Markov switching multifractal

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

Let $P\_t$ denote the price of a financial asset, and let $r\_t\; =\; \backslash ln\; (P\_t\; /\; P\_\{t-1\})$ denote the return over two consecutive periods. In MSM, returns are specified as

:$r\_t\; =\; \backslash mu\; +\; \backslash bar\{\backslash sigma\}(M\_\{1,t\}M\_\{2,t\}...M\_\{\backslash bar\{k\},t\})^\{1/2\}\backslash epsilon\_t,$

where $\backslash mu$ and $\backslash sigma$ are constants and {$\backslash epsilon\_t$} are independent standard Gaussians. Volatility is driven by the first-order latent Markov state vector:

:$M\_t\; =\; (M\_\{1,t\}M\_\{2,t\}\backslash dots\; M\_\{\backslash bar\{k\},t\})\; \backslash in\; R\_+^\backslash bar\{k\}.$

Given the volatility state $M\_t$, the next-period multiplier $M\_\{k,t+1\}$ is drawn from a fixed distribution {{mvar|M}} with probability $\backslash gamma\_k$, and is otherwise left unchanged.

:

The transition probabilities are specified by

:$\backslash gamma\_k\; =\; 1\; -\; (1\; -\; \backslash gamma\_1)^\{(b^\{k-1\})\}$.

The sequence $\backslash gamma\_k$ is approximately geometric $\backslash gamma\_k\; \backslash approx\; \backslash gamma\_1b^\{k-1\}$ at low frequency. The marginal distribution {{mvar|M}} has a unit mean, has a positive support, and is independent of {{mvar|k}}.

In empirical applications, the distribution {{mvar|M}} is often a discrete distribution that can take the values $m\_0$ or $2-m\_0$ with equal probability. The return process $r\_t$ is then specified by the parameters $\backslash theta\; =\; (m\_0,\backslash mu,\backslash bar\{\backslash sigma\},b,\backslash gamma\_1)$. Note that the number of parameters is the same for all $\backslash bar\{k\}>1$.

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

:$\backslash frac\{dP\_t\}\{P\_t\}\; =\; \backslash mu\; dt\; +\; \backslash sigma(M\_t)\backslash ,dW\_t,$

where $\backslash sigma(M\_t)\; =\; \backslash bar\{\backslash sigma\}(M\_\{1,t\}\backslash dots\; M\_\{\backslash bar\{k\},t\})^\{1/2\}$, $W\_t$ is a standard Brownian motion, and $\backslash mu$ and $\backslash bar\{\backslash sigma\}$ are constants. Each component follows the dynamics:

:

The intensities vary geometrically with {{mvar|k}}:

:$\backslash gamma\_k\; =\; \backslash gamma\_1b^\{k-1\}.$

When the number of components $\backslash bar\{k\}$ 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.

When $M$ has a discrete distribution, the Markov state vector $M\_t$ takes finitely many values $m^1,...,m^d\; \backslash in\; R\_+^\{\backslash bar\{k\}\}$. For instance, there are $d\; =\; 2^\{\backslash bar\{k\}\}$ possible states in binomial MSM. The Markov dynamics are characterized by the transition matrix $A\; =\; (a\_\{i,j\})\_\{1\backslash leq\; i,j\backslash leq\; d\}$ with components $a\_\{i,j\}\; =\; P\backslash left(M\_\{t+1\}\; =\; m^j|\; M\_t\; =\; m^i\backslash right)$.

Conditional on the volatility state, the return $r\_t$ has Gaussian density

:$f(\; r\_t\; |\; M\_t\; =\; m^i)\; =\; \backslash frac\{1\}\; \{\backslash sqrt\{2\backslash pi\backslash sigma^2(m^i)\}\}\backslash exp\backslash left[-\backslash frac\{(r\_t-\backslash mu)^2\}\{2\backslash sigma^2(m^i)\}\backslash right]\; .$

The log likelihood function has the following analytical expression:

:$\backslash ln\; L(r\_1,\backslash dots,r\_T;\backslash theta)\; =\; \backslash sum\_\{t=1\}^\{T\}\backslash ln[\backslash omega(r\_t).(\backslash Pi\_\{t-1\}A)].$

Maximum likelihood provides reasonably precise estimates in finite samples.

When $M$ has a continuous distribution, estimation can proceed by simulated method of moments,{{cite journal |title=Regime-switching and the estimation of multifractal processes |last1=Calvet |first1=Laurent |first2=Adlai |last2=Fisher |date=July 2003 |journal=NBER Working Paper No. 9839 |doi=10.3386/w9839 |doi-access=free }}{{Cite journal | last1 = Lux | first1 = T. | title = The Markov-Switching Multifractal Model of Asset Returns | doi = 10.1198/073500107000000403 | journal = Journal of Business & Economic Statistics | volume = 26 | issue = 2 | pages = 194–210 | year = 2008 | s2cid = 55648360 }} or simulated likelihood via a particle filter.{{Cite journal | last1 = Calvet | first1 = L. E. | last2 = Fisher | first2 = A. J. | last3 = Thompson | first3 = S. B. | title = Volatility comovement: A multifrequency approach | doi = 10.1016/j.jeconom.2005.01.008 | journal = Journal of Econometrics | volume = 131 | issue = 1–2 | pages = 179–215 | year = 2006 | citeseerx = 10.1.1.331.152 }}

