Forecasting daily variability of the S&P 100 stock index using historical,realised and implied volatility measurements

The increasing availability of financial market data at intraday frequencies has not only led to the development of improved volatility measurements but has also inspired research into their potential value as an information source for volatility forecasting. In this paper, we explore the forecasting value of historical volatility (extracted from daily return series), of implied volatility (extracted from option pricing data) and of realised volatility (computed as the sum of squared high frequency returns within a day). First, we consider unobserved components (UC-RV) and long memory models for realised volatility which is regarded as an accurate estimator of volatility. The predictive abilities of realised volatility models are compared with those of stochastic volatility (SV) models and generalised autoregressive conditional heteroskedasticity (GARCH) models for daily return series. These historical volatility models are extended to include realised and implied volatility measures as explanatory variables for volatility. The main focus is on forecasting the daily variability of the Standard & Poor's 100 (S&P 100) stock index series for which trading data (tick by tick) of almost 7 years is analysed. The forecast assessment is based on the hypothesis of whether a forecast model is outperformed by alternative models. In particular, we will use superior predictive ability tests to investigate the relative forecast performances of some models. Since volatilities are not observed, realised volatility is taken as a proxy for actual volatility and is used for computing the forecast error. A stationary bootstrap procedure is required for computing the test statistic and its p-value. The empirical results show convincingly that realised volatility models produce far more accurate volatility forecasts compared to models based on daily returns. Long memory models seem to provide the most accurate forecasts.     

Bayesian realized-GARCH models for financial tail risk forecasting incorporating the two-sided Weibull distribution

The realized-GARCH framework is extended to incorporate the two-sided Weibull distribution, for the purpose of volatility and tail risk forecasting in a financial time series. Further, the realized range, as a competitor for realized variance or daily returns, is employed as the realized measure in the realized-GARCH framework. Sub-sampling and scaling methods are applied to both the realized range and realized variance, to help deal with inherent micro-structure noise and inefficiency. A Bayesian Markov Chain Monte Carlo (MCMC) method is adapted and employed for estimation and forecasting, while various MCMC efficiency and convergence measures are employed to assess the validity of the method. In addition, the properties of the MCMC estimator are assessed and compared with maximum likelihood, via a simulation study. Compared to a range of well-known parametric GARCH and realized-GARCH models, tail risk forecasting results across seven market indices, as well as two individual assets, clearly favour the proposed realized-GARCH model incorporating the two-sided Weibull distribution; especially those employing the sub-sampled realized variance and sub-sampled realized range.     

VIX Forecast Under Different Volatility Specifications

Stochastic volatility (SV) models are theoretically more attractive than the GARCH type of models as it allows additional randomness. The classical SV models deduce a continuous probability distribution for volatility so that it does not admit a computable likelihood function. The estimation requires the use of Bayesian approach. A recent approach considers discrete stochastic autoregressive volatility models for a bounded and tractable likelihood function. Hence, a maximum likelihood estimation can be achieved. This paper proposes a general approach to link SV models under the physical probability measure, both continuous and discrete types, to their processes under a martingale measure. Doing so enables us to deduce the close-form expression for the VIX forecast for the both SV models and GARCH type models. We then carry out an empirical study to compare the performances of the continuous and discrete SV models using GARCH models as benchmark models.     

Stock market momentum, business conditions, and GARCH option pricing models

This paper examines the forecasting performance of GARCH option pricing models from a market momentum perspective, and the possible impacts of financial crises and business conditions are also examined. The empirical results demonstrate that market momentum impacts the forecasting performance of GARCH option pricing models. The EGARCH model performs better under downward market momentum, while the standard GARCH performs better under upward market momentum. In addition, parsimonious models generally outperform richly parameterized ones. The above findings are robust to financial crises, and the results further demonstrate that business conditions influence the forecasting performance of GARCH option pricing models.     

Forecasting variance using stochastic volatility and GARCH

This paper estimates the conditional variance of daily Swedish OMX-index returns with stochastic volatility (SV) models and GARCH models and evaluates the in-sample performance as well as the out-of-sample forecasting ability of the models. Asymmetric as well as weekend/holiday effects are allowed for in the variance, and the assumption that errors are Gaussian is released. Evidence is found of a leverage effect and of higher variance during weekends. In both in-sample and out-of-sample comparisons SV models outperform GARCH models. However, while asymmetry, weekend/holiday effects and non-Gaussian errors are important for the in-sample fit, it is found that these factors do not contribute to enhancing the forecasting ability of the SV models.     

