Probability two hedged assets both rise

Question

Two assets have daily returns that are jointly normal, each with mean $0$ , and a correlation of $\rho = -\tfrac12$ (they tend to move opposite, as a hedge would).

On a given day, what is the probability that both assets finish up?

Accepted Answer

Zero-mean returns being positive is the same event as a standard normal being positive, so use the orthant identity for jointly normal pairs: P(	ext{both up}) = \frac14 + \frac{\arcsin ho}{2\pi}. With ho = -	frac12 and \arcsin(-	frac12) = -	frac{\pi}{6}: P = \frac14 + \frac{-\pi/6}{2\pi} = \frac14 - \frac{1}{12} = \frac{3}{12} - \frac{1}{12} = \frac{2}{12} = \frac16. So the two hedged assets finish up together only about 17\% of the time, less than the 25\% you would see if they were independent. That is exactly the point of a negative correlation: when one is up, the other is more of…

Probability two hedged assets both rise

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