Practitioners tend to wear Black-Scholes glasses when dealing with European options: to them, quoting a certain option price today $V(S_0;T,K)$ is equivalent to quoting the forward price of the underlying $F(0,T)$ along with a relevant Black-Scholes volatility figure $\sigma(T,K)$(*)
That being said, when you are asked to price a European option on a stock $A$, there is typically 2 possible situations in practice: [A] vs [B].
There is a liquid market of listed vanilla options written on $A$. Basically, this means that a finite set of European options with very specific characteristics (i.e. specific times to maturity $T$ and strike levels $K$) trade on an exchange meaning that their prices are public. In that case, your job is to perform an "arbitrage free inter/extrapolation" of sorts, i.e. you should use the observed prices to infer the price you are looking for under the assumption that there is no free lunch. There are different approaches:
- Choose a particular (jump-)diffusion model and calibrate it so that it reproduces observed option prices as best as possible, then use this model to infer the price of the option you want (or any other option for that matter). As you mention, different models have different performances, both regarding the static fit of the vanilla market (which matters when pricing European options, by definition), but also in terms of forward volatility dynamics (which matters when pricing exotics).
- Wear Black-Scholes glasses and directly interpolate the market implied forward curve and volatility surface. Different interpolation/extrapolation assumptions can be used, see for instance the SVI paradigm when it comes to fitting the implied volatility surface. The complexity here consists in finding interpolation/extrapolation methods that preclude arbitrage opportunities with yet sufficient degrees of freedom.
Typically, the fit of the vanilla market using method (2) will be almost perfect compared to (1) (actually the fit using Local Volatility models à la Dupire with method (1) should be perfect as well, but the same fit can be relatively out for poorly designed Stochastic Volatility models). Still, method (2) is a purely descriptive one: it is not a "model" per se and it will not help you to price options that are not European.
No similar options trade on stock $A$. Various possibilities here also, but much more "freestyle" because you are "in the dark". Some ideas (usually these are proprietary methods that few are willing to discuss in details, which I won't do here either),
- Use a proxy underlying which behaves similarly to $A$ (here the complexity is transposed to identifying what a good proxy is)
- Use historical time series (e.g. calibrate an econometric model) along with a stochastic discount factor (e.g. model the market risk premium)
- Use the break-even volatility (i.e. the volatility which, historically and on average, makes the P&L of your delta-hedged option position zero over a range of different estimation windows).
(*) This assumes that the discount curve is fixed using e.g. EONIA for derivatives delivering a payoff in EUR.