Modeling Three Correlated Assets with Cholesky Decomposition

Summary

The document addresses how to represent correlated Brownian shocks for a payoff depending on three underlying assets. The answer recommends applying a Cholesky decomposition to the assets’ covariance matrix. For a symmetric covariance matrix, the lower triangular Cholesky factor multiplied by its transpose reconstructs the matrix.

The resulting factor can be used to transform a vector of independent Brownian increments into correlated shocks in a multivariate stochastic differential equation. This generalizes the familiar two-asset correlation setup to three assets and provides a standard way to encode pairwise correlations and volatilities. The exchange gives the matrix relationship and its role in the dynamics, but does not derive the full triple-product forward price or specify drift assumptions, covariance inputs, or payoff conventions; those are still needed for a complete risk-neutral valuation.

Key ideas

• A three-asset model represents correlated Brownian shocks using a covariance matrix.
• Cholesky decomposition factors a symmetric covariance matrix as a lower triangular matrix times its transpose.
• Multiplying the Cholesky factor by independent Brownian increments produces correlated asset shocks.
• The decomposition specifies the correlation structure but does not alone determine the triple-product forward price.

Tags

# Multiple underlying brownian motions


# 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 :)

## Answer by Kermittfrog (score 2)

https://quant.stackexchange.com/a/59723

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

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.