I'm currently doing a course in derivatives pricing and I'm having some trouble wrapping my head around the sweet spot where theory meets reality in terms of Risk Neutral Pricing.
I know that the first and second fundamental theorems of asset pricing lays the foundation for the normal risk neutral pricing argument, and that part is fine. However my lecturer said that for example if you have certain types of stochastic volatility, or the asset price follows a jump process you can no longer hedge all the risks from holding the option. But the way I understand it you need to be able to hedge all the risks from the asset price process in order to use the RNP framework.
So my question is this; when building a model for an asset price process (that you ideally want to be as realistic as possible), what specific characteristics (assumptions about sources of risk etc.) need to be in place in order to keep you within the RNP framework. Conversely, at what point does this machinery break down?