Well, question is in the title. Assume I have two different models (for example a local volatility model and a stochastic volatility model such as a Dupire model and a SABR model for example) and I am looking at European options only. Let's assume that both models have a parameter set in which they produce the exact same volatility smile, e.g. for a give strike, they agree on a European option price. For simplicity, let's really focus on EOs here.
Does that also imply their sensitivities coincide? I know that a LV delta and for example a SABR delta can both differ for an option but I can't really find much information about whether that comes with a different surface (so there is SOME dynamic which would be different) or not.
My thinking so far tells me that no, they must be equal. Assume I take a FD estimator of any greek. As the prices produced from both models coincide, both are also the same and thus in the limit the same quantity.
I'm currently thinking about the greeks of straddles or strangles or other option strategies. For those, I have option prices quoted and I also have different volatilities per component. I would technically just add their BS greeks naively if I wanted to have the "overall delta exposure" of the product but I know that of course skew plays a role.
Am I overthinking this? Would you in practice really go an say "here, these three models fit the market perfectly and all their greeks here are slightly different so pick what's interesting for you" if they were in fact different?