Markov-switching Multifractal Model Research Articles

Forecasting the variability of stock index returns with the multifractal random walk model for realized volatilities

We adapt the multifractal random walk model by Bacry et al. (2001) to realized volatilities (denoted RV-MRW) and take stock of recent theoretical insights on this model in Duchon et al. (2012) to derive forecasts of financial volatility. Moreover, we propose a new extension of the binomial Markov-switching multifractal (BMSM) model by Calvet and Fisher (2001) to the RV framework. We compare the predictive ability of the two against 10 classical and multifractal volatility models. Forecasting performance is evaluated out-of-sample based on the empirical MSE, MAE, and QLIKE criteria as well as using model confidence sets following the methodology of Hansen et al. (2011). Overall, our empirical study for 14 international stock market indices has a clear message: The RV-MRW is the best model throughout under the MAE criterion, i.e., for all indices and forecast horizons between one day and 100 days, with uniform results for forecast evaluations against both realized volatilities and squared returns or during tranquil/turbulent market periods. Based on the evaluation of MSE and QLIKE forecast errors, the RV-MRW, RV-BMSM, and RV-ARFIMA provide the most accurate forecasts during our tranquil sample from 2016–2018, where we can observe a transition from RV-MRW dominating long-term forecasts to RV-BMSM and RV-ARFIMA dominating in the short term. The new RV-BMSM takes the lead in phases of more turbulent market dynamics (sample 2010–2012), when it appears throughout in the 90% model confidence set at horizons ≤10 days and for 13 out of 14 indices at 20 days. These results are very promising if we consider that this is the first empirical application of the RV-MRW and RV-BMSM. Moreover, whereas RV-ARFIMA forecasts are often a time-consuming task, the RV-MRW stands out due to its fast execution and straightforward implementation.

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.

Forecasting volatility in bitcoin market

In this paper, we revisit the stylized facts of bitcoin markets and propose various approaches for modeling the dynamics governing the mean and variance processes. We first provide the statistical properties of our proposed models and study in detail their forecasting performance and adequacy by means of point and density forecasts. We adopt two loss functions and the model confidence set test to evaluate the predictive ability of the models and the likelihood ratio test to assess their adequacy. Our results confirm that bitcoin markets are characterized by regime shifting, long memory and multifractality. We find that the Markov switching multifractal and FIGARCH models outperform other GARCH-type models in forecasting bitcoin returns volatility. Furthermore, combined forecasts improve upon forecasts from individual models.

Inference for Nonlinear State Space Models: A Comparison of Different Methods applied to Markov-Switching Multifractal Models

Nonlinear, non-Gaussian state space models have found wide applications in many areas. These models usually do not allow for an analytical representation of their likelihood function and thus, sequential Monte Carlo or particle filter methods are mostly applied to estimate their parameters. Finding the best-fitting parameters of a model is a non-trivial task since stochastic approximations lead to non-smooth likelihood functions. Recently proposed iterative filtering algorithms developed for this purpose are compared with simpler on-line filters and more traditional methods of inference. A highly nonlinear class of Markov-switching models, the so called Markov-switching multifractal model (MSM) is used as illustrative example in the comparison of different optimisation routines. Besides the well-established univariate discrete-time MSM, univariate and multivariate continuous-time versions of MSM are considered. Monte Carlo simulation experiments indicate that across a variety of MSM specifications, the classical Nelder-Mead or simplex algorithm appears still as more efficient and robust compared to a number of online and iterated filters. A very close competitor is one of the recently proposed iterated filters while other alternatives are mostly dominated by these two algorithms. An empirical application of both discrete and continuous-time MSM to seven financial time series shows that both models dominate GARCH and FIGARCH models in terms of in-sample goodness-of-fit. Out-of-sample forecast comparisons show in the majority of cases a clear dominance of the continuous-time MSM under a mean absolute error criterion, and less conclusive results under a mean squared error criterion.

