed-rank tests.     

Volatility transmission between oil prices and equity sector returns

This paper employs bivariate GARCH models to simultaneously estimate the mean and conditional variance between five different US sector indexes and oil prices. Since many different financial assets are traded based on these market sector returns, it is important for financial market participants to understand the volatility transmission mechanism over time and across these series in order to make optimal portfolio allocation decisions. We examine weekly returns from January 1, 1992 to April 30, 2008 and find evidence of significant transmission of shocks and volatility between oil prices and some of the examined market sectors. The findings support the idea of cross-market hedging and sharing of common information by investors.     

Long-Range Dependence in U.S. Home Price Volatility

The existence of GARCH effects in a financial price series means that the probability of large losses is much higher than standard mean-variance analysis suggests. Accordingly, several recent papers have investigated whether GARCH effects exist in the U.S. housing market, as changes in house prices can have far-ranging impacts on defaults, foreclosures, tax revenues and the values of mortgage-backed securities. Some research in finance indicates that the conditional variance of some assets exhibits far greater persistence, or even “long memory”, than is accounted for in standard GARCH models. If house prices do indeed have this very persistent volatility, properly estimating the conditional variance to allow for such persistence is crucial for optimal portfolio management. We examine a number of U.S. metropolitan areas, and find that, for those with significant GARCH effects, more than half indeed exhibit the very high persistence found in other assets such as equities. We also find that, for those markets exhibiting such persistent volatility, C-GARCH models typically do a better job in forecasting than standard GARCH models. Moreover, there is some tentative evidence that metro areas with the fastest appreciation may be most likely to have such long memory conditional variance. These findings should help in improving risk management, through, for instance the construction of better-specified value-at-risk models.     

Aggregational Gaussianity and barely infinite variance in financial returns

This paper aims at reconciling two apparently contradictory empirical regularities of financial returns, namely, the fact that the empirical distribution of returns tends to normality as the frequency of observation decreases (aggregational Gaussianity) combined with the fact that the conditional variance of high frequency returns seems to have a (fractional) unit root, in which case the unconditional variance is infinite. We provide evidence that aggregational Gaussianity and infinite variance can coexist, provided that all the moments of the unconditional distribution whose order is less than two exist. The latter characterizes the case of Integrated and Fractionally Integrated GARCH processes. Finally, we discuss testing for aggregational Gaussianity under barely infinite variance. Our empirical motivation derives from commodity prices and stock indices, while our results are relevant for financial returns in general.     

Conditional Dependence in Precious Metal Prices

This study investigates the time-series properties of gold and silver spot prices. Both precious metal price series are found to exhibit time dependence and pronounced generalized autoregressive conditional heteroscedastic (GARCH) effects. Splitting the data into similar economic subperiods provides superior explanation of these effects because of the observed long-run nonconstancy of the unconditional variance. Further, the power exponential distribution, as opposed to the Student-t, is found to portray accurately the thick-tailed conditional variance that remains after the GARCH effects are removed. These findings imply that constant variance pricing models are inappropriate for securities that are based on precious metal prices.