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In this paper we study a general framework of American put option withstochastic volatility whose value function is associated with a 2-dimensionalparabolic variational inequality with degenerate boundaries. We apply PDEmethods to analyze the existences of the strong solution and the properties ofthe 2-dimensional manifold for the free boundary. Thanks to the regularityresult on the solution of the underlying PDE, we can also provide theuniqueness of the solution by the argument of the verification theorem togetherwith the generalized Ito's formula even though the solution may not be secondorder differentiable in the space variable across the free boundary.

American option of stochastic volatility model with negative Fichera function on degenerate boundary