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.
Often when using multifactor models, the model requires correlated sources of noise. A popular choice is to use a multidimensional Brownian motion.
The proposed algorithm uses two properties of BM:
- Increments of a BM are normally distributed.
- Assuming n independent BM's whose increments are generated from a standard normal distribution (denoted N(0,1)), a derived process. 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).
- Vector 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.
- Matrix of samples where each column represents a BM and each row a new increment.
The user is interested in generating samples from 2 Brownian motions with a correlation of 0.8. Additionally, 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 480 samples.
import numpy as np
from CorBM import *
mu = [1,0]
VarCovar = np.matrix('1.5, 0.8; 0.8, 2')
sampleSize = 480
out = CorBrownian(mu, VarCovar, sampleSize)
print(out)
>> [out] = [[ 3.01432184 0.21246299]
[ 0.98350335 2.68478661]
[ 1.42922771 -0.9489711 ]
[-0.49778143 -3.0404678 ]
[ 0.29396451 0.64626096]
[ 0.80811248 -0.4542567 ]
[ 0.29441548 0.41880222]
[ 2.11883666 1.0262339 ]
[ 3.17777954 -1.94769166]
[ 0.02836758 -0.89723843]
[ 1.41159275 -0.91685134]
[ 0.80002199 0.94750505]
[ 2.10351657 -0.97580137]
[-0.67431779 -2.1930151 ]
[-0.10431932 -0.00665984]
[ 1.18779711 0.41885266]
[ 1.54634453 -3.74537725]
[ 4.10357654 3.50137711]
[ 1.27666983 -0.202701 ]
[ 1.45607394 -1.29394992]
[ 4.42056013 1.17314064]
[ 0.17981926 0.02718553]
[ 2.08776471 -1.2151297 ]
[ 0.00975734 -1.87426259]
[ 0.44370952 -0.84889546]
[-0.91702991 -0.64974021]
[ 2.41707492 -0.48260669]
[ 0.26420335 -1.08896695]
[ 2.62254181 -0.06424235]
[ 2.17341372 1.99310141]
[ 2.71013259 -1.83770762]
[-0.48606994 -0.92329548]
[ 0.62851221 -2.64537566]
[ 1.78415689 2.56601775]
[ 0.85502579 1.50565467]
[ 1.16226566 -0.49576818]
[ 0.97005964 1.3292272 ]
[ 0.74111532 -2.0000361 ]
[ 1.52058537 0.32836389]
[ 2.45704707 1.73679504]
[ 1.46771852 1.08691729]
[-1.23507574 -1.16338738]
[ 0.16330948 -1.72058513]
[-0.11886678 -0.71182892]
[ 1.64520848 1.89947365]
[-0.29259006 -0.13394478]
[-0.32839732 -0.83890525]
[-1.00189062 -0.14614664]
[ 1.37039228 0.16268565]
[ 3.35019224 -1.41428558]
[ 2.35659306 -0.65411604]
[ 1.04461038 1.3945269 ]
[ 0.46508655 0.93780721]
[-0.28913945 -0.60518967]
[ 1.80189922 0.35159355]
[ 0.57657657 -1.39084704]
[ 1.27479344 0.27996933]
[-0.30639903 2.54723502]
[ 2.5373733 1.87532831]
[-1.14445785 -2.47072282]
[-0.59016974 0.66626821]
[ 0.7555812 -1.30411159]
[-1.08346996 2.02117262]
[-0.41431095 1.37450613]
[-1.06265565 -1.18989157]
[ 1.80578244 1.79412479]
[ 4.19777057 0.99893666]
[ 0.50213584 -0.77556348]
[ 1.9186039 1.09613311]
[ 1.6930982 2.03285367]
[-0.27571345 -0.98032212]
[ 2.81264489 -1.780791 ]
[ 0.06394456 -1.71073406]
[ 1.73889537 1.51100972]
[ 0.39641242 -0.351381 ]
[ 2.99119662 -0.23606054]
[ 2.93104271 1.63527194]
[-0.53147698 -0.67081085]
[ 1.6547926 0.16459858]
[ 0.43974066 -0.0947692 ]
[ 1.74082625 1.67718711]
[ 0.99803465 -1.11834038]
[ 0.20050859 0.25441171]
[ 1.04611722 0.92303653]
[ 0.77831377 0.25247936]
[ 0.15764237 -1.45322145]
[ 2.32716896 1.50761654]
[ 0.46371323 -0.89645604]
[ 2.08381869 2.13579417]
[ 1.56593025 2.4389585 ]
[-0.81187929 0.60117895]
[ 0.32764279 -0.01306386]
[ 1.41249816 0.24986421]
[ 2.06642759 1.30855174]
[-0.19649758 -0.63859554]
[ 1.19242652 0.89506971]
[ 0.35556785 -3.65657223]
[ 1.74584652 0.79949231]
[ 2.21807447 -0.14098937]
[ 2.81308771 2.65884627]]