Two correlated bets both finishing in the money

Question

Two trading strategies have returns X and Y that are jointly normal, each with mean 0, and correlation ho between them. What is the probability that both strategies finish with a positive return, i.e. P(X 0 	ext{ and } Y 0), as a function of ho?

Accepted Answer

Since means are 0, only the angle between X and Y matters, rescaling each to a standard normal doesn't change the sign pattern. There is a famous closed form (the "orthant probability" for the bivariate normal): P(X 0,\ Y 0) = \frac{1}{4} + \frac{\arcsin\rho}{2\pi}. Where it comes from. Standardize to unit-variance normals. A slick geometric picture: write Y = \rho X + \sqrt{1-\rho^2}\,Z with Z independent of X. The pair (X, Y) is a Gaussian scatter whose contour lines are ellipses tilted according to \rho. The probability of the first quadrant equals the "opening angle" of that quadrant as s…

Two correlated bets both finishing in the money

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