Markov-switching Multifractal Research Articles

Short-term volatility forecasting with kernel support vector regression and Markov switching multifractal model

In volatility forecasting literature, Markov switching multifractal (MSM) models are well known for capturing many important stylized facts such as long memory and fat tails. MSM delivers stronger performance both in- and out-of-sample than GARCH-type models in long-term forecasts. However, the literature shows that MSM forecasts only slightly improve on GARCH(1,1) at short-term intervals. This indicates that there may exist certain patterns to be discovered in the innovation part . To enhance MSM's prediction accuracy at the short-term level with higher frequency data, a hybrid model of the MSM model and support vector regression (SVR) is proposed, in which a particle swarm optimization (PSO) algorithm is applied to optimize hyperparameters of the support vector regression in the scope of constraint permission. The method is referred to as MSM-PSO-SVR. Further, we introduce the Fourier kernel MSM-PSO-SVR and evaluate the performance of various MSM-PSO-SVR models in terms of mean absolute error (MAE) and the mean squared error (MSE) with one-minute data of the exchange traded fund (ETF) SPDR S&P 500 Trust ETF (ticker symbol: SPY). The experimental results show that the proposed approach outperforms the other competing peer models and in particular, the selection of SVR kernel might yield significant boosts in forecasting ability. Results of Hansen's Superior Predictive Ability test further validate the conclusion.

Investors’ Uncertainty and Forecasting Stock Market Volatility

This article examines whether incorporating investors’ uncertainty, as captured by the conditional volatility of sentiment, can help forecasting volatility of stock markets. In this regard, using the Markov-switching multifractal (MSM) model, we find that investors’ uncertainty can substantially increase the accuracy of the forecasts of stock market volatility according to the forecast encompassing test. We further provide evidence that the MSM outperforms the dynamic conditional correlation-generalized autoregressive conditional heteroskedasticity (DCC-GARCH) model.

Are multifractal processes suited to forecasting electricity price volatility? Evidence from Australian intraday data

Abstract We analyze Australian electricity price returns and find that they exhibit volatility clustering, long memory, structural breaks, and multifractality. Consequently, we let the return mean equation follow two alternative specifications, namely (i) a smooth transition autoregressive fractionally integrated moving average (STARFIMA) process, and (ii) a Markov-switching autoregressive fractionally integrated moving average (MSARFIMA) process. We specify volatility dynamics via a set of (i) short- and long-memory GARCH-type processes, (ii) Markov-switching (MS) GARCH-type processes, and (iii) a Markov-switching multifractal (MSM) process. Based on equal and superior predictive ability tests (using MSE and MAE loss functions), we compare the out-of-sample relative forecasting performance of the models. We find that the (multifractal) MSM volatility model keeps up with the conventional GARCH- and MSGARCH-type specifications. In particular, the MSM model outperforms the alternative specifications, when using the daily squared return as a proxy for latent volatility.

High-Frequency Volatility Forecasting of US Housing Markets

We propose a logistic smooth transition autoregressive fractionally integrated [STARFI (p, d)] process for modeling and forecasting US housing price volatility. We discuss the statistical properties of the model and investigate its forecasting performance by assuming various specifications for the dynamics underlying the variance process in the model. Using a unique database of daily data on price indices from ten major US cities, and the corresponding daily Composite 10 Housing Price Index, and also a housing futures price index, we find that using the Markov-switching multifractal (MSM) and FIGARCH frameworks for modeling the variance process helps improving the gains in forecast accuracy.

Comparing null models for testing multifractality in time series

The behaviors of fat-tailed distribution, linear long memory, and nonlinear long memory are considered as possible sources of apparent multifractality. Which behavior should be preserved in null models plays an important role in statistical tests of empirical multifractality. In this paper, we compare the performance of two null models on testing the existence of multifractality in fractional Brownian motions (fBm), Markov-switching multifractal (MSM) model, and financial returns. One null model is obtained by shuffling the original data, which keeps the distribution unchanged. The other null model is generated by the iterative amplitude adjusted Fourier transform (IAAFT) algorithm, which insures that the surrogate data and the original data sharing the same distribution and linear long memory behavior. We find that the tests based on the shuffle null model only reject the multifractality in fBm with and the tests based on the IAAFT null model reject the multifractality in fBms (except for ). And the multifractality in MSM and financial returns are significantly supported by the tests based on both null models. Our findings also shed light on the necessity of choosing suitable null models to test multifractality in other complex systems.