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 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 the volatility of the Dow Jones Islamic Stock Market Index: Long memory vs. regime switching

The financial crisis has fueled interest in alternatives to traditional asset classes that might be less affected by large market gyrations and, thus, provide for a less volatile development of a portfolio. One attempt at selecting stocks that are less prone to extreme risks is obeyance of Islamic Sharia rules. In this light, we investigate the statistical properties of the Dow Jones Islamic Stock Market Index (DJIM) and explore its volatility dynamics using a number of up-to-date statistical models allowing for long memory and regime-switching dynamics. We find that the DJIM shares all stylized facts of traditional asset classes, and estimation results and forecasting performance for various volatility models are also in line with prevalent findings in the literature. With this proximity to standard asset classes, investments in the DJIM could hardly provide a cushion against extreme market fluctuations. Among the various models, the relatively new Markov-switching multifractal model performs best under the majority of time horizons and loss criteria. Long memory GARCH-type models (FIGARCH and FITVGARCH) always improve upon the short-memory GARCH specification and additionally allowing for regime changes can further improve their performance, and also enhance the accuracy of value-at-risk forecast.

Binomial Markov-Switching Multifractal model with Skewed t innovations and applications to Chinese SSEC Index

Abstract This paper presents the Binomial Markov-switching Multifractal (BMSM) model of asset returns with Skewed t innovations (BMSM-Skewed t for short), which considers the fat tails, skewness and multifractality in asset returns simultaneously. The parameters of BMSM-Skewed t model can be estimated by Maximum Likelihood (ML) methods, and volatility forecasting can be accomplished via Bayesian updating. In order to evaluate the performance of BMSM-Skewed t model, BMSM model with Normal innovations (BMSM-N), BMSM model with Student-t innovations (BMSM-t) and GARCH(1,1) models (GARCH-N, GARCH-t and GARCH-Skewed t) are chosen for comparison. Through empirical studies on Shanghai Stock Exchange Composite Index (SSEC), we find that for sample estimation, BMSM models outperform the GARCH(1,1) models through BIC and AIC rules, and BMSM-Skewed t performs the best among all the models due to its fat tails, skewness and multifractality. In addition, BMSM-Skewed t model dominates other models at most forecasting horizons for out-of-sample volatility forecasts in terms of MSE, MAE and SPA test.

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.

Non-homogeneous volatility correlations in the bivariate multifractal model

In this paper, we consider an extension of the recently proposed bivariate Markov-switching multifractal model of Calvet, Fisher, and Thompson [2006. “Volatility Comovement: A Multifrequency Approach.” Journal of Econometrics 131: 179–215]. In particular, we allow correlations between volatility components to be non-homogeneous with two different parameters governing the volatility correlations at high and low frequencies. Specification tests confirm the added explanatory value of this specification. In order to explore its practical performance, we apply the model for computing value-at-risk statistics for different classes of financial assets and compare the results with the baseline, homogeneous bivariate multifractal model and the bivariate DCC-GARCH of Engle [2002. “Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models.” Journal of Business & Economic Statistics 20 (3): 339–350]. As it turns out, the multifractal model with heterogeneous volatility correlations provides more reliable results than both the homogeneous benchmark and the DCC-GARCH model.

Discrete stochastic autoregressive volatility

We use Markov chain methods to develop a flexible class of discrete stochastic autoregressive volatility (DSARV) models. Our approach to formulating the models is straightforward, and readily accommodates features such as volatility asymmetry and time-varying volatility persistence. Moreover, it produces models with a low-dimensional state space, which greatly enhances computational tractability. We illustrate the proposed methodology for both individual stock and stock index returns, and show that simple first- and second-order DSARV models outperform generalized autoregressive conditional heteroscedasticity and Markov-switching multifractal models in forecasting volatility.

