In financial econometrics (the application of statistical methods to economic data), the Markov-switching multifractal (MSM) is a model of asset returns developed by Laurent E. Calvet and Adlai J. Fisher that incorporates stochastic volatility components of heterogeneous durations. 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 for different types of forecasts.

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

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:

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.

The transition probabilities are specified by

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