Forecasting market risk of portfolios: copula-Markov switching multifractal approach

ABSTRACTThis paper proposes a new methodology for modeling and forecasting market risks of portfolios. It is based on a combination of copula functions and Markov switching multifractal (MSM) processes. We assess the performance of the copula-MSM model by computing the value at risk of a portfolio composed of the NASDAQ composite index and the S&P 500. Using the likelihood ratio (LR) test by Christoffersen [1998. “Evaluating Interval Forecasts.” International Economic Review 39: 841–862], the GMM duration-based test by Candelon et al. [2011. “Backtesting Value at Risk: A GMM Duration-based Test.” Journal of Financial Econometrics 9: 314–343] and the superior predictive ability (SPA) test by Hansen [2005. “A Test for Superior Predictive Ability.” Journal of Business and Economic Statistics 23, 365–380] we evaluate the predictive ability of the copula-MSM model and compare it to other common approaches such as historical simulation, variance–covariance, RiskMetrics, copula-GARCH and constant conditional correlation GARCH (CCC-GARCH) models. We find that the copula-MSM model is more robust, provides the best fit and outperforms the other models in terms of forecasting accuracy and VaR prediction.

Forecasting crude-oil market volatility: Further evidence with jumps

This paper analyzes volatility models and their forecasting abilities in the presence of jumps in two crude-oil markets - Brent and West Texas Intermediate (WTI) - between January 6th 1992 and December 31st 2014. We compare a number of GARCH-type models that capture short memory as well as asymmetry (GARCH, GJR-GARCH and EGARCH), estimated on raw returns, to three competing approaches that deal with the presence of jumps: GARCH-type models estimated on jump-filtered returns, and two new classes of volatility models, called Generalized Autoregressive Score (GAS) and Markov-switching multifractal (MSM) models, estimated using raw returns. The forecasting performance of these volatility models is evaluated using the model confidence set approach, which allows us to identify a subset of models that outperform all the other competing models. We find that asymmetric models estimated on filtered returns provide better out-of-sample forecasts than do GARCH-, GAS-type and MSM models estimated on raw return series for Brent and WTI returns.

Forecasting crude oil price volatility and value-at-risk: Evidence from historical and recent data

This paper adopts the Markov-switching multifractal (MSM) model and a battery of generalized autoregressive conditional heteroscedasticity (GARCH)-type models to model and forecast oil price volatility. Extending previous work by Wei et al., (2010) and Wang et al., (2016), we evaluate the forecasting performance of all these models via a superior predictive ability (SPA) test. We go beyond previous research by (i) considering oil price volatility in the nineteenth century along with recent data, (ii) applying different types of MSM models and (iii) considering value-at-risk predictions besides our forecasting of volatility. Confirming its successful performance in other studies, the new MSM model comes out as the model that most often across forecasting horizons and subsamples cannot be outperformed by other models. This superiority also applies to forecasting of value-at-risk.

Forecasting crude oil market volatility: A Markov switching multifractal volatility approach

We use a Markov switching multifractal (MSM) volatility model to forecast crude oil return volatility. Not only can the model capture stylized facts of multiscaling, long memory, and structural breaks in volatility, it is also more parsimonious in parameterization, after allowing for hundreds of regimes in the volatility. Our in-sample results suggest that MSM models fit oil return data better than the traditional GARCH-class models. The out-of-sample results show that MSM models generate more accurate volatility forecasts than either popular GARCH-class models or the historical volatility model.

Dynamic Futures Hedging with a Vector Error Correction Markov Switching Multifractal Model

This paper proposes and estimates dynamic hedge ratios using a Vector Error Correction Markov Switching Multifractal (VEC-MSM) model, and examines its hedging effectiveness using conditional VaR coverage. The VEC-MSM model introduces an error correction mechanism in the mean equation that incorporates the long-run relationship between spot and futures prices, and combines this with a Markov Switching Multifractal (MSM) volatility process. Using data from commodities, foreign exchange and stock markets, in and out-of-sample tests indicate that the model is able to provide significant reduction in downside portfolio risk.

Asset Prices in an Economy Where Volatility Comes with the Intensity of Rare Disaster

This paper proposes a regime-switching asset pricing equilibrium model where volatility and the intensity of a rare disaster comove with MSM(Markov Switching Multifractal, a generalization of two-state regime-switching of Hamilton (1989) into multifrequencies) dynamics. Using monthly returns from 1927 to 2011, I estimate the joint evolution of volatility and the intensity of a rare disaster with five frequencies, generating a sizable amount of risk premium with a reasonable level of risk aversion as well as excessive volatility of market returns and predictability of long-horizon returns through price to dividend ratio. Yield curve of dividend strips reveals investor's forecasts on the future path of volatility. With in inflation jump in the event of rare disaster, the model demonstrates features of nominal bond return predictability documented in Fama and Bliss (1987) and Cochrane and Piazzesi (2005). Jumps due to regime-switching and rare disaster explain the substantial size of Variance Risk Premium. Deep out-of-the-money puts are highly priced as hedging instruments for downside tail-risk, yielding a smirk on the implied volatility surface.

