# Multifractal Random Walk¶

This module implements the fractal approach to understanding financial markets that was pioneered by Mandelbrot. In particular, it implements the multifractal random walk model of asset returns as developed by Bacry, Kozhemyak, and Muzy, 2006, Continuous cascade models for asset returns and many other papers by Bacry et al. See http://www.cmap.polytechnique.fr/~bacry/ftpPapers.html

See also Mandelbrot’s The Misbehavior of Markets for a motivated introduction to the general idea of using a self-similar approach to modeling asset returns.

One of the main goals of this implementation is that everything is highly optimized and ready for real world high performance simulation work.

AUTHOR:

• William Stein (2008)
sage.finance.fractal.fractional_brownian_motion_simulation(H, sigma2, N, n=1)

Returns the partial sums of a fractional Gaussian noise simulation with the same input parameters.

INPUT:

• H – float; 0 < H < 1; the Hurst parameter.
• sigma2 - float; innovation variance (should be close to 0).
• N – positive integer.
• n – positive integer (default: 1).

OUTPUT:

List of n time series.

EXAMPLES:

sage: set_random_seed(0)
sage: finance.fractional_brownian_motion_simulation(0.8,0.1,8,1)
[[-0.0754, 0.1874, 0.2735, 0.5059, 0.6824, 0.6267, 0.6465, 0.6289]]
sage: set_random_seed(0)
sage: finance.fractional_brownian_motion_simulation(0.8,0.01,8,1)
[[-0.0239, 0.0593, 0.0865, 0.1600, 0.2158, 0.1982, 0.2044, 0.1989]]
sage: finance.fractional_brownian_motion_simulation(0.8,0.01,8,2)
[[-0.0167, 0.0342, 0.0261, 0.0856, 0.1735, 0.2541, 0.1409, 0.1692],
[0.0244, -0.0153, 0.0125, -0.0363, 0.0764, 0.1009, 0.1598, 0.2133]]

sage.finance.fractal.fractional_gaussian_noise_simulation(H, sigma2, N, n=1)

Return n simulations with N steps each of fractional Gaussian noise with Hurst parameter H and innovations variance sigma2.

INPUT:

• H – float; 0 < H < 1; the Hurst parameter.
• sigma2 - positive float; innovation variance.
• N – positive integer; number of steps in simulation.
• n – positive integer (default: 1); number of simulations.

OUTPUT:

List of n time series.

EXAMPLES:

We simulate a fractional Gaussian noise:

sage: set_random_seed(0)
sage: finance.fractional_gaussian_noise_simulation(0.8,1,10,2)
[[-0.1157, 0.7025, 0.4949, 0.3324, 0.7110, 0.7248, -0.4048, 0.3103, -0.3465, 0.2964],
[-0.5981, -0.6932, 0.5947, -0.9995, -0.7726, -0.9070, -1.3538, -1.2221, -0.0290, 1.0077]]


The sums define a fractional Brownian motion process:

sage: set_random_seed(0)
sage: finance.fractional_gaussian_noise_simulation(0.8,1,10,1)[0].sums()
[-0.1157, 0.5868, 1.0818, 1.4142, 2.1252, 2.8500, 2.4452, 2.7555, 2.4090, 2.7054]


ALGORITHM:

See Simulating a Class of Stationary Gaussian Processes using the Davies-Harte Algorithm, with Application to Long Meoryy Processes, 2000, Peter F. Craigmile for a discussion and references for why the algorithm we give – which uses the stationary_gaussian_simulation() function.

sage.finance.fractal.multifractal_cascade_random_walk_simulation(T, lambda2, ell, sigma2, N, n=1)

Return a list of n simulations of a multifractal random walk using the log-normal cascade model of Bacry-Kozhemyak-Muzy 2008. This walk can be interpreted as the sequence of logarithms of a price series.

INPUT:

• T – positive real; the integral scale.
• lambda2 – positive real; the intermittency coefficient.
• ell – a small number – time step size.
• sigma2 – variance of the Gaussian white noise eps[n].
• N – number of steps in each simulation.
• n – the number of separate simulations to run.

OUTPUT:

List of time series.

EXAMPLES:

sage: set_random_seed(0)
sage: a
[[-0.0096, 0.0025, 0.0066, 0.0016, 0.0078, 0.0051, 0.0047, -0.0013, 0.0003, -0.0043],
[0.0003, 0.0035, 0.0257, 0.0358, 0.0377, 0.0563, 0.0661, 0.0746, 0.0749, 0.0689],
[-0.0120, -0.0116, 0.0043, 0.0078, 0.0115, 0.0018, 0.0085, 0.0005, 0.0012, 0.0060]]


The corresponding price series:

sage: a[0].exp()
[0.9905, 1.0025, 1.0067, 1.0016, 1.0078, 1.0051, 1.0047, 0.9987, 1.0003, 0.9957]


MORE DETAILS:

The random walk has n-th step $$\text{eps}_n e^{\omega_n}$$, where $$\text{eps}_n$$ is gaussian white noise of variance $$\sigma^2$$ and $$\omega_n$$ is renormalized gaussian magnitude, which is given by a stationary gaussian simulation associated to a certain autocovariance sequence. See Bacry, Kozhemyak, Muzy, 2006, Continuous cascade models for asset returns for details.

sage.finance.fractal.stationary_gaussian_simulation(s, N, n=1)

Implementation of the Davies-Harte algorithm which given an autocovariance sequence (ACVS) s and an integer N, simulates N steps of the corresponding stationary Gaussian process with mean 0. We assume that a certain Fourier transform associated to s is nonnegative; if it isn’t, this algorithm fails with a NotImplementedError.

INPUT:

• s – a list of real numbers that defines the ACVS. Optimally s should have length N+1; if not we pad it with extra 0’s until it has length N+1.
• N – a positive integer.

OUTPUT:

A list of n time series.

EXAMPLES:

We define an autocovariance sequence:

sage: N = 2^15
sage: s = [1/math.sqrt(k+1) for k in [0..N]]
sage: s[:5]
[1.0, 0.7071067811865475, 0.5773502691896258, 0.5, 0.4472135954999579]


We run the simulation:

sage: set_random_seed(0)
sage: sim = finance.stationary_gaussian_simulation(s, N)[0]


Note that indeed the autocovariance sequence approximates s well:

sage: [sim.autocovariance(i) for i in [0..4]]
[0.98665816086255..., 0.69201577095377..., 0.56234006792017..., 0.48647965409871..., 0.43667043322102...]


Warning

If you were to do the above computation with a small value of N, then the autocovariance sequence would not approximate s very well.

REFERENCES:

This is a standard algorithm that is described in several papers. It is summarized nicely with many applications at the beginning of Simulating a Class of Stationary Gaussian Processes Using the Davies-Harte Algorithm, with Application to Long Memory Processes, 2000, Peter F. Craigmile, which is easily found as a free PDF via a Google search. This paper also generalizes the algorithm to the case when all elements of s are nonpositive.

The book Wavelet Methods for Time Series Analysis by Percival and Walden also describes this algorithm, but has a typo in that they put a $$2\pi$$ instead of $$\pi$$ a certain sum. That book describes exactly how to use Fourier transform. The description is in Section 7.8. Note that these pages are missing from the Google Books version of the book, but are in the Amazon.com preview of the book.

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