Relative forecasting performance of volatility models: Monte Carlo evidence

A Monte Carlo (MC) experiment is conducted to study the forecasting performance of a variety of volatility models under alternative data-generating processes (DGPs). The models included in the MC study are the (Fractionally Integrated) Generalized Autoregressive Conditional Heteroskedasticity models ((FI)GARCH), the Stochastic Volatility model (SV), the Long Memory Stochastic Volatility model (LMSV) and the Markov-switching Multifractal model (MSM). The MC study enables us to compare the relative forecasting performance of the models accounting for different characterizations of the latent volatility process: specifications that incorporate short/long memory, autoregressive components, stochastic shocks, Markov-switching and multifractality. Forecasts are evaluated by means of mean squared errors (MSE), mean absolute errors (MAE) and value-at-risk (VaR) diagnostics. Furthermore, complementarities between models are explored via forecast combinations. The results show that (i) the MSM model best forecasts volatility under any other alternative characterization of the latent volatility process and (ii) forecast combinations provide systematic improvements upon most single misspecified models, but are typically inferior to the MSM model even if the latter is applied to data governed by other processes.

Discrete Stochastic Autoregressive Volatility

We use Markov chain methods to develop a flexible class of discrete stochastic autoregressive volatility (DSARV) models. Our approach to formulating the models is straightforward, and readily accommodates features such as volatility asymmetry and time-varying volatility persistence. Moreover, it produces models with a low-dimensional state space, which greatly enhances computational tractability. We illustrate the proposed methodology for both individual stock and stock index returns, and show that simple first and second order DSARV models outperform generalized autoregressive conditional heteroscedasticity and Markov-switching multifractal models in forecasting volatility.

A fractal version of the Hull–White interest rate model

This paper develops a new version of the Hull–White's model of interest rates, in which the volatility of the short term rate is driven by a Markov switching multifractal model. The interest rate dynamics is still mean reverting but the constant volatility of the Brownian motion is replaced by a multifractal process so as to capture persistent volatility shocks. In this setting, we infer properties of the short term rate distribution, a semi-closed form expression for bond prices and their dynamics under a forward measure. Finally, our work is illustrated by a numerical application in which we assess the exposure of a bonds portfolio to the interest risk.

Component-Driven Regime-Switching Volatility

We develop a new class of regime-switching volatility models that are characterized by high-dimensional state spaces, parsimonious transition matrices, and ARMA dynamics for the log volatility process. This combination of features is achieved by assuming that we can decompose the Markov chain that describes regime dynamics into a number of two-state component chains that evolve independently through time. Using daily data for S&P 500 index and IBM shares, we show that our component-driven regime-switching (CDRS) models are capable of outperforming GARCH, component GARCH, regime-switching GARCH, and Markov-switching multifractal models in forecasting realized variances out of sample. Interestingly, we find that CDRS models with simple AR(1) dynamics perform well across the board.

Component-Driven Regime-Switching Volatility

We develop a new class of regime-switching volatility models that are characterized by high-dimensional state spaces, parsimonious transition matrices, and ARMA dynamics for the log volatility process. This combination of features is achieved by assuming that we can decompose the Markov chain that describes regime dynamics into a number of two-state component chains that evolve independently through time. Using daily data for S&P 500 index and IBM shares, we show that our component-driven regime-switching (CDRS) models are capable of outperforming GARCH, component GARCH, regime-switching GARCH, and Markov-switching multifractal models in forecasting realized variances out of sample. Interestingly, we find that CDRS models with simple AR(1) dynamics perform well across the board. Copyright The Author, 2012. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org, Oxford University Press.

A Fractal Version of the Hull-White Interest Rate Model

This paper develops a new version of the Hull-White's model of interest rates, in which the volatility of the short term rate is driven by a Markov switching multifractal model. The interest rate dynamics is still mean reverting but the constant volatility of the Brownian motion is replaced by a multifractal process so as to capture persistent volatility shocks. In this setting, we infer properties of the short term rate distribution, a semi closed form expression for bond prices and their dynamics under a forward measure. Finally, our work is illustrated by a numerical application in which we assess the exposure of a bonds portfolio to the interest risk.