Predicting volatility using the Markov-switching multifractal model: Evidence from S&P 100 index and equity options

In this paper, we evaluate the performance of the ability of Markov-switching multifractal (MSM), implied, GARCH, and historical volatilities to predict realized volatility for both the S&P 100 index and equity options. Some important findings are as follows. First, we find that the ability of MSM and GARCH volatilities to predict realized volatility is better than that of implied and historical volatilities for both the index and equity options. Second, equity option volatility is more difficult to be forecast than index option volatility. Third, both index and equity option volatilities can be better forecast during non-global financial crisis periods than during global financial crisis periods. Fourth, equity option volatility exhibits distinct patterns conditional on various equity and option characteristics and its predictability by MSM and implied volatilities depends on these characteristics. And finally, we find that MSM volatility outperforms implied volatility in predicting equity option volatility conditional on various equity and option characteristics.

A support vector machine based MSM model for financial short-term volatility forecasting

Financial time series forecasting has become a challenge because of its long-memory, thick tails and volatility persistence. Multifractal process has recently been proposed as a new formalism for this problem. An iterative Markov-Switching Multifractal (MSM) model was introduced to the literature. It is able to capture many of the important stylized features of the financial time series, including long-memory in volatility, volatility clustering, and return outliers. The model delivers stronger performance both in- and out-of-sample than GARCH-type models in long-term forecasts. To enhance MSM’s short-term prediction accuracy, this paper proposes a support vector machine (SVM) based MSM approach which exploits MSM model to forecast volatility and SVM to model the innovations. To verify the effectiveness of the proposed approach, two stock indexes in the Chinese A-share market are chosen as the forecasting targets. Comparing with some existing state-of-the-art models, the proposed approach gives superior results. It indicates that the proposed model provides a promising alternative to financial short-term volatility prediction.

MULTIFRACTALITY AND LONG-RANGE DEPENDENCE OF ASSET RETURNS: THE SCALING BEHAVIOR OF THE MARKOV-SWITCHING MULTIFRACTAL MODEL WITH LOGNORMAL VOLATILITY COMPONENTS

In this paper, we consider daily financial data from various sources (stock market indices, foreign exchange rates and bonds) and analyze their multiscaling properties by estimating the parameters of a Markov-switching multifractal (MSM) model with Lognormal volatility components. In order to see how well estimated models capture the temporal dependency of the empirical data, we estimate and compare (generalized) Hurst exponents for both empirical data and simulated MSM models. In general, the Lognormal MSM models generate "apparent" long memory in good agreement with empirical scaling provided that one uses sufficiently many volatility components. In comparison with a Binomial MSM specification [11], results are almost identical. This suggests that a parsimonious discrete specification is flexible enough and the gain from adopting the continuous Lognormal distribution is very limited.

True and apparent scaling: The proximity of the Markov-switching multifractal model to long-range dependence

In this paper, we consider daily financial data of a collection of different stock market indices, exchange rates, and interest rates, and we analyze their multi-scaling properties by estimating a simple specification of the Markov-switching multifractal (MSM) model. In order to see how well the estimated model captures the temporal dependence of the data, we estimate and compare the scaling exponents H ( q ) (for q = 1 , 2 ) for both empirical data and simulated data of the MSM model. In most cases the multifractal model appears to generate ‘apparent’ long memory in agreement with the empirical scaling laws.

Volatility comovement: a multifrequency approach

We implement a multifrequency volatility decomposition of three exchange rates and show that components with similar durations are strongly correlated across series. This motivates a bivariate extension of the Markov-Switching Multifractal (MSM) introduced in Calvet and Fisher (J. Econ. 105 (2001) 27, J. Financ. Econ. 2 (2004) 49). Bivariate MSM is a stochastic volatility model with a closed-form likelihood. Estimation can proceed by maximum likelihood for state spaces of moderate size, and by simulated likelihood via a particle filter in high-dimensional cases. We estimate the model and confirm its main assumptions in likelihood ratio tests. Bivariate MSM compares favorably to a standard multivariate GARCH both in- and out-of-sample. A parsimonious multifrequency factor structure is finally proposed for multivariate settings with potentially many assets.

Volatility Comovement: A Multifrequency Approach

We implement a multifrequency volatility decomposition of three exchange rates and show that components with similar durations are strongly correlated across series. This motivates a bivariate extension of the Markov-Switching Multifractal (MSM) introduced in Calvet and Fisher (2001, 2004). Bivariate MSM is a stochastic volatility model with a closed-form likelihood. Estimation can proceed by ML for state spaces of moderate size, and by simulated likelihood via a particle filter in high-dimensional cases. We estimate the model and confirm its main assumptions in likelihood ratio tests. Bivariate MSM compares favorably to a standard multivariate GARCH both in- and out-of-sample. We extend the model to multivariate settings with a potentially large number of assets by proposing a parsimonious multifrequency factor structure.