We study the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models . The standard Feynman-Kac theorem cannot be directly applied because the diffusion coefficients may  degenerate on the boundaries of the state space and grow faster than linearly . 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 an extension of the standard Feynman-Kac theorem in the context of stochastic volatility models. Sharp results on the existence and uniqueness of classical solutions to the valuation equation are obtained using a combination of probabilistic and analytical techniques. The role of boundary conditions is also discussed .
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 .
R
. The
where the
A
grow faster than linearly and
D
and grow faster than linearly
R
an extension of the standard Feynman-Kac theorem in the context of stochastic volatility models. Sharp results on the
the
R
are obtained using a combination of probabilistic and analytical techniques. The role of boundary conditions is also discussed
. 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