Regime-Switching and the Estimation of Multifractal Processes

We propose a discrete-time stochastic volatility model in which regimeswitching serves three purposes. First, changes in regimes capture low frequency variations, which is their traditional role. Second, they specify intermediate frequency dynamics that are usually assigned to smooth autoregressive processes. Finally, high frequency switches generate substantial outliers. Thus, a single mechanism captures three important features of the data that are typically addressed as distinct phenomena in the literature. Maximum likelihood estimation is developed and shown to perform well in finite sample. We estimate on exchange rate data a version of the process with four parameters and more than a thousand states. The estimated model compares favorably to earlier specifications both in- and out-of-sample. Multifractal forecasts slightly improve on GARCH(1,1) at daily and weekly intervals, and provide considerable gains in accuracy at horizons of 10 to 50 days.

Regime-Switching and the Estimation of Multifractal Processes

We propose a discrete-time stochastic volatility model in which regime switching serves three purposes. First, changes in regimes capture low frequency variations, which is their traditional role. Second, they specify intermediate frequency dynamics that are usually assigned to smooth autoregressive processes. Finally, high frequency switches generate substantial outliers. Thus, a single mechanism captures three important features of the data that are typically addressed as distinct phenomena in the literature. Maximum likelihood estimation is developed and shown to perform well in finite sample. We estimate on exchange rate data a version of the process with four parameters and more than a thousand states. The estimated model compares favorably to earlier specifications both in- and out-of-sample. Multifractal forecasts slightly improve on GARCH(1,1) at daily and weekly intervals, and provide considerable gains in accuracy at horizons of 10 to 50 days.

Multifrequency Jump-Diffusions: An Equilibrium Approach

This paper proposes that equilibrium valuation is a powerful method to generate endogenous jumps in asset prices, which provides a structural alternative to traditional reduced-form specifications with exogenous discontinuities. We specify an economy with continuous consumption and dividend paths, in which endogenous price jumps originate from the market impact of regime-switches in the drifts and volatilities of fundamentals. We parsimoniously incorporate shocks of heterogeneous durations in consumption and dividends while keeping constant the number of parameters. Equilibrium valuation creates an endogenous relation between a shock's persistence and the magnitude of the induced price jump. As the number of frequencies driving fundamentals goes to infinity, the price process converges to a novel stochastic process, which we call a multifractal jump-diffusion.

Multifractality of Deutschemark/US Dollar Exchange Rates

This paper presents the first empirical investigation of the Multifractal Model of Asset Returns ("MMAR"). The MMAR, developed in Mandelbrot, Fisher, and Calvet (1997), is an alternative to ARCH-type representations for modelling temporal heterogeneity in financial returns. Typically, researchers introduce temporal heterogeneity through time-varying conditional second moments in a discrete time framework. Multifractality introduces a new source of heterogeneity through time-varying local regularity in the price path. The concept of local Holder exponent describes local regularity. Multifractal processes bridge the gap between locally Gaussian (Ito) diffusions and jump-diffusions by allowing a multiplicity of Holder exponents. This paper investigates multifractality in Deutschemark/US Dollar currency exchange rates. After finding evidence of multifractal scaling, we show how to estimate the multifractal spectrum via the Legendre transform. The scaling laws found in the data are replicated in simulations. Further simulation experiments test whether alternative representations, such as FIGARCH, are likely to replicate the multifractal signature of the Deutschemark/US Dollar data. On the basis of this evidence, the MMAR hypothesis appears more likely. Overall, the MMAR is quite successful in uncovering a previously unseen empirical regularity. Additionally, the model generates realistic sample paths, and opens the door to new theoretical and applied approaches to asset pricing and risk valuation. We conclude by advocating further empirical study of multifractality in financial data, along with more intensive study of estimation techniques and inference procedures.Multifractal model of asset returns, multifractal process, compound stochastic process, trading time, time deformation, scaling laws, multiscaling, self-similarity, self-affinity

Multifrequency News and Stock Returns

Recent research documents that aggregate stock prices are driven by shocks with persistence levels ranging from daily intervals to several decades. Building on these insights, we introduce a parsimonious equilibrium model in which regime-shifts of heterogeneous durations affect the volatility of dividend news. We estimate tightly parameterized specifications with up to 256 discrete states on daily U.S. equity returns. The multifrequency equilibrium has significantly higher likelihood than the classic Campbell and Hentschel (1992) specification, while generating volatility feedback effects 6 to 12 times larger. We show in an extension that Bayesian learning about stochastic volatility is faster for bad states than good states, providing a novel source of endogenous skewness that complements the "uncertainty" channel considered in previous literature (e.g., Veronesi, 1999). Furthermore, signal precision induces a tradeoff between skewness and kurtosis, and economies with intermediate investor information best match the data.

