Sampled increments from two or more correlated Brownian motions (BM)#
Popular algorithm for generating a matrix of increments from a multidimensional Brownian motion (BM) with a given vector of means and a Variance-Covariance matrix.
Problem#
Offten when using multifactor models, the model requires correlated sources of noise. A popular choice is to use a multidimensional Brownian motion.
Solution#
The proposed algorithm uses two propoerties of BM:
Increments of a BM are normaly distributed.
assuming n independent BM’s whose increments are generated from a standard normal distribution (denoted N(0,1)), a derived proces Y = μ + L*z has its increments distributed as N(μ, E) where μ is the vector of means and L is the square root of the Variance-Covariance matrix (denoted E in the code).
Inputs#
Vecor of means for each BM
mu.Variance-Covariance matrix whose diagonal elements describe the volatility of each BM and the off-diagonal elements describe the covariance
E.Number of samples needed
sampleSize.
Output#
Matrix of samples where each column represents a BM and each row a new increment.
Getting started#
The user is interested in generating samples from 2 Brownian motions with a correlation of 0.8. Additionaly, the first BM has a mean of 1 and a variance of 1.5. The second BM has a mean of 0 and a variance of 2. The user is interested in 100 samples.
>>> import numpy as np
>>> from CorBM import CorBrownian
>>> mu = [1,0]
>>> VarCovar = np.matrix('1.5, 0.8; 0.8, 2')
>>> sampleSize = 480
>>> out = CorBrownian(mu, VarCovar, sampleSize)
>>> print(out)
[[ 3.01432184 0.21246299]
[ 0.98350335 2.68478661]
[ 1.42922771 -0.9489711 ]
...
[ 1.74584652 0.79949231]
[ 2.21807447 -0.14098937]
[ 2.81308771 2.65884627]]