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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:

  1. When we move an element in the correlation matrix, how do we ensure the matrix is positive definite?
  2. We just ignore $\rho_{ii}$, right?
  3. What if the correlation between different assets is 1? How do we adjust it?
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  • $\begingroup$ To add to the answer below: correlation = 1 (or -1) is a tricky thing, and something you may want to avoid. In fact if two assets are perfectly correlated, conditions need to be imposed on their drifts to ensure consistency. The question whether the matrix is still positive definite after shocks, i.e. how to maintain positive definiteness, is a good one. Honestly I do not know the answer to that, perhaps someone else does? $\endgroup$ – ilovevolatility Jun 16 at 12:08
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  1. 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

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

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

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