GARCH vs. stochastic volatility: Option pricing and risk management

In this paper we compare the out-of-sample performance of two common extensions of the Black–Scholes option pricing model, namely GARCH and stochastic volatility (SV). We calibrate the three models to intraday FTSE 100 option prices and apply two sets of performance criteria, namely out-of-sample valuation errors and Value-at-Risk (VaR) oriented measures. When we analyze the fit to observed prices, GARCH clearly dominates both SV and the benchmark Black–Scholes model. However, the predictions of the market risk from hypothetical derivative positions show sizable errors. The fit to the realized profits and losses is poor and there are no notable differences between the models. Overall, we therefore observe that the more complex option pricing models can improve on the Black–Scholes methodology only for the purpose of pricing, but not for the VaR forecasts.     

Bayesian Risk Management for Equity-Linked Insurance

This paper describes how to apply Markov Chain Monte Carlo (MCMC) techniques to a regime switching model of the stock price process to generate a sample from the joint posterior distribution of the parameters of the model. The MCMC output can be used to generate a sample from the predictive distribution of losses from equity linked contracts, assuming first an actuarial approach to risk management and secondly a financial economics approach. The predictive distribution is used to show the effect of parameter uncertainty on risk management calculations. We also explore model uncertainty by assuming a GARCH model in place of the regime switching model. The results indicate that the financial economics approach to risk management is substantially more robust to parameter uncertainty and model uncertainty than the actuarial approach.     

Volatility Forecasting in the Hang Seng Index using the GARCH Approach

The aim of this paper is to add to the literature on volatility forecasting using data from the Hong Kong stock market to determine if forecasts from GARCH based models can outperform simple historical averaging models. Overall, unlike previous studies we find that the GARCH models with non-Normal distributions show a robust volatility forecasting performance in comparison to the historical models. The results indicate that although not all models outperform simple historical averaging, the EGARCH based models, with non-normal conditional volatility, tend to produce more accurate out-of-sample forecasts using both standard measures of forecast accuracy and financial loss functions. In addition we test for asymmetric adjustment in the Hang Seng, finding strong evidence of asymmetries due to the domination of financial and property firms in this market.     

Hierarchical GARCH

There is strong empirical evidence that the GARCH estimates obtained from panels of financial time series cluster. In order to capture this empirical regularity, this paper introduces the Hierarchical GARCH (HG) model. The HG is a nonlinear panel specification in which the coefficients of each series are modeled as a function of observed series characteristic and an unobserved random effect. A joint panel estimation strategy is proposed to carry out inference for the model. A simulation study shows that when there is a strong degree of coefficient clustering panel estimation leads to substantial accuracy gains in comparison to estimating each GARCH individually. The HG is applied to a panel of U.S. financial institutions in the 2007–2009 crisis, using firm size and leverage as characteristics. Results show evidence of coefficient clustering and that the characteristics capture a significant portion of cross sectional heterogeneity. An out-of-sample volatility forecasting application shows that when the sample size is modest coefficient estimates based on the panel estimation approach perform better than the ones based on individual estimation.     

Forecasting the weekly time-varying beta of UK firms: GARCH models vs. Kalman filter method

This paper investigates the forecasting ability of three different Generalised Autoregressive Conditional Heteroscedasticity (GARCH) models and the Kalman filter method. The three GARCH models applied are: bivariate GARCH, BEKK GARCH, and GARCH-GJR. Forecast errors based on 20 UK company's weekly stock return (based on time-varying beta) forecasts are employed to evaluate the out-of-sample forecasting ability of both the GARCH models and the Kalman method. Measures of forecast errors overwhelmingly support the Kalman filter approach. Among the GARCH models, GJR appears to provide somewhat more accurate forecasts than the two other GARCH models.     

SV—N与SV—M模型测量股市波动性的应用

与传统的GARCH类模型一样,SV模犁(随机波动模型)是用来捕捉股市波动特征的一个较好的模型,该模型在国外得到广泛的应用.实证研究表明:利用SV模型的两个子类,即基于正态分布下的SV模型(SV-N)和均值SV模型(SV-M)来测量我国沪深股市波动性明显优于GARCH类模型,能够更好地描述其统计特征.     

American option pricing with discrete and continuous time models: An empirical comparison

This paper considers discrete time GARCH and continuous time SV models and uses these for American option pricing. We first of all show that with a particular choice of framework the parameters of the SV models can be estimated using simple maximum likelihood techniques. We then perform a Monte Carlo study to examine their differences in terms of option pricing, and we study the convergence of the discrete time option prices to their implied continuous time values. Finally, a large scale empirical analysis using individual stock options and options on an index is performed comparing the estimated prices from discrete time models to the corresponding continuous time model prices. The results show that, while the overall differences in performance are small, for the in the money put options on individual stocks the continuous time SV models do generally perform better than the discrete time GARCH specifications.     

