Estimating Correlation Delta with a Valid Correlation Shock

Summary

The document discusses estimating correlation delta, also called cega, for a multi-asset derivative by repricing after a small change in an asset-pair correlation. It identifies practical constraints when perturbing a correlation matrix: the matrix is symmetric, diagonal entries represent each asset’s correlation with itself, and correlations at the boundary of one cannot be shocked upward while remaining valid.

The suggested procedure is to change both symmetric off-diagonal entries together, leave diagonal values unchanged, and use a downward shock when the correlation is already one. The resulting price difference can then be used in a finite-difference estimate of sensitivity. The response gives a specific downward perturbation from one to 0.99, but it does not discuss how to ensure the shocked matrix remains positive definite in all cases, nor does it specify a general shock size or finite-difference scheme. Those details may matter for numerical implementation.

Key ideas

• Cega measures the derivative of derivative value with respect to an asset-pair correlation.
• A correlation matrix is symmetric, so perturbations must update both corresponding off-diagonal entries.
• Diagonal correlations remain equal to one and should not be changed.
• At a correlation of one, use a downward shock and estimate sensitivity with a finite difference.
• The response does not give a general guarantee that the shocked matrix remains positive definite.

Tags

# Cega - Correlation Delta from multi-asset derivative


# Cega - Correlation Delta from multi-asset derivative












I want to calculate the Cega, i.e. correlation delta, for a multi-asset derivative numerically (the difference of the price from a tiny move in correlation). However, I found it is difficult to follow the definition of Cega from wikipedia. As my understanding, Cega is defined there as the 1st derivative of the price with respect to the correlation, i.e. $\frac{\partial C}{\partial \rho_{ij}}$.

My questions are:

- When we move an element in the correlation matrix, how do we ensure the matrix is positive definite?

- We just ignore $\rho_{ii}$, right?

- What if the correlation between different assets is 1? How do we adjust it?

## Answer by Appel Moi Dougy (score 1)

https://quant.stackexchange.com/a/54953

- Since the correlation matrix is symetric, if you move the term (i,j), you have to do it for the term (j,i) as well

- Of course -> the correlation of an asset with itself is equal to 1... so it should not change

- You apply a downward shock (1 to 0.99) and you use the formula of finite differences

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