# Markov Switching Multifractal model¶

REFERENCE:

How to Forecast Long-Run Volatility: Regime Switching and the Estimation of Multifractal Processes, Calvet and Fisher, 2004.

AUTHOR:

• William Stein, 2008

TESTS:

sage: msm = finance.MarkovSwitchingMultifractal(8,1.4,1.0,0.95,3)
True

class sage.finance.markov_multifractal.MarkovSwitchingMultifractal(kbar, m0, sigma, gamma_kbar, b)

INPUT:

• kbar – positive integer
• m0 – float with 0 <= m0 <= 2
• sigma – positive float
• gamma_kbar – float with 0 <= gamma_kbar < 1
• b – float > 1

EXAMPLES:

sage: msm = finance.MarkovSwitchingMultifractal(8,1.4,0.5,0.95,3); msm
Markov switching multifractal model with m0 = 1.4, sigma = 0.5, b = 3.0, and gamma_8 = 0.95
sage: yen_usd = finance.MarkovSwitchingMultifractal(10,1.448,0.461,0.998,3.76)
sage: dm = finance.MarkovSwitchingMultifractal(10,1.326,0.643,0.959,2.7)

b()

Return parameter b of Markov switching multifractal model.

EXAMPLES:

sage: msm = finance.MarkovSwitchingMultifractal(8,1.4,1,0.95,3)
sage: msm.b()
3.0

gamma()

Return the vector of the kbar transitional probabilities.

OUTPUT:

• gamma – a tuple of self.kbar() floats.

EXAMPLES:

sage: msm = finance.MarkovSwitchingMultifractal(8,1.4,1.0,0.95,3)
sage: msm.gamma()
(0.001368852970712986, 0.004100940201672509, 0.012252436441829..., 0.03630878209190..., 0.10501923017634..., 0.28312883556311..., 0.6315968501359..., 0.95000000000000...)

gamma_kbar()

Return parameter gamma_kbar of Markov switching multifractal model.

EXAMPLES:

sage: msm = finance.MarkovSwitchingMultifractal(8,1.4,0.01,0.95,3)
sage: msm.gamma_kbar()
0.95

kbar()

Return parameter kbar of Markov switching multifractal model.

EXAMPLES:

sage: msm = finance.MarkovSwitchingMultifractal(8,1.4,0.01,0.95,3)
sage: msm.kbar()
8

m0()

Return parameter m0 of Markov switching multifractal model.

EXAMPLES:

sage: msm = finance.MarkovSwitchingMultifractal(8,1.4,1,0.95,3)
sage: msm.m0()
1.4

sigma()

Return parameter sigma of Markov switching multifractal model.

EXAMPLES:

sage: msm = finance.MarkovSwitchingMultifractal(8,1.4,1,0.95,3)
sage: msm.sigma()
1.0

simulation(n)

Same as self.simulations, but run only 1 time, and returns a time series instead of a list of time series.

INPUT:

• n – a positive integer.

EXAMPLES:

sage: msm = finance.MarkovSwitchingMultifractal(8,1.4,1.0,0.95,3)
sage: msm.simulation(5)
[0.0059, -0.0097, -0.0101, -0.0110, -0.0067]
sage: msm.simulation(3)
[0.0055, -0.0084, 0.0141]

simulations(n, k=1)

Return k simulations of length n using this Markov switching multifractal model for n time steps.

INPUT:

• n – positive integer; number of steps.
• k – positive integer (default: 1); number of simulations.

OUTPUT:

list – a list of TimeSeries objects.

EXAMPLES:

sage: cad_usd = finance.MarkovSwitchingMultifractal(10,1.278,0.262,0.644,2.11); cad_usd
Markov switching multifractal model with m0 = 1.278, sigma = 0.262, b = 2.11, and gamma_10 = 0.644


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Multifractal Random Walk