Stochastic volatility and time-varying country risk in emerging markets

This study suggests an alternative method to estimate time-varying country risk. We first apply a new multivariate stochastic volatility (SV) model to a set of emerging stock markets. To estimate the SV model, we use a Bayesian Markov chain Monte Carlo simulation procedure. By applying the deviance information criterion, we show that the new model performs well relative to alternative multivariate SV models. We then compute the conditional betas for the different markets and compare the results with an often-used procedure based on multivariate GARCH models. We show that the new multivariate SV model more accurately captures the time-varying nature of country risk. The conditional betas show signs of large variations, indicating the importance of taking time-varying country risk into consideration when managing emerging market portfolios.     

Analysis of efficiency for Shenzhen stock market based on multifractal detrended fluctuation analysis

We divided the whole series of Shenzhen stock market into two sub-series at the criterion of the date of a reform and their scale behaviors are investigated using multifractal detrended fluctuation analysis (MF-DFA). Employing the method of rolling window, we find that Shenzhen stock market was becoming more and more efficient by analyzing the change of Hurst exponent and a new efficient measure, which is equal to multifractality degree sometimes. We also study the change of Hurst exponent and multifractality degree of volatility series. The results show that the volatility series still have significantly long-range dependence and multifractality indicating that some conventional models such as GARCH and EGARCH cannot be used to forecast the volatilities of Shenzhen stock market. At last, the abnormal phenomenon of multifractality degrees for return series is discussed. The results have very important implications for analyzing the influence of policies, especially under the environment of financial crisis.     

Forecasting VaR using analytic higher moments for GARCH processes

It is widely accepted that some of the most accurate Value-at-Risk (VaR) estimates are based on an appropriately specified GARCH process. But when the forecast horizon is greater than the frequency of the GARCH model, such predictions have typically required time-consuming simulations of the aggregated returns distributions. This paper shows that fast, quasi-analytic GARCH VaR calculations can be based on new formulae for the first four moments of aggregated GARCH returns. Our extensive empirical study compares the Cornish–Fisher expansion with the Johnson SU distribution for fitting distributions to analytic moments of normal and Student t, symmetric and asymmetric (GJR) GARCH processes to returns data on different financial assets, for the purpose of deriving accurate GARCH VaR forecasts over multiple horizons and significance levels.     

Level changes in volatility models

Level shifts confound the estimation of persistence. This paper shows analytically, in simulations, and using high-frequency stock price data that models for financial volatility that feature a separate source of randomness in the volatility equation are less susceptible to this effect. Such models include recently proposed time series models for realized volatility, as opposed to GARCH models for daily observations, which are highly sensitive to unknown shifts, as has been shown before.     

Evolving fuzzy modelling in risk analysis

Traditionally, forecast methodologies emphasize precise point-forecasts of stationary data. Risk analysis demands forecasts that, in practice, must be developed using imprecise and nonstationary data. Currently, value-at-risk (VaR) is widely employed in risk analysis. VaR requires a form of interval forecasts. Generalized autoregressive conditional heteroskedasticity (GARCH) models are stochastic recursive systems commonly adopted in financial prediction. This paper addresses a new approach to handle imprecise and nonstationary data using evolving fuzzy modelling translated into a recursive, adaptive forecasting procedure. VaR analysis is conducted to compare the performance and robustness of evolving fuzzy forecasting against GARCH using São Paulo Stock Exchange data. Copyright © 2009 John Wiley & Sons, Ltd.     

A New Approach to Markov-Switching GARCH Models

The use of Markov-switching models to capture the volatilitydynamics of financial time series has grown considerably duringpast years, in part because they give rise to a plausible interpretationof nonlinearities. Nevertheless, GARCH-type models remain ubiquitousin order to allow for nonlinearities associated with time-varyingvolatility. Existing methods of combining the two approachesare unsatisfactory, as they either suffer from severe estimationdifficulties or else their dynamic properties are not well understood.In this article we present a new Markov-switching GARCH modelthat overcomes both of these problems. Dynamic properties arederived and their implications for the volatility process discussed.We argue that the disaggregation of the variance process offeredby the new model is more plausible than in the existing variants.The approach is illustrated with several exchange rate returnseries. The results suggest that a promising volatility modelis an independent switching GARCH process with a possibly skewedconditional mixture density.     

Model-free versus Model-based Volatility Prediction

The well-known ARCH/GARCH models for financial time series havebeen criticized of late for their poor performance in volatilityprediction, that is, prediction of squared returns.¹ Focusingon three representative data series, namely a foreign exchangeseries (Yen vs. Dollar), a stock index series (the S&P500index), and a stock price series (IBM), the case is made thatfinancial returns may not possess a finite fourth moment. Takingthis into account, we show how and why ARCH/GARCH models—whenproperly applied and evaluated—actually do have nontrivialpredictive validity for volatility. Furthermore, we show howa simple model-free variation on the ARCH theme can performeven better in that respect. The model-free approach is basedon a novel normalizing and variance–stabilizing transformation(NoVaS, for short) that can be seen as an alternative to parametricmodeling. Properties of this transformation are discussed, andpractical algorithms for optimizing it are given.