I was reading the answers to this question: Long Gamma vs Vega , but I still I feel I am missing a bit of context.
Let's say I am long an European call today. From the plots shown in the second answer, it is clear that I am both long gamma and long vega, in a Black-Scholes model. But how do we define vega for a stochastic volatility or LSV model? We would have to fit a volatility surface to observed implied vols corresponding to vanilla prices and in this case we have a vega for every point on the vol surface. Will we still be long all vegas?
Another question that I have is: assuming some SV or LSV dynamics, how does my PnL evolve when I am short or long options (or using option strategies) as a function of future realized vol? Is there a standard reference on this?