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werner06
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Joined: June 8th, 2004, 12:10 pm

Correlation Matrix

June 16th, 2004, 11:30 am

Hi,Assume with have a correlation matrix M (sym, semi-positive..).I would like to find an example of matrix whose nature (sym, semi-positive) changes when we apply a bump on some terms (ie M_ij=M_ij*(1+0.01) and the same for M_ji).If we bump one of the M_ij (and M_ji) term of matrix by 1 percent, is that possible that the new matrix should not be a correlation matrix. Is that is the case, could you give me an example, please?Thanks a lot
 
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SPAAGG
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Joined: March 21st, 2003, 1:31 pm

Correlation Matrix

June 16th, 2004, 11:36 am

Often, you do the contrary. That is, from a matrix which is constructed pair-wized (i.e you compute the correlation for each pairs, and then construct a sym matrix) which is not semi-pos definite, you change a little bit the components in order to obtain a "good" correlation matrix. This is done numerically by changing the eigenvalues. Check the literature in that subject.good luckDavid
 
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Aaron
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Correlation Matrix

June 17th, 2004, 12:29 am

That's not hard. Try:This is positive definite, but if you change either of the 0.9's to 0.909 it will lose that property.For a 3x3 correlation matrix if the correlation of x and y is p1 and the correlation of x and z is p2, the correlation of y and z must be in the interval:The original matrix is consistent with this, but the bumped matrix is not.
 
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werner06
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Correlation Matrix

June 17th, 2004, 9:27 am

Thanks Aaron for the answer. Then I have another questions.Imagine you have a basket options, on three underlying, S_1, S_2 and S_3.Assume we have a 3*3 correlation matrix. If I want to measure the impact of a change of 1% on p_12 (correlation between S_1 and S_2), the logical way to do that is to bump p_12 and p_21 about 1% in M and to recalculate the option price with this new matrix. The fact is that, doing this perturbation on matrix terms, I’m not sure to still have a correlation matrix.What kind of adjustments I must do on the new matrix? What is the process?If you could tell me your meaning about that, it will be helpful for me.Thanks a lotDavid
 
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jimmy
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Correlation Matrix

June 17th, 2004, 1:09 pm

see the RiskMetrics Monitor Q4 1997 (www.riskmetrics.com)there is an article explaining how to stress correlation matrices while still maintaining their "integrity"
 
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Aaron
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Joined: July 23rd, 2001, 3:46 pm

Correlation Matrix

June 18th, 2004, 1:12 am

I do not know Jimmy's article, but it sounds like a good place to look.The natural theoretical answer is to sample from a Wishart distribution. This is the distribution of the covariance matrix if you sample from a multivariate Normal. Not only does it guarantee positive semi definiteness, the stresses will be equally probable. The trouble with stressing individual entries is you can get some very unlikely patterns.A simpler approach, and more robust, would be to bootstrap the data you used to generate the original correlation matrix. If you didn't get it from data, you could instead sample from a multivariate Normal with your original matrix, and compute the sample covariance matrix from the results.
 
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David
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Joined: September 13th, 2001, 4:05 pm

Correlation Matrix

June 18th, 2004, 3:16 pm

QuoteImagine you have a basket options, on three underlying, S_1, S_2 and S_3.It’s usually more convenient to measure correlations of three financial variables via geometric approach. Imagine a triangle in which the angles are labeled as a, b and c. The law of cosines is:c^2 = a^2 + b^2 - 2cos(y)Remember that, if correlations are positive the degrees of angles will be between 0 to 90. If correlations are negetive the degrees will be between 90 to 180. Then if you’re using any bumps at whatever percentage the arcus cosines must equal to 180 degrees. As:arccos(P1) + arccos(P2) + arccos(P3) = n (radian)So we can express one correlation as a function of the other two correlations as: P3 = cos(n – arccos[P1] – arccos[P2]) Hope this may help
 
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SPAAGG
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Correlation Matrix

June 22nd, 2004, 7:48 am

 
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Napo
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Correlation Matrix

September 11th, 2008, 5:33 pm

QuoteOriginally posted by: AaronThat's not hard. Try:For a 3x3 correlation matrix if the correlation of x and y is p1 and the correlation of x and z is p2, the correlation of y and z must be in the interval:What is the general mathematics for this formular? I only know that d/dx arccos(x) = -1/sqrt(1 - x^2)Is this the right direction to search and what would be the relationship for a (n x n) matrix?
 
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amit7ul
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Correlation Matrix

September 12th, 2008, 9:31 am

we assume x, e1, e2 ~ N(0,1) and also assume E(x.e1)=E(x.e2)=0y=x.p1+e1.p1' because correlation(x,y) = p1 and where p1' = sqrt(1-p1^2)z=x.p2+e2.p2' because correlation(x,z) = p2 and where p2' = sqrt(1-p2^2)=> y, z ~ N(0,1) you could easily verify this with simple assumptions aboveNow as correlation between y and z = E(y.z) = E(p1.p2.x^2+e1.e2.p1'.p2'+something+something) = p1.p2 + p1'.p2'.E(e1.e2)+0+0As we know -1<E(e1.e2)<1Aaron's result follows
 
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Napo
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Correlation Matrix

September 12th, 2008, 12:20 pm

I'm sorry, but I don't get it. First, what does your notation mean? x.p1 = x * p1 ?Why do you transform p1' = sqrt(1 - p1^2)? What is the motivation in algebraic or geometric terms?
 
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amit7ul
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Correlation Matrix

September 13th, 2008, 4:56 am

x.p1 = x * p1 and p1' = sqrt(1 - p1^2)both for ease of writing.... no ulterior motive..algebraic, geometric or political
 
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ronm
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Correlation Matrix

September 13th, 2008, 2:09 pm

Quotehttp://www.rebonato.com/CorrelationMatrix.pdf This link is not working. Can you pls provide an updated link?
 
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ACD
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Joined: April 19th, 2004, 8:09 am

Correlation Matrix

September 13th, 2008, 2:22 pm

QuoteOriginally posted by: ronmQuotehttp://www.rebonato.com/CorrelationMatrix.pdf This link is not working. Can you pls provide an updated link?Should be able to get it here:http://www.quarchome.org/correlationmatrix.pdf