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We establish higher-order weighted Sobolev and Holder regularity forsolutions to variational equations defined by the elliptic Heston operator, alinear second-order degenerate-elliptic operator arising in mathematicalfinance. Furthermore, given $C^\infty$-smooth data, we prove$C^\infty$-regularity of solutions up to the portion of the boundary where theoperator is degenerate. In mathematical finance, solutions to obstacle problemsfor the elliptic Heston operator correspond to value functions for perpetualAmerican-style options on the underlying asset.

Degenerate-elliptic operators in mathematical finance and higher-order regularity for solutions to variational equations