Given $r\_1,\backslash dots,r\_t$, the conditional distribution of the latent state vector at date $t+n$ is given by:

:$\backslash hat\{\backslash Pi\}\_\{t,n\}\; =\; \backslash Pi\_tA^n.\backslash ,$

MSM often provides better volatility forecasts than some of the best traditional models both in and out of sample. Calvet and Fisher report considerable gains in exchange rate volatility forecasts at horizons of 10 to 50 days as compared with GARCH(1,1), Markov-Switching GARCH,{{Cite journal | last1 = Gray | first1 = S. F. | doi = 10.1016/0304-405X(96)00875-6 | title = Modeling the conditional distribution of interest rates as a regime-switching process | journal = Journal of Financial Economics | volume = 42 | pages = 27–77 | year = 1996 }}{{Cite journal | last1 = Klaassen | first1 = F. | title = Improving GARCH volatility forecasts with regime-switching GARCH | doi = 10.1007/s001810100100 | journal = Empirical Economics | volume = 27 | issue = 2 | pages = 363–394 | year = 2002 | s2cid = 29571612 | url = https://pure.uva.nl/ws/files/3524906/21100_v60.pdf }} and Fractionally Integrated GARCH.{{Cite journal | last1 = Bollerslev | first1 = T. | last2 = Ole Mikkelsen | first2 = H. | doi = 10.1016/0304-4076(95)01736-4 | title = Modeling and pricing long memory in stock market volatility | journal = Journal of Econometrics | volume = 73 | pages = 151–184 | year = 1996 }} Lux 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.

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.{{cite book|last1= Calvet |first1=Laurent E.|last2=Fisher |first2=Adlai J.|title=Multifractal volatility theory, forecasting, and pricing|year=2008|publisher=Academic Press|location=Burlington, MA|isbn=9780080559964}}

MSM is a stochastic volatility model{{cite book|last=Taylor|first=Stephen J|title=Modelling financial time series|year=2008|publisher=World Scientific|location=New Jersey|isbn=9789812770844|edition=2nd}}{{Cite journal | last1 = Wiggins | first1 = J. B. | title = Option values under stochastic volatility: Theory and empirical estimates | doi = 10.1016/0304-405X(87)90009-2 | journal = Journal of Financial Economics | volume = 19 | issue = 2 | pages = 351–372 | year = 1987 | url = http://dml.cz/bitstream/handle/10338.dmlcz/135564/Kybernetika_39-2003-6_1.pdf }} with arbitrarily many frequencies. MSM builds on the convenience of regime-switching models, which were advanced in economics and finance by James D. Hamilton.{{cite journal | last1 = Hamilton | first1 = J. D. | year = 1989| title = A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle | journal = Econometrica | volume = 57 | issue = 2 | pages = 357–384 | jstor = 1912559 | doi = 10.2307/1912559| citeseerx = 10.1.1.397.3582 }}{{cite book|chapter=Regime-Switching Models | last=Hamilton | first=James |title=New Palgrave Dictionary of Economics |edition=2nd | year=2008 |publisher=Palgrave Macmillan Ltd. |isbn= 9780333786765 }}

MSM is closely related to the Multifractal Model of Asset Returns.{{cite journal|title=A multifractal model of asset returns | last3=Calvet | first3=Laurent | first2=Adlai |last2=Fisher |first1=Benoit |last1=Mandelbrot |journal=Cowles Foundation Discussion Paper No. 1164 |date=September 1997 |ssrn=78588 }} 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.{{Cite journal | last1 = Mandelbrot | first1 = B. B. | title = Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier | doi = 10.1017/S0022112074000711 | journal = Journal of Fluid Mechanics | volume = 62 | issue = 2 | pages = 331–358 | year = 2006 | s2cid = 222375985 }}{{cite book|last=Mandelbrot|first=Benoit B.|title=The fractal geometry of nature|year=1983|publisher=Freeman|location=New York|isbn=9780716711865|edition=Updated and augm.|url-access=registration|url=https://archive.org/details/fractalgeometryo00beno}}{{cite book|first=Benoit B.|last=Mandelbrot|author2=J.M. Berger|title=Multifractals and 1/f noise : wild self-affinity in physics (1963 - 1976).|year=1999|publisher=Springer|location=New York, NY [u.a.]|isbn=9780387985398|edition=Repr.|display-authors=etal|url-access=registration|url=https://archive.org/details/multifractals1fn0000mand}}

• Brownian motion
• Rogemar Mamon
• Markov chain
• Multifractal model of asset returns
• Multifractal
• Stochastic volatility

{{Reflist}}

• [https://web.archive.org/web/20120402215902/http://wiki.wstein.org/2008/simuw Financial Time Series, Multifractals and Hidden Markov Models]

{{DEFAULTSORT:Markov Switching Multifractal}}

Category:Mathematical finance

Category:Markov models

Category:Fractals