Markov switching multifractal

MSM specification

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

Discrete time

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.

Binomial MSM

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 ${\displaystyle {\bar {k}}>1}$.

Continuous time

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.

Inference and closed-form likelihood

When ${\displaystyle M}$ has a

${\displaystyle M_{t}}$

${\displaystyle m^{1},...,m^{d}\in R_{+}^{\bar {k}}}$

${\displaystyle d=2^{\bar {k}}}$

${\displaystyle A=(a_{i,j})_{1\leq i,j\leq d}}$

${\displaystyle a_{i,j}=P\left(M_{t+1}=m^{j}|M_{t}=m^{i}\right)}$

${\displaystyle r_{t}}$

${\displaystyle f(r_{t}|M_{t}=m^{i})={\frac {1}{\sqrt {2\pi \sigma ^{2}(m^{i})}}}\exp \left[-{\frac {(r_{t}-\mu )^{2}}{2\sigma ^{2}(m^{i})}}\right].}$

Conditional distribution

Closed-form Likelihood

The log likelihood function has the following analytical expression:

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

Forecasting

Given ${\displaystyle r_{1},\dots ,r_{t}}$, the conditional distribution of the latent state vector at date ${\displaystyle t+n}$ is given by:

${\displaystyle {\hat {\Pi }}_{t,n}=\Pi _{t}A^{n}.\,}$

See also