In financial econometrics (the application of statistical methods to economic data), the Markov-switching multifractal (MSM) is a model of asset returns developed by Laurent E. Calvet and Adlai J. Fisher that incorporates stochastic volatility components of heterogeneous durations.
[1][2] MSM captures the outliers, log-memory-like volatility persistence and power variation of financial returns.
In currency and equity series, MSM compares favorably with standard volatility models such as GARCH(1,1) and FIGARCH both in- and out-of-sample.
MSM is used by practitioners in the financial industry for different types of forecasts.
denote the price of a financial asset, and let
denote the return over two consecutive periods.
Volatility is driven by the first-order latent Markov state vector: Given the volatility state
is drawn from a fixed distribution M with probability
The transition probabilities are specified by The sequence
The marginal distribution M has a unit mean, has a positive support, and is independent of k. In empirical applications, the distribution M is often a discrete distribution that can take the values
MSM is similarly defined in continuous time.
The price process follows the diffusion: where
goes to infinity, continuous-time MSM converges to a multifractal diffusion, whose sample paths take a continuum of local Hölder exponents on any finite time interval.
has a discrete distribution, the Markov state vector
The Markov dynamics are characterized by the transition matrix
Conditional on the volatility state, the return
has Gaussian density The log likelihood function has the following analytical expression: Maximum likelihood provides reasonably precise estimates in finite samples.
has a continuous distribution, estimation can proceed by simulated method of moments,[3][4] or simulated likelihood via a particle filter.
, the conditional distribution of the latent state vector at date
is given by: MSM often provides better volatility forecasts than some of the best traditional models both in and out of sample.
Calvet and Fisher[2] report considerable gains in exchange rate volatility forecasts at horizons of 10 to 50 days as compared with GARCH(1,1), Markov-Switching GARCH,[6][7] and Fractionally Integrated GARCH.
[8] Lux[4] obtains similar results using linear predictions.
Extensions of MSM to multiple assets provide reliable estimates of the value-at-risk in a portfolio of securities.
[5] In financial economics, MSM has been used to analyze the pricing implications of multifrequency risk.
The models have had some success in explaining the excess volatility of stock returns compared to fundamentals and the negative skewness of equity returns.
[9] MSM is a stochastic volatility model[10][11] with arbitrarily many frequencies.
MSM builds on the convenience of regime-switching models, which were advanced in economics and finance by James D.
[12][13] MSM is closely related to the Multifractal Model of Asset Returns.
[14] MSM improves on the MMAR's combinatorial construction by randomizing arrival times, guaranteeing a strictly stationary process.
MSM provides a pure regime-switching formulation of multifractal measures, which were pioneered by Benoit Mandelbrot.