Large Deviations and the Distribution of Price Changes

The Multifractal Model of Asset Returns ("MMAR," see Mandelbrot, Fisher, and Calvet, 1997) proposes a class of multifractal processes for the modelling of financial returns. In that paper, multifractal processes are defined by a scaling law for moments of the processes' increments over finite time intervals. In the present paper, we discuss the local behavior of multifractal processes. We employ local Holder exponents, a fundamental concept in real analysis that describes the local scaling properties of a realized path at any point in time. In contrast with the standard models of continuous time finance, multifractal processes contain a multiplicity of local Holder exponents within any finite time interval. We characterize the distribution of Holder exponents by the multifractal spectrum of the process. For a broad class of multifractal processes, this distribution can be obtained by an application of Cramer's Large Deviation Theory. In an alternative interpretation, the multifractal spectrum describes the fractal dimension of the set of points having a given local Holder exponent. Finally, we show how to obtain processes with varied spectra. This allows the applied researcher to relate an empirical estimate of the multifractal spectrum back to a particular construction of the Stochastic process.Multifractal model of asset returns, multifractal spectrum, compound stochastic process, subordinated stochastic process, time deformation, scaling laws, self-similarity, self-affinity

Multivariate Stock Returns Around Extreme Events: A Reassessment of Economic Fundamentals and the 1987 Market Crash

This paper reassesses the role of economic fundamentals in the 1987 stock market crash using a two factor common-component model of returns. The model decomposes returns into idiosyncratic components, a common white noise component, and a common source of Poisson jumps. Among three two-year sample periods for Major Market Index stocks, only a 1987-88 sample results in an estimated jump component with low frequency and large size. Using Bayes' rule, we infer ex post jump probabilities for each sample day. In contrast to an analogous univariate model for an index return, the multivariate model captures information in the cross-section of returns. Leading financial news on the most likely jump days from the multivariate model is compared with news on a control group of high index return days. Days with high jump probabilities under the multivariate model contain systematically more news related to the dollar, trade deficits, and financing of the U. S. budget deficit. This suggest that the common jump component proxies for economic fundaments related to this cluster of news events, and that the unexpectedly large U.S. trade deficit news released on the Wednesday prior to the crash provided an economic catalyst for the event

Multivariate Stock Returns Around Extreme Events: A Reassessment of Economic Fundamentals and the 1987 Market Crash

This paper reassesses the role of economic fundamentals in the 1987 stock market crash using a two factor common-component model of returns. The model decomposes returns into idiosyncratic components, a common white noise component, and a common source of Poisson jumps. Among three two-year sample periods for Major Market Index stocks, only a 1987-88 sample results in an estimated jump component with low frequency and large size. Using Bayes' rule, we infer ex post jump probabilities for each sample day. In contrast to an analogous univariate model for an index return, the multivariate model captures information in the cross-section of returns. Leading financial news on the most likely jump days from the multivariate model is compared with news on a control group of high index return days. Days with high jump probabilities under the multivariate model contain systematically more news related to the dollar, trade deficits, and financing of the U. S. budget deficit. This suggest that the common jump component proxies for economic fundaments related to this cluster of news events, and that the unexpectedly large U.S. trade deficit news released on the Wednesday prior to the crash provided an economic catalyst for the event

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.

Multivariate Stock Returns Around Extreme Events: A Reassessment of Economic Fundamentals and the 1987 Market Crash

This paper reassesses the role of economic fundamentals in the 1987 stock market crash using a two factor common-component model of returns. The model decomposes returns into idiosyncratic components, a common white noise component, and a common source of Poisson jumps. Among three two-year sample periods for Major Market Index stocks, only a 1987-88 sample results in an estimated jump component with low frequency and large size. Using Bayes' rule, we infer ex post jump probabilities for each sample day. In contrast to an analogous univariate model for an index return, the multivariate model captures information in the cross-section of returns. Leading financial news on the most likely jump days from the multivariate model is compared with news on a control group of high index return days. Days with high jump probabilities under the multivariate model contain systematically more news related to the dollar, trade deficits, and financing of the U. S. budget deficit. This suggest that the common jump component proxies for economic fundaments related to this cluster of news events, and that the unexpectedly large U.S. trade deficit news released on the Wednesday prior to the crash provided an economic catalyst for the event