Multifractal Model of Asset Returns


(B) The price of a stock or any other asset trading on the stock market is a multifractal process with fat tails and long-term dependency. The Multifractal Model of Asset Returns (MMAR) provides the price of the asset by compounding a Fractional Brownian Model with a Trading Time. The Trading Time is a multifractal deformation of the time. Following is a summary of the mathematical model of the MMAR imagined by Benoit Mandelbrot that was described qualitatively in the previous Blog article:

Multifractal Measure
A random measure μ defined on [0,1] is multifractal, if for all q ∈ Q:

      As ∆t->0                         E [ μ (t, t+ Δt) ]q ~ C(q) ∆tζ(q) + 1

Q is an interval containing [0,1] and c(q) and ζ(q) are defined on Q. ζ(q) is the scaling function of the fractal.

Multifractal Process
Extending multifractal measures to stochastic processes lead us to the following definition:

A stochastic process X(t) is multifractal if it has stationary increments and satisfies:

As ∆t->0                         E [|X(t+ Δt) – X(t)|q] ~ C(q) ∆tζ(q) + 1

H or Hurst or Holder Exponent
The Hurst or Holder exponent measures the long-term dependency of stock prices P(t) across different time scales:

       As ε -> 0                        H(t) = lim log [ P(t+ε) – P(t)]/log ε

As a consequence, the infinitesimal variations of the stock price P are a function of H:

dP ~ F(t) dtH(t)

Self Affine Process
H is also involved in the definition of a self-affine process.

Given X(t) = 0, a random process X(t) that satisfies:

{X(ct1),…….,X(ctk)} =d {cHX(t1),…….,cHX(tk)}

for some H > 0 and all c, k, t1,…….,tk ≥ 0 is called self-affine.

Self affine processes are fractal. So knowing that:

X(t) = tHX(1) => E [|X(t)|q] = tHq E[ |X(1)|q ]

E [|X(t)|q] = C(q) tζ(q) + 1 = tHq E[|X(1)|q ]


ζ (t) = Hq – 1 and c(q) = E[|X(1)|q ]

When the scaling function ζ(q) is linear, the fractal is uniscaling or unifractal. When ζ(q) is concave, the fractal is multiscale or multifractal.

Fractional Brownian Motion (FBM)
Given 0 < H < 1, the Fractional Brownian Motion BH(t) is defined as follows

– Its increments are Gaussian and satisfies:

E [BH(θ) – BH(0)] = 0  and  E [BH(θ) – BH(0)]2 =  θ2H

– And for q > -1:

E [|BH(θ) – BH(0)|q ] =  C(q) θqH

BH(t) is a Brownian motion when BH(0) = 0 and H = 1/2, is antipersistent when 0 < H < 1/2 and persistent with long memory returns when 1/2 < H < 1. Note that BH(t) is unifractal.

Multifractal Trading Time (MTT)                                                                              Given t the clock time, the Multifractal Trading Time θ(t) is the cumulative distribution function of a multifractal measure μ defined on [0,1]:

θ(t) = μ ([0,1])

And, the moments of θ(t) take the form:

E [θ(t)]q = E Ωq tζ(q) + 1

Ω is an important variable, that we will see later in the Multifractal Model of Asset Returns, fully determines the tail behavior of stock prices.

Multifractal Model of Asset Returns (MMAR)
Given θ (t) a Multifractal Trading Time, and BH(t) a Fractal Brownian Motion of a time series of stock prices P(t).
The Multifractal Model of Asset Returns defines P(t) as the compound process of θ (t) and BH(t):

In P(t) ≡ BH [ θ (t) ]

And so:

E [|P(t)|q ] ~ C(q) tζ(q) + 1

Both θ (t) and BH(t) are self-affine and self-affinity is preserved when they are compounded.
Note that both P(t) and θ(t) are multifractal while BH(t) is unifractal.

MMAR Properties
The beauty of the MMAR is that it can accommodate the wide variety of tail behaviors exhibited by the stock market. The trading time θ(t) controls the moment of the stock price P(t). And the moments of θ(t) have very different properties whether μ is randomly microcanonical or canonical. Microcanonical measures generate “mild” volatility of stock prices with relatively thin tails and finite moments. On the other hand, canonical measures generate “wild” volatility of stock prices with fat tails and diverging moments.

The tail behavior of stock prices is fully determined by the variable Ω in the moment of θ(t). Ω has generally Paretian tails and infinite moments. Infinite moments have a Pareto power law distributions where P(X>x) = x-qcrit with qcrit such as 1 < qcrit < ∞

The moment E [θ(t)]q is finite when 0 ≤ q < qcrit and infinite when q ≥ qcrit. Furthermore, the scaling function ζ(q) is negative when 0 < q < 1 and positive when 1< q < qcrit.

Ce qu’il fallait demontrer (CQFD)!

Books from Benoit Mandelbrot:

  • Mandelbrot, Benoit and Hudson, Richard (2004). The (Mis) Behavior of Markets. A Fractal View of Risk, Ruin and Reward. Profile Books.
  • Mandelbrot Benoit (1997), Fractal and Scaling in Finance, Springer

Publications from Benoit Mandelbrot:

  • Mandelbrot, Benoit (2003) Heavy tails in finance for independent or multifractal price increments.  Handbook of Heavy Tailed Distributions in Finance.  Edited by Svetlozar T. Rachev.
  • Mandelbrot, Benoit (2001) Scaling in financial prices, I: Tails and dependence. Quantitative Finance: 1, 113-123.
  • Mandelbrot, Benoit (2001) Scaling in financial prices, II: Multifractals and the star equation. Quantitative Finance: 1, 124-130.
  • Mandelbrot, Benoit (2001) Scaling in financial prices, III: Cartoon Brownian motions in multifractal time.  Quantitative Finance: 1, 427-440.
  • Mandelbrot, Benoit (2001) Scaling in financial prices, IV: Multifractal concentration. Quantitative Finance: 1, 641-649.
  • Mandelbrot, Benoit with Calvet Laurent, & Fisher Adlai (1997). The multifractal model of asset returns.  Cowles Foundation Discussion Papers.
  • Mandelbrot, Benoit (1963) The Variation of Certain Speculative Prices. Journal of Business, 36, 394-419.

Note: Those three kids are from the island of Huahine in Polynesia and showed me one afternoon the beauty of the multifractality of Huahine.

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Categories: Financial Engineering