Correlation stress testing Could you suggest good articles / books / online materials on correlation stress testing?
Correlation stress testing: say, we have a positive semidefinite correlation matrix. We might want to explore the impact on the results if some or all elements of the matrix are increased or a decrease by a given value/s. Of course, there are limits to stress testing due to the requirement of the correlation matrix to remain positive semidefinite.
Thank you in advance for your suggestions!
 A: Of course, there are limits to stress testing due to the requirement of the correlation matrix to remain positive semidefinite.
finding these limits is, imho, the critical part of what you are trying to do.
The answer to this problem depends on how many of the correlation coefficient you want to stress test simultaneously.
if you want to stress test only one coeficient at a time (say the entry $a_{ij},i\neq j$ of your correlation matrix), then you can use the Gershgorin theorem to place explicit bounds of the values that $a_{ij}$ can take. Let $a_{ij}^+$ and $a_{ij}^-$ be these bounds. Then you can stress test by computing you measure of risk for a grid of values in $(a_{ij}^-,a_{ij}^+)$
If you want to stress test more than one (say $k>1$) coefficient simultaneously, then there is no closed form solution for the bounds on these correlations coefficients. An exact solution for this problem exists however (i.e. the $k$-dimensional ellipse inside which your $k$ correlation coefficient are allowed to reside)  but this requires a higher level of mathematical sophistication. If this is the case you are interested in, let me know in the comments.
Edit:
Exact solution for $k>1$: this is can be recasted as an SDP problem. 
say $a=(a_1,...,a_k)$ are $k$ correlation coefficients you want to vary and $w=(w_1,...,w_k)$ are strictly positive numbers and $C(a)$ is the $p$ by $p$ correlation matrix with $p\geq k$. Then, 
$\underset{a|w}{\min.}\; a'w$
$s.t. C(a)\in \mathbb{R}_{++}$
where $\mathbb{R}_{++}$ is the set of all SDP symmetric matrices. 
Imposing this constraint requires SDP programing. It can be shown that (via the Shur complement of $C(a)$) this is equivalent to imposing $p$ linear inequality and $p$ quadratic inequality on the values of $a$.
Now, we know the set of all solutions is a $k$ dimensional ellipse. Such ellipse is defined by $k+1$ points on it's boundary. Each solution to this SDP (corresponding to a given vector $w$) will be one point on the boundary of this ellipse.
Finally, each run of the minimization problem has time complexity $O(p^3)$ where $p$ is the number of assets.
there is a nice packages to solve SDP in Matlab. I think there is also one in R, but  last i checked it was not as nice.
A: The answer may depend on why you want to do stress testing but here is one partial idea:
You can do a eigendecompostion of your current matrix, The eigenvectors are aligned with the axis of the ellipsoid corresponding to the correlation matrix and the eigenvalues are a measure of how "long" the ellipsoid is along the corresponding vector. So one way to perform stress testing is to vary the eigenvalues so that the shape of the ellipsoid changes from the original one.
