📚 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 480 samples.

mu = [1; 0];
E = [1.5, 0.8; 0.8, 2];
sampleSize = 480;
CorBrownian(mu, E, sampleSize)