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I have a question regarding the relationship between correlations and probabilities. Assuming you have calculated the correlation coefficient for two products, say Stock A & Stock B, can you then convert the coefficient into a probability of some sort? If so, how is this done?Thanks.

If you have calculated the correlation coefficient, then presumably you have calculated the volatilities for the two stocks. Therefore, using the bivariate normal distribution you can work out the probability that stock A is below a certain value and stock B is below a certain value. (The bivariate normal distribution requires the distributions of the two stocks, along with the correlation).

There is also probability associated with the confidence intervals for the estimation of the correlation coefficient , but I don't know if that probability distribution can be accounted for in the bivariate distribution.eg., I have 2 random variables X1 and Y1 , the correlation coefficient estimate is 0.5 and sigma = 0.2I have other 2 random variables X2 and Y2 , the correlation coefficient of 0.5 and sigma = 0.4Should the resulting value of the bivariate cumulative normal distribution function , be different for the 2 sets of variables ?Regards,asd

if your linear correlation is 0.8, then, you have 80% chances that, if stock X increases, then Y increases also. You don't know the magnitude, but you know with 80% chances, the direction of the movement.

that isn't the case - if the correlation is 0, you certainly couldn't say that the probability of share B falling is 0%.

in that case you are right, if the correl is 0, you cannot say anything. However, my example is correct. (Of course, under the assumption that the linear correlation describes world od dependencies, i.e. Gaussian framework)

The probability of one stock rising, given the other stock rises, = ND2(0,0,.8)/N(0)=0.398/.5 is not quite equal to the correlation. The closer you get to a correlation of one, the closer they get.(note ND2 is the cumulative bivariate normal dist)

If the correlation is between binary events, you can convert it directly to a probability. For example, suppose bond A has probability p_A of defaulting and bond B has probability p_B. If the correlation coefficient is r then the probability of both bonds defaulting is:p_A*p_B + r*[p_A*p_B*(1-p_A)*(1-p_B)]^.5If p_A and p_B are small and r is not near or below zero, then this is approximately equal to r*(p_A*p_B)^0.5, so the probability of a double default is approximately the correlation coefficient times the geometric mean of the probabilities of single default.

Assume the world is normal and all we know is correlation…If (X,Y) is N(0,0,sX,sY,r) and N1, N2 are iid N(0,1) then we have the (distribution) equality(X,Y)=( sX*N1 , sY(r*N1 + sqrt(1-r^2)*N2) )If |r|<1, thenP(Y>0 | X>0) = P( sY(r*N1 + sqrt(1-r^2)*N2)>0 | sX*N1>0)= P( r*N1 + sqrt(1-r^2)*N2>0 | N1>0)= P( N2 > -N1* r/sqrt(1-r^2) | N1>0)=f(r)and solving for f(r)=r numerically could give r around 0.8 while the true value probably involves pi. >if the correl is 0, you cannot say anythingWell, if r=0 then P(Y>0 | X>0) = P(Y>0)=1/2SPAAGGs original statement isn’t that wrong. If r is 0.8 then P(Y>0|X>0) is approximately equal to r. Not an answer to QuietStorms question however, who should look at f(r) closer or a generalization if the means are to be non-zero. I don’t have time to look closer at f(r) but I expect that it has a similar curvature as arcsine, shifted up by 1/2. Prove me wrong.p.s. how beautiful life would be if Wilmott.com had Tex…

If (X,Y) is N(0,0,sX,sY,r)P(Y>0 | X>0)=1/2 + 1/pi * arcsin(r)

Hi Freyzi,r 0.5+ASIN(r)/PI()1 10.5 0.6666666670.8 0.7951672350.7 0.746816689-0.7 0.253183311-0.5 0.333333333-1 0seems a quite simple and usefull formula for calculate conditional propabilities... However did you make some intraday test with using a 5 min, 15 min interval? I reckon r = covar(x,y)/(var(x)*var(y)) but what the ideal size of your time series? 5 days, 20 days, 1 year?