Building three correlated normals

Idea. This is the Cholesky factorisation of the correlation matrix, built one row at a time. Each new variable takes what it must share with the earlier ones and adds fresh independent noise for the rest.

Step 1: $x_2$. $\mathrm{corr}(x_1,x_2)=a=\frac{1}{2}$, and $a^2+b^2=1$ gives $b=\frac{\sqrt{3}}{2}$.

Step 2: $x_3$.

• $\mathrm{corr}(x_1,x_3)=c=\frac{1}{2}$.
• $\mathrm{corr}(x_2,x_3)=ac+bd=\frac{1}{4}+\frac{\sqrt{3}}{2}d=\frac{1}{2}$, so $d=\frac{1}{2\sqrt{3}}$.

Step 3: the rest. $$f^2=1-c^2-d^2=1-\frac{1}{4}-\frac{1}{12}=\frac{2}{3},$$ so $f=\sqrt{2/3}\approx 0.816$.

A shortcut. $f^2$ is the variance of $x_3$ left unexplained by $x_1$ and $x_2$, which equals the ratio of determinants of the $3\times 3$ and the top-left $2\times 2$ correlation matrices: $\frac{1/2}{3/4}=\frac{2}{3}$.

Use. This is how correlated returns are simulated in Monte Carlo pricing: multiply a vector of independent normals by the Cholesky factor of the target covariance matrix.