In what scenarios does one have a closed form equation for the Pearson Correlation - especially s.t. changing parameters changes correlation?

Solution 1:

What you're looking for is possibly the concept of copula. Copulas make multivariate modelling very flexible, especially in terms of correlation/dependence. A copula $C$ is defined as a joint distribution of two (or more, but I'll keep it to $d=2$) uniform random variables $U,V \sim \textrm{U}[0,1]$. A famous theorem due to Abe Sklar states that $(1)$ for random variables $X\sim F_X,Y \sim F_Y,(X,Y)\sim F_{X,Y}$ $$F_{X,Y}(x,y)=C(F_X(x),F_Y(y))$$ for some copula $C$ (it is an existence statement). Continuity of the rvs makes the copula unique. $(2)$ given marginals $F_X,F_Y$ and some copula $C(u,v)$, then $C(F_X(x),F_Y(y))$ is a valid joint df with marginals $F_X,F_Y$. This is a very powerful result: you can model marginals and joint separately.

There is a plethora of copulas, in fact you can make up your own, with their own parameters, using any decreasing, continuous, convex $\psi:\mathbb{R}^+\to [0,1],\,\psi(0)=1$ and $\psi(x)\to0$ for $x \to \infty$. This class of functions is called Archimedean copula generators. Some of the copulas are particularly famous because they have closed form solutions for rank correlations (a more sophisticated form of correlation if compared to $\rho$) in terms of the copula parameters (an example is the Gumbel copula). Rank correlations can be computed as integrals involving the copula of $X,Y$, and don't have the attainability restrictions of $\rho$.

A good introductory reference for all this is Alexander J. McNeil, Rudiger Frey and Paul Embrechts (2005) "Quantitative Risk Management: Concepts, Techniques, and Tools".