I am looking at the difference if you calibrated the heston from market data using objective function minimisation.
In scenario 1, I calibrate all the parameters from market data In scenario 2, I calculate the implied volatility using market data for strikes close to ATM and 1-2 weeks expiry, then I use that as my parameter for $v_0$ in the Heston and calibrate the rest of the parameters.
In other words, scenario 2 is calibrating with 1 less unknown parameter.
For scenario 2, when calculating the implied vol's, if I use options that expire in 1-2 weeks with strikes 1 above and below of $S_0$ (So if $S_0=4002.15$, then use strikes $K=4000$ and $K=4005$, calculate those implied vols to get an estimation of implied vol for $S_0 = 4002.15$).
Would I then only need to linearly interpolate with degree=1 because the vol surface is quite flat in that area? Or would I need to use options further away from ATM to get a better fit for the implied vol at $S_0$ (Since implied vol is curved in reality and not flat).
(I'm not calculating implied vol of ITM options, I'm just using put options with put-call parity to get implied vols for $K<S_0$)