We know of plenty ways to extract risk neutral distirbutions from option prices (for example Breeden Litzberger) but there is no real analysis on how to interpret negative state prices (Haug 2007 for example). State prices are Arrow Debreu securities, so it is the price an agent is willing to pay to get $1\$$ in a particular state and $0$ else.
Doing equilibirum and utility maximization we know that the agents smoothen their consumption by marginal rate of substitution which describes their risk-aversion.
Coming back to the negative probabilties: Could it not be possible there exist actual economic states where the agent is receiving money for the contract?