The stochastic dominance (SD) approach is applied to the valuation of index options in frictionless markets for a wide class of stochastic volatility (SV) processes. SD allows for the derivation of a unique, exponential option pricing kernel based on the physical underlying return and volatility dynamics. A lower bound and an upper bound on option prices are obtained, for a wide class of stochastic volatility jump (SVJ) processes that feature jumps in addition to diffusion. Using parameter estimates for the physical process from high-profile studies, the bounds are shown to be remarkably tight, especially for the empirically important class of short-term near-the-money options. The bounds are in many cases inconsistent with separate parameter estimates for the risk-neutral process that are extracted from observed option prices: for many option series, the risk-neutral value exceeds the SD upper bound. This inconsistency points at the possibility that the distributional shape of the risk-neutral process is mis-specified or that the parameters are estimated without properly taking the option bid-ask spread into account.