Coverage for local/lib/python2.7/site-packages/sage/finance/fractal.pyx : 86%

r""" 

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) 

""" 

from __future__ import absolute_import 

from sage.rings.all import RDF, CDF, Integer 

from sage.modules.all import vector 

I = CDF.gen() 

from .time_series cimport TimeSeries 

cdef extern from "math.h": 

double exp(double) 

double log(double) 

double pow(double, double) 

double sqrt(double) 

################################################################## 

# Simulation 

################################################################## 

def stationary_gaussian_simulation(s, N, n=1): 

r""" 

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. 

""" 

N = Integer(N) 

if N < 1: 

raise ValueError("N must be positive") 

if not isinstance(s, TimeSeries): 

s = TimeSeries(s) 

# Make sure s has length N+1. 

if len(s) > N + 1: 

s = s[:N+1] 

elif len(s) < N + 1: 

s += TimeSeries(N+1-len(s)) 

# Make symmetrized vector. 

v = s + s[1:-1].reversed() 

# Compute its fast Fourier transform. 

cdef TimeSeries a 

a = v.fft() 

# Take the real entries in the result. 

a = TimeSeries([a[0]]) + a[1:].scale_time(2) 

# Verify the nonnegativity condition. 

if a.min() < 0: 

raise NotImplementedError("Stationary Gaussian simulation only implemented when Fourier transform is nonnegative") 

sims = [] 

cdef Py_ssize_t i, k, iN = N 

cdef TimeSeries y, Z 

cdef double temp, nn = N 

for i from 0 <= i < n: 

Z = TimeSeries(2*iN).randomize('normal') 

y = TimeSeries(2*iN) 

y._values[0] = sqrt(2*nn*a._values[0]) * Z._values[0] 

y._values[2*iN-1] = sqrt(2*nn*a._values[iN]) * Z._values[2*iN-1] 

for k from 1 <= k < iN: 

temp = sqrt(nn*a._values[k]) 

y._values[2*k-1] = temp*Z._values[2*k-1] 

y._values[2*k] = temp*Z._values[2*k] 

y.ifft(overwrite=True) 

sims.append(y[:iN]) 

return sims 

def fractional_gaussian_noise_simulation(double H, double sigma2, N, n=1): 

r""" 

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. 

""" 

if H <= 0 or H >= 1: 

raise ValueError("H must satisfy 0 < H < 1") 

if sigma2 <= 0: 

raise ValueError("sigma2 must be positive") 

N = Integer(N) 

if N < 1: 

raise ValueError("N must be positive") 

cdef TimeSeries s = TimeSeries(N+1) 

s._values[0] = sigma2 

cdef Py_ssize_t k 

cdef double H2 = 2*H 

for k from 1 <= k <= N: 

s._values[k] = sigma2/2 * (pow(k+1,H2) - 2*pow(k,H2) + pow(k-1,H2)) 

return stationary_gaussian_simulation(s, N, n) 

def fractional_brownian_motion_simulation(double H, double 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]] 

""" 

return [a.sums() for a in fractional_gaussian_noise_simulation(H,sigma2,N,n)] 

def multifractal_cascade_random_walk_simulation(double T, 

double lambda2, 

double ell, 

double sigma2, 

N, 

n=1): 

r""" 

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 = finance.multifractal_cascade_random_walk_simulation(3770,0.02,0.01,0.01,10,3) 

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. 

""" 

if ell <= 0: 

raise ValueError("ell must be positive") 

if T <= ell: 

raise ValueError("T must be > ell") 

if lambda2 <= 0: 

raise ValueError("lambda2 must be positive") 

N = Integer(N) 

if N < 1: 

raise ValueError("N must be positive") 

# Compute the mean of the Gaussian stationary process omega. 

# See page 3 of Bacry, Kozhemyak, Muzy, 2008 -- "Log-Normal 

# Continuous Cascades..." 

cdef double mean = -lambda2 * (log(T/ell) + 1) 

# Compute the autocovariance sequence of the Gaussian 

# stationary process omega. [Loc. cit.] We have 

# tau = ell*i in that formula (6). 

cdef TimeSeries s = TimeSeries(N+1) 

s._values[0] = lambda2*(log(T/ell) + 1) 

cdef Py_ssize_t i 

for i from 1 <= i < N+1: 

if ell*i >= T: 

s._values[i] = lambda2 * log(T/(ell*i)) 

else: 

break # all covariance numbers after this point are 0 

# Compute n simulations of omega, but with mean 0. 

omega = stationary_gaussian_simulation(s, N+1, n) 

# Increase each by the given mean. 

omega = [t.add_scalar(mean) for t in omega] 

# For each simulation, create corresponding multifractal 

# random walk, as explained on page 6 of [loc. cit.] 

sims = [] 

cdef Py_ssize_t k 

cdef TimeSeries eps, steps, om 

for om in omega: 

# First compute N Gaussian white noise steps 

eps = TimeSeries(N).randomize('normal', 0, sigma2) 

# Compute the steps of the multifractal random walk. 

steps = TimeSeries(N) 

for k from 0 <= k < N: 

steps._values[k] = eps._values[k] * exp(om._values[k]) 

sims.append(steps.sums()) 

return sims