Multifractal Models of Returns (MMAR and MRW)
Which features of market returns require a multifractal model to explain them?
Compare the model with observed scaling and volatility clustering across several horizons, using held-out periods and simple alternatives. A multifractal fit can describe multi-scale variation without proving that the market literally follows that mechanism at every horizon.
A multifractal specification is a model, not proof of realized structure at every horizon.
Evidence map
| Aspect | Finding |
|---|---|
| What it is | A family of models in which volatility follows a cascade of fluctuations across scales: fluctuations at one scale are the product of random multipliers from all coarser scales, so the scaling exponent of the q-th moment of returns is a nonlinear function of q. |
| Key result / formula | The signature is E|r(Δ)|^q ∝ Δ^ζ(q) with ζ(q) concave rather than linear in q. |
| Why it matters for backtesting | Multifractal scaling is the null competing with "this market has a special time scale". |
What it is
The picture comes from turbulence and was imported by Mandelbrot.
Key result / formula
Mandelbrot, Fisher & Calvet (1997) built the Multifractal Model of Asset Returns as a Brownian or fractional Brownian motion subordinated to a multifractal trading time θ(t): X(t) = B_H[θ(t)], θ the cumulative measure of a multiplicative cascade. The construction delivers fat tails, long memory in absolute returns and a nonlinear ζ(q) from few parameters; the Cowles paper is the theory, with calibration left to companion work. Its weaknesses are that a grid-based cascade is not stationary and singles out a scale ratio. Bacry, Delour & Muzy (2001) removed both with the Multifractal Random Walk: r_t = ε_t exp(ω_t), with ε Gaussian and ω a Gaussian process whose covariance decays logarithmically, Cov(ω_t, ω_(t+τ)) = λ² ln(T/τ) up to an integral scale T. It is the first multifractal process with continuous dilation invariance and stationary increments, and it is fixed by a few parameters — the variance, the intermittency coefficient λ² and T — which is why it is the version that actually gets estimated.
Why it matters for backtesting
Whenever a strategy is claimed to work at one bar size and not at others, or a volatility feature is found at one horizon, the multifractal alternative holds that the same structure exists across scales with different amplitude — and the MRW reproduces volatility clustering, fat tails and the slow decay of the volatility autocorrelation from three numbers with no directional content at all. What makes the claim fail is estimation: the concavity of ζ(q) is read from high moments, and with a tail exponent near 3 those moments do not exist in the population, so an estimated spectrum must be compared with the one obtained from surrogate data of the same length before the word "multifractal" is used.
Source
Mandelbrot, Fisher & Calvet, "A Multifractal Model of Asset Returns", Cowles Foundation Discussion Paper 1164, Yale University, 1997; Bacry, Delour & Muzy, "Multifractal random walk", Physical Review E 64(2), 2001, 026103. Primary source