Markov switching multifractal

In financial econometrics, the Markov-switching multifractal (MSM) is a model of asset returns developed by Laurent E. Calvet and Adlai J. Fisher that incorporates 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 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.

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

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