Building a Multifractal Market Model and Comparing Its Volatility Forecasts

Summary

This article builds a Multifractal Model of Asset Returns (MMAR) from previously estimated scaling parameters and compares its volatility forecasts with a GARCH(1,1) model. It creates multifractal trading time with a normalized multiplicative cascade, then generates Fractional Brownian Motion (FBM) to supply long-memory structure. Composing the FBM with the nonlinear trading-time clock produces the MMAR price process. The article describes using Monte Carlo simulations to forecast volatility and evaluating forecasts against realized volatility.

The cascade concentrates activity unevenly across intervals, while the FBM component captures persistence or anti-persistence according to its Hurst exponent. FBM simulation uses the Davies–Harte method with an alternative numerical approach if its covariance construction fails validation. The proposed comparison includes forecast error measures and rolling realized-volatility assessments. These procedures provide a modeling and evaluation framework, but the supplied material does not establish that MMAR consistently outperforms GARCH. Results depend on fitted parameters, distribution choices, simulation design, and the selected forecast horizon.

Key ideas

• MMAR combines a multifractal time change with Fractional Brownian Motion to represent clustered activity and long-memory structure.
• A multiplicative cascade allocates uneven mass across intervals and its cumulative measure defines trading time.
• The Hurst exponent determines whether FBM increments exhibit persistence or anti-persistence.
• Monte Carlo simulation can generate MMAR volatility forecasts for comparison with realized volatility and GARCH.
• Model rankings depend on parameter estimates and evaluation choices, so comparison results are not universal.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.