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
but those bounds are not necessarily sharp, given distributions of 
and 
. From Fréchet-Hoeffding bounds, we know that
where 
. Further, correlation matrices are necessarily positive-semidefinite matrices.
To go further to find possible bounds, recall that
and therefore
i.e.
Similarity, 
will be a function of 
.
Thus, given marginal distributions of and , covariances (and also correlations 
and 
) are functions of 
,
and 
.
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 and 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 
, respectively describing global dependence (that can be related to and tail asymmetry (that should be related either to and depending 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.
- draw
from 
(but actually any positive distribution should work) and 
- do 250 times the following (index
)
- draw
with a Bernoulli 
distribution - draw&nbs
p;
from copula 
if 
and 
if 
- set
and 
, and 
- estimate the correlation of sample
In dimension 3, I have the following graph,
In order to get a better understanding, I look at slices, for instance when 
. 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 
and 
 |
 |
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 
is simply
where 
.