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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