# Multiple underlying brownian motions

I'm trying to find a way to price a triple product forward with payoff XYZ at time T using risk-neutral pricing. But I don't really have a math background and I have trouble finding a way to account for correlation with 3 assets.

I know that for 2 assets with SDEs:

dX= a1dt + b1dz1

dY = a2dt + b2dz2

We have:

But how could we translate this expression when we have 3 assets instead of 2?

I looked for it online but examples were always given with 2 assets.

Thanks a lot :)

What you need is the Cholesky decomposition of the covariance matrix.

For a symmetric matrix $$\Sigma$$, the Cholesky matrix $$L$$ has the property

$$\Sigma = LL^T$$ where $$L$$ is a matrix with zeros above the main diagonal.

$$d\begin{pmatrix}X\\Y\\Z\end{pmatrix}=\ldots + Ldz$$ where $$L$$ is the lower Cholesky matrix.
You can find a general $$3x3$$ example at Rosetta code and a $$3x3$$ online calculator at Wolfram alpha