Markov-Switching Multifractal Model

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In financial econometrics, the Markov-Switching multifractal (MSM) is a model of asset returns 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 to forecast volatility, compute value-at-risk, and price derivatives.

MSM Specification

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The MSM model can be specified in both discrete time and continuous time.

Discrete time

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Let   denote the price of a financial asset, and let   denote the return over two consecutive periods. In MSM, returns are specified as

 

where   and   are constants and { } are independent standard Gaussians. Volatility is driven by the first-order latent Markov state vector:

 

Given the volatility state  , the next-period multiplier   is drawn from a fixed distribution M with probability  , and is otherwise left unchanged.


  drawn from distribution   with probability  
  with probability  

The transition probabilities are specified by

 .

The sequence   is approximately geometric   at low frequency. The marginal distribution   has a unit mean, has a positive support, and is independent of  .


Binomial MSM

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In empirical applications, the distribution   is often a discrete distribution that can take the values   or   with equal probability. The return process   is then specified by the parameters  . Note that the number of parameters is the same for all  .

Continuous Time

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MSM is similarly defined in continuous time. The price process follows the diffusion:

 ,

where  ,   is a standard Brownian motion, and  and   are constants. Each component follows the dynamics:

  drawn from distribution   with probability  
  with probability  

The intensities vary geometrically with  :

 

When the number of components   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.

Inference and Closed-Form Likelihood

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When   has a discrete distribution, the Markov state vector   takes finitely many values  . For instance, there are   possible states in binomial MSM. The Markov dynamics are characterized by the transition matrix   with components  . Conditional on the volatility state, the return   has Gaussian density

 

Conditional Distribution

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We do not directly observe the latent state vector  . Given past returns, we can define the conditional probabilities:


 .


The vector   is computed recursively:


 '


where  ,   for any   , and


 


The initial vector   is set equal to the ergodic distribution of  . For binomial MSM,   for all  .

Closed-Form Likelihood

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The log likelihood function has the following analytical expression:

 .

Maximum likelihood provides reasonably precise estimates in finite samples [3].

Other Estimation Methods

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When   has a continuous distribution, estimation can proceed by simulated method of moments [4] [5], or simulated likelihood via a particle filter [6].

Forecasting

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Given  ,…, , 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[7] report considerable gains in exchange rate volatility forecasts at horizons of 10 to 50 days as compared with GARCH(1,1), Markov-Switching GARCH [8][9], and Fractionally Integrated GARCH [10]. [11] obtains similar results using linear predictions.

Applications

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Multiple Assets and Value-at-Risk

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Extensions of MSM to multiple assets provide reliable estimates of the value-at-risk in a portfolio of securities [12].

Asset Pricing

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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. They have also been used to generate multifractal jump-diffusions [13].

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MSM is a stochastic volatility model [14][15] with arbitrarily many frequencies. MSM builds on the convenience of regime-switching models, which were advanced in economics and finance by James D. Hamilton [16], [17]. MSM is closely related to the MMAR [18]. 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 [19][20][21].

See Also

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References

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  1. ^ Calvet, Laurent (2001). "Forecasting multifractal volatility". Journal of Econometrics. 105: 27–58. doi:10.1016/S0304-4076(01)00069-0. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  2. ^ Calvet, Laurent (2004). "How to Forecast long-run volatility: regime-switching and the estimation of multifractal processes". Journal of Financial Econometrics. 2: 49–83. doi:10.1093/jjfinec/nbh003. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  3. ^ Calvet, Laurent (2004). "How to Forecast long-run volatility: regime-switching and the estimation of multifractal processes". Journal of Financial Econometrics. 2: 49–83. doi:10.1093/jjfinec/nbh003. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  4. ^ Calvet, Laurent (2002). "Regime-switching and the estimation of multifractal processes". {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help) [1] (Available at ),
  5. ^ Lux, Thomas (2008). "The Markov-switching multifractal model of asset returns: GMM estimation and linear forecasting of volatility". Journal of Business and Economic Statistics. 26: 194–210. doi:10.1198/073500107000000403.
  6. ^ Calvet, Laurent (2006). "Volatility comovement: a multifrequency approach". Journal of Econometrics. 131 (1–2): 179–215. doi:10.1016/j.jeconom.2005.01.008. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  7. ^ Calvet, Laurent (2004). "How to Forecast long-run volatility: regime-switching and the estimation of multifractal processes". Journal of Financial Econometrics. 2: 49–83. doi:10.1093/jjfinec/nbh003. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  8. ^ Gray, Stephen (1996). "Modeling the conditional distribution of interest rates as a regime-switching process". Journal of Financial Economics. 42: 27–62. doi:10.1016/0304-405X(96)00875-6.
  9. ^ Klaassen, Franc (2002). "Improving GARCH volatility forecasts with regime-switching GARCH". Empirical Economics. 27 (2): 363–394. doi:10.1007/s001810100100.
  10. ^ *Bollerslev, Tim (1996). "Modeling and pricing long memory in stock market volatility". Journal of Econometrics. 73: 151–184. doi:10.1016/0304-4076(95)01736-4. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  11. ^ Lux, Thomas (2008). "The Markov-switching multifractal model of asset returns: GMM estimation and linear forecasting of volatility". Journal of Business and Economic Statistics. 26: 194–210. doi:10.1198/073500107000000403.
  12. ^ Calvet, Laurent (2006). "Volatility comovement: a multifrequency approach". Journal of Econometrics. 131 (1–2): 179–215. doi:10.1016/j.jeconom.2005.01.008. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  13. ^ Calvet, Laurent (2008). "Multifractal Volatility: Theory, Forecasting and Pricing". Elsevier - Academic Press. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  14. ^ Taylor, Stephen (1986). "Modelling Financial Time Series". New York: Wiley.
  15. ^ Wiggins, James (1987). "Option values under stochastic volatility: theory and empirical estimates". Journal of Financial Economics. 19 (2): 351–372. doi:10.1016/0304-405X(87)90009-2.
  16. ^ Hamilton, James (1989). "A new approach to the economic analysis of nonstationary time series and the business cycle". Econometrica. 57: 357–84. doi:10.2307/1912559. JSTOR 1912559.
  17. ^ Hamilton, James (2008). "Regime-Switching Models". New Palgrave Dictionary of Economics. 2nd edition, Palgrave McMillan Ltd: 1–7. doi:10.1057/978-1-349-95121-5_2459-1. ISBN 978-1-349-95121-5.
  18. ^ Calvet, Laurent (1997). "A multifractal model of asset returns". Discussion Papers , Cowles Foundation Yale University.: 1164–1166. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  19. ^ Mandelbrot, B. (1974). "Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier". Journal of Fluid Mechanics. 62 (2): 331–58. doi:10.1017/S0022112074000711.
  20. ^ Mandelbrot, B. (1982). "The Fractal Geometry of Nature". New York: Freeman.
  21. ^ Mandelbrot, B. (1999). "Multifractals and 1/f Noise: Wild Self-Affinity in Physics". Springer.