Markov switching multifractal

Let ${\displaystyle P_{t}}$ denote the price of a financial asset, and let ${\displaystyle r_{t}=\ln(P_{t}/P_{t-1})}$ denote the return over two consecutive periods. In MSM, returns are specified as

${\displaystyle r_{t}=\mu +{\bar {\sigma }}(M_{1,t}M_{2,t}...M_{{\bar {k}},t})^{1/2}\epsilon _{t},}$

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

${\displaystyle M_{t}=(M_{1,t}M_{2,t}\dots M_{{\bar {k}},t})\in R_{+}^{\bar {k}}.}$

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

${\displaystyle M_{k,t}}$ drawn from distribution M	with probability ${\displaystyle \gamma _{k}}$
${\displaystyle M_{k,t}=M_{k,t-1}}$	with probability ${\displaystyle 1-\gamma _{k}}$

The transition probabilities are specified by

${\displaystyle \gamma _{k}=1-(1-\gamma _{1})^{(b^{k-1})}}$.

The sequence ${\displaystyle \gamma _{k}}$ is approximately geometric ${\displaystyle \gamma _{k}\approx \gamma _{1}b^{k-1}}$ at low frequency. 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 ${\displaystyle m_{0}}$ or ${\displaystyle 2-m_{0}}$ with equal probability. The return process ${\displaystyle r_{t}}$ is then specified by the parameters ${\displaystyle \theta =(m_{0},\mu ,{\bar {\sigma }},b,\gamma _{1})}$. Note that the number of parameters is the same for all ${\displaystyle {\bar {k}}>1}$.

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

${\displaystyle {\frac {dP_{t}}{P_{t}}}=\mu dt+\sigma (M_{t})\,dW_{t},}$

where ${\displaystyle \sigma (M_{t})={\bar {\sigma }}(M_{1,t}\dots M_{{\bar {k}},t})^{1/2}}$, ${\displaystyle W_{t}}$ is a standard Brownian motion, and ${\displaystyle \mu }$ and ${\displaystyle {\bar {\sigma }}}$ are constants. Each component follows the dynamics:

${\displaystyle M_{k,t}}$ drawn from distribution M	with probability ${\displaystyle \gamma _{k}dt}$
${\displaystyle M_{k,t+dt}=M_{k,t}}$	with probability ${\displaystyle 1-\gamma _{k}dt}$

The intensities vary geometrically with k:

${\displaystyle \gamma _{k}=\gamma _{1}b^{k-1}.}$

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

The log likelihood function has the following analytical expression:

${\displaystyle \ln L(r_{1},\dots ,r_{T};\theta )=\sum _{t=1}^{T}\ln[\omega (r_{t}).(\Pi _{t-1}A)].}$

Maximum likelihood provides reasonably precise estimates in finite samples.[2]

When ${\displaystyle M}$ has a continuous distribution, estimation can proceed by simulated method of moments,[3][4] or simulated likelihood via a particle filter.[5]

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. They have also been used to generate multifractal jump-diffusions.[9]