Exchange rates and correlation matrices

https://blogperso.univ-rennes1.fr/arthur.charpentier/public/perso2/.euro_dollar_yen_s.jpg I wanted to upload here a small problem I started to work on…. unfortunatley, I could not find (yet) a proper answer. Any comments and suggestions are welcomed. The problem is simple: consider 3 random variables that can be interpreted as exchange rates. Thus X is the exchange rate USD versus EUR, Y is the exchange rate EUR versus YEN, while Z is the exchange rate USD versus YEN. We then have the following constraint XY=Z. Given that constraint, what are the implied constraints on the correlation matrix of random vector (X,Y,Z) ?

  • Bounds on correlation

There are classically standard constraints on correlation. For instance, from Cauchy-Schwarz’s inequality

https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-01.png

but those bounds are not necessarily sharp, given distributions of https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-02.png and https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-03.png. From Fréchet-Hoeffding bounds, we know that

https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-04.png

where https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-05.png. Further, correlation matrices are necessarily positive-semidefinite matrices.
To go further to find possible bounds, recall that

https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-06.png

and therefore

https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-07.png

i.e.

https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-08.png

Similarity, https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-09.png will be a function of https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-10.png.
Thus, given marginal distributions of https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-02.png and https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-03.png, covariances (and also correlations https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-11.png and https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-12.png) are functions of L'image “https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-13.png” ne peut être affichée car elle contient des erreurs.,https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-14.png and https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-15.png.
Somehow, we have obtained 2 degrees of freedom: if there are no additional constraints on those two coefficients, there should be no other constraints than having positive-semidefinite matrices.
I guess that https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-14.png and https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-15.png should not be connected. Those quantities are related to the co-skewness* and they should be different when dependence is asymmetric.
Given a 3 parameter copula for pair https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-16.png, respectively describing global dependence (that can be related to L'image “https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-13.png” ne peut être affichée car elle contient des erreurs. and tail asymmetry (that should be related either to https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-14.png and https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-15.pngdepending on the corner) it should be possible to derive any kind of triplets.

  • Running simulations

I have tried to generate a lot of samples. The idea is to draw pairs from mixtures of Marshall-Olkins’ copula (wich are asymmetric), where parameters where generated randomly. I draw 250 values according to that copula. Then, the third variable is obtained taking the product of normalized pairs (or Studentized). And then calculate the associated correlation matrix.

  1.  draw https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-17.png from https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-18.png (but actually any positive distribution should work) and https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-19.png
  2. do 250 times the following (index https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-20.png)
  • draw https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-21.png with a Bernoulli https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-22.png distribution
  • draw&nbs
    p;https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-23.png from copula https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-24.png if https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-25.png and https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-27.png if https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-26.png
  • set https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-28.png and https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-29.png, and https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-30.png
  1. estimate the correlation of sample https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-31.png

In dimension 3, I have the following graph,

In order to get a better understanding, I look at slices, for instance when https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-32.png. On the graph on the right, to dark area are values that can not be obtained since correlation have to be positive-semidefinite. Dots are points that have been obtained in a particular scenario. I have also included the convex hull if the points obtained with all the scenarios (here 50,000 scenarios), since I assume that the set of positive correlation is necessarily convex. Graphs below have been obtained when https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-33.png and https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-34.png

So far I have not been able to obtain all possible correlations, I guess…  If anyone has suggestions…
* Recall that the co-skewness matrix, in dimension 2, for a pair https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-35.png is simply

https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-36.png

where https://perso.univ-rennes1.fr/arthur.charpentier/latex/change-corr-37.png.



Cite this blog post
Arthur Charpentier (2010, March 18). Exchange rates and correlation matrices. Freakonometrics. Retrieved February 25, 2024, from https://doi.org/10.58079/oudh

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