We study the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models where the standard Feynman-Kac theorem cannot be directly applied because 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 ; for example, the volatility process in our model can potentially reach zero and either stay there or instantaneously reflect, and asset-price processes may be strict local martingales under a given risk-neutral measure . Our main result is the existence and uniqueness of classical solutions to the valuation equation . It is shown that the value function is the smallest nonnegative classical solution to the valuation equation . Moreover, when the payoff is of strictly sublinear growth, the uniqueness holds among the same class of functions ; when the payoff is of linear growth , the uniqueness holds if and only if the asset-price process is a martingale.
We analyze the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models where the standard Feynman-Kac theorem cannot be directly applied because 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 asset-price process may be 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 the functions with at most linear growth if and only if the asset-price is a martingale.
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