I take implied volatility as the positive floating point number which lets the BS formula match an observed option price (assuming we have some useful interest rate, some underlying, etc).
How useful is that information really? I know all about the mathematical side of things but through the recent events with GameStop and Wirecard, I kept thinking about how useful volatility is here at all if you wanted to deviate from Black-Scholes. Is there in practice a concept of 'model-dependent' implied volatility? Let's assume I model stock-borrowing fees, that turns the linear BS PDE into a non-linear PDE which you can handle but it's fundamentally a different equation (that I would solve numerically then). Given some interest rate and some borrowing rate I now can't use the implied volatility anymore from BS and have to imply it from my other model.
How do you combine these issues? I mean with interest-rate risk of some sort I might also back out volatility smiles but they are difficult to compare with 'standard' smiles. For stock borrowing fees, you have up to three interest rates you have to consider and while you could for example come up with a "interest rate smile" (... shiver.) such that the BS impl. vola smile together with the IR smile fits for example the prices that consider a model with borrowing rates, then you have to ask yourself what's the point. Do you really fudge-factor everything just to keep on using BS? That sounds terrifying. Going back to the replicating portfolio aspect, if you borrow with one rate and lend with the other and have to pay stock lending fees which count towards the IR income, you can't just assume that there is some hypothetical interest rate that you can use for all three. You totally can and it will match prices but what the heck.
How is that handled in practice? I mean stock borrowing fees are a good example for something not being a problem at all until it is. I also don't like these arguments in the manner of "yeah that's just traded then by ball-feeling, it's useless anyways (but the same person will tell me a "precise" DV01 estimate down to the 7th digit on the phone which is reeeally important for his swap trade to get his cashflow right down to the digit because he is a "precise person and likes to have order in his book" hurr durr)" or "implied parameters normally mean nothing and have no connection to the real world (in that case, as a numerical analyst, please use high-order polynomials to just interpolate everything, inferring parameters here is linear so much easier and by Weierstrass you can even interpolate option prices! Great.)"