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:

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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 :)


1 Answer 1


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.

In your case,

$$ d\begin{pmatrix}X\\Y\\Z\end{pmatrix}=\ldots + Ldz $$ 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

  • $\begingroup$ Thanks, I'll look into it :) $\endgroup$ Commented Dec 3, 2020 at 15:15

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