Let us say that I want to calibrate for example the Heston model to some observed prices of European call options, and that I will use some different strikes and some different maturities to do the calibration.
From what I have understood, it seems that you have to take a "snapshot" at some time $t$ of the state of the market, so that you have the underlying price $S_t$ and the prices of the different call options $C_{t, T_i, K_j}$, all given at the same time $t$, to be able to run the calibration.
So, practically, how do I get such prices ?
Are the option prices supposed to be indeed collected at the same time, for example from the limit order book ? What if there is a large spread in the limit order book, how should the prices of the options be computed ?
What if I do not have access to limit order book data, but just historical trades, which implies that option prices are collected at different times, hence with different underlying prices and different remaining time to maturity. What is then the recommended approach ?
