Why do we fit volatility surfaces implied from a option pricing model to the empirical data?

Question

As far as I understand volatility surface. It is made of 2 components, the volatility skew/smile and the volatility term structure. Together they form something like

Implied volatility is essentially an output from Black-Scholes model when it is matched with the existing option price. Implied volatility also implies that the differences of distribution used in Black-Scholes model (but that's not important in this question).

Here comes the questions:

1. Since the model is relying on the same underlying asset, thus it must have the same underlying volatility. So, IF the model is right, i.e. all assumptions of the model are correct, it will return a flat volatility surface. No smile, no skew, no curvature implied by the volatility term structure? The Figure 1 will be plain flat like a flat paper.
2. After going through some texts related to volatility surfaces and some article on advancement in option pricing, I have noticed that the authors are comparing the model's implied volatility surface to the implied volatility in the first figure in this thread, i.e. comparing the if the model fits the empirical implied volatility surface. My question is this: shouldn't the authors attempt to obtain a flat volatility surface such as describe in the first question, since it'll indicate that the model is consistent with the market prices? I am definitely missing something here :(
3. Another confusion that might be cleared if I understand the first 2 questions (idk?) is the way of fitting an option pricing model. For example, to fit the Heston model, we normally extract the empirical prices from the market itself and use some algorithms to minimize the errors from the model to obtain the required parameters. From here we can compare the models by comparing the volatility surfaces that it had generated from the Heston model vs the empirical volatility surfaces generated from Black Scholes. Since Heston model is a stochastic volatility model, by definition of it, shouldn't the volatility can't be attained from its model. A similar question is from Confusion with volatility smiles implied by different models . I don't understand one of the answer that says:

In the context of option pricing, "implied volatility" always refers to the equivalent diffusion coefficient in the geometric Brownian motion (GBM) dynamics that is necessary to match an observed European plain vanilla price for a given strike and maturity.

You choose some pricing model such as the ones mentioned in your question. You fix the model's parameters. Under the given model dynamics and parameters you value European plain vanilla option prices for all strikes of the maturity of interest. Finally, you then treat these model prices as the observed inputs to your implied volatility computation. I.e. you compute the equivalent GBM volatility such that the previously computed model prices are matched.

Here's a page from "The volatility surface, A Practitioner's Guide by Jim Gatheral" to illustrate my 2nd question

Can someone please clear my confusions on these several questions. I will be much appreciated in any effort or attempt to clear my confusion on this topic :(

Answer 1

1. Yes, that's what we wish to see from the correctly-specified model.

Now, let me try to answer your 2nd and 3rd questions together as they are based on the same confusion. There are two different concepts: model-implied volatility and model-implied BSIV (Black-Scholes Implied Volatility). I think you are confused because of mixing them up.

So yes, people attempt to obtain a flat volatility surface of the model-implied volatilities, but not of the model-implied BSIV. In fact, the BSIV surface is based on market observables, thus it is kind of empirically fixed at a given time-point.

Regarding the fitting. In order to obtain model parameters, you basically want to minimize the distance between the observed option prices and model-implied option prices, that is, option prices generated by your model for a given set of parameters. However, some option prices are low and some are large for the same underlying stock (for example, OTM options typically worth less than their ITM counterparts). Therefore, when minimizing the distance between observed and model-implied option prices you will put more weights on more expensive options, which bias your parameter estimates. So, what people do is that they minimize the distance between the BSIVs of the observed and model-implied option prices. That is fitting a BSIV surface (but not a model-implied volatility surface) generated from the model to the one we "observe" in the market. Why? Because BSIV is a monotonic transformation of option prices in money terms, so it is equivalent to minimize the distance between regular option prices. Furthermore, BSIV transformation is a way of standardizing option prices in the sense that you don't have such a considerable difference between model-implied BSIV relative to differences in option prices.

Coming back to your 2nd point. For the same reason, people compare the model-implied BSIV surface (but not the model-implied volatility surface) to the market-observed BSIV as it is akin to comparing how well a model fits option prices, but expressed in BSIV terms rather than in money-terms. This is especially convenient when you want, for example, to compare a fit of option pricing model to options traded in different currencies as expressed in BSIV terms they would be currency-free.

Hope this helps