  We analyze the valuation partial differential equation for European
contingent claims in a general framework of stochastic volatility models where
the diffusion coefficients may grow faster than linearly and degenerate on the
boundaries of the state space. We allow for various types of model behavior:
the volatility process in our model can potentially reach zero and either stay
there or instantaneously reflect, and the asset-price process may be a strict
local martingale. Our main result is a necessary and sufficient condition on
the uniqueness of classical solutions to the valuation equation: the value
function is the unique nonnegative classical solution to the valuation equation
among functions with at most linear growth if and only if the asset-price is a
martingale.
