# Markov switching multifractal

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

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

### Discrete time

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

$r_t = \mu + \bar{\sigma}(M_{1,t}M_{2,t}...M_{\bar{k},t})^{1/2}\epsilon_t,$

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

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

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

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

The transition probabilities are specified by

$\gamma_k = 1 - (1 - \gamma_1)^{(b^{k-1})}$.

The sequence $\gamma_k$ is approximately geometric $\gamma_k \approx \gamma_1b^{k-1}$ 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 $m_0$ or $2-m_0$ with equal probability. The return process $r_t$ is then specified by the parameters $\theta = (m_0,\mu,\bar{\sigma},b,\gamma_1)$. Note that the number of parameters is the same for all $\bar{k}>1$.

### Continuous time

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

$\frac{dP_t}{P_t} = \mu dt + \sigma(M_t)\,dW_t,$

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

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

The intensities vary geometrically with $k$:

$\gamma_k = \gamma_1b^{k-1}.$

When the number of components $\bar{k}$ 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 $M$ has a discrete distribution, the Markov state vector $M_t$ takes finitely many values $m^1,...,m^d \in R_+^{\bar{k}}$. For instance, there are $d = 2^{\bar{k}}$ possible states in binomial MSM. The Markov dynamics are characterized by the transition matrix $A = (a_{i,j})_{1\leq i,j\leq d}$ with components $a_{i,j} = P\left(M_{t+1} = m^j| M_t = m^i\right)$. Conditional on the volatility state, the return $r_t$ has Gaussian density

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

### Conditional distribution

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

$\Pi_t^j \equiv P(M_t = m^j | r_1,...,r_t).$

The vector $\Pi_t = (\Pi_t^1,...,\Pi_t^d)$ is computed recursively:

$\Pi_t = \frac{\omega(r_t)*(\Pi_{t-1}A)}{[\omega(r_t)*(\Pi_{t-1}A)]\textbf{1}'}.$

where $\textbf{1} = (1,\dots,1) \in R^d$, $x*y = \left(x_1 y_1,\dots,x_d y_d\right)$ for any $x,y \in R^d$, and

$\omega(r_t)=\left[f(r_t|M_t=m^1);\dots;f(r_t|M_t=m^d)\right].$

The initial vector $\Pi_0$ is set equal to the ergodic distribution of $M_t$. For binomial MSM, $\Pi_0^j = 1/d = 2^{-\bar{k}}$ for all $j$.

### Closed-form Likelihood

The log likelihood function has the following analytical expression:

$\ln L(r_1,\dots,r_T;\theta) = \sum_{t=1}^{T}\ln[\omega(r_t).(\Pi_{t-1}A)].$

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

### Other estimation methods

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

## Forecasting

Given $r_1,\dots,r_t$, the conditional distribution of the latent state vector at date $t+n$ is given by:

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

MSM often provides better volatility forecasts than some of the best traditional models both in and out of sample. Calvet and Fisher[2] report considerable gains in exchange rate volatility forecasts at horizons of 10 to 50 days as compared with GARCH(1,1), Markov-Switching GARCH,[6][7] and Fractionally Integrated GARCH.[8] Lux[4] obtains similar results using linear predictions.

## Applications

### Multiple assets and value-at-risk

Extensions of MSM to multiple assets provide reliable estimates of the value-at-risk in a portfolio of securities.[5]

### Asset pricing

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]

## Related approaches

MSM is a stochastic volatility model[10][11] with arbitrarily many frequencies. MSM builds on the convenience of regime-switching models, which were advanced in economics and finance by James D. Hamilton.[12][13] MSM is closely related to the Multifractal Model of Asset Returns.[14] 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.[15][16] [17]

## References

1. ^ Calvet, L.; Fisher, A. (2001). "Forecasting multifractal volatility". Journal of Econometrics 105: 27. doi:10.1016/S0304-4076(01)00069-0. edit
2. ^ a b c Calvet, L. E. (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. edit
3. ^ Calvet, Laurent; Fisher, Adlai (July 2003). Regime-switching and the estimation of multifractal processes (Technical report 9839). Working Paper Series. National Bureau of Economic Research.
4. ^ a b Lux, T. (2008). "The Markov-Switching Multifractal Model of Asset Returns". Journal of Business & Economic Statistics 26 (2): 194–210. doi:10.1198/073500107000000403. edit
5. ^ a b Calvet, L. E.; Fisher, A. J.; Thompson, S. B. (2006). "Volatility comovement: A multifrequency approach". Journal of Econometrics 131: 179. doi:10.1016/j.jeconom.2005.01.008. edit
6. ^ Gray, S. F. (1996). "Modeling the conditional distribution of interest rates as a regime-switching process". Journal of Financial Economics 42: 27–77. doi:10.1016/0304-405X(96)00875-6. edit
7. ^ Klaassen, F. (2002). "Improving GARCH volatility forecasts with regime-switching GARCH". Empirical Economics 27 (2): 363–394. doi:10.1007/s001810100100. edit
8. ^ Bollerslev, T.; Ole Mikkelsen, H. (1996). "Modeling and pricing long memory in stock market volatility". Journal of Econometrics 73: 151. doi:10.1016/0304-4076(95)01736-4. edit
9. ^ Calvet, Laurent E.; Fisher, Adlai J. (2008). Multifractal volatility theory, forecasting, and pricing. Burlington, MA: Academic Press. ISBN 9780080559964.
10. ^ Taylor, Stephen J (2008). Modelling financial time series (2nd ed.). New Jersey: World Scientific. ISBN 9789812770844.
11. ^ Wiggins, J. B. (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. edit
12. ^ Hamilton, J. D. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle". Econometrica (The Econometric Society) 57 (2): 357–384. doi:10.2307/1912559. JSTOR 1912559. edit
13. ^ Hamilton, James (2008). "Regime-Switching Models". New Palgrave Dictionary of Economics (2nd ed.). Palgrave McMillan Ltd. ISBN 9780333786765.
14. ^ Mandelbrot, Benoit; Fisher, Adlai; Calvet, Laurent (1997-09-15). "A multifractal model of asset returns". Discussion Papers , Cowles Foundation Yale University.: 1164–1166.
15. ^ Mandelbrot, B. B. (2006). "Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier". Journal of Fluid Mechanics 62 (2): 331. doi:10.1017/S0022112074000711. edit
16. ^ Mandelbrot, Benoit B. (1983). The fractal geometry of nature (Updated and augm. ed.). New York: Freeman. ISBN 9780716711865.
17. ^ Mandelbrot, Benoit B.; J.M. Berger et.al. (1999). Multifractals and 1/f noise : wild self-affinity in physics (1963 - 1976). (Repr. ed.). New York, NY [u.a.]: Springer. ISBN 9780387985398.