# Calibration of Heston using implied vol as $v_0$

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)

Using the ATM implied vol of short term options is indeed a common practice for $$v_0$$ as in your Scenario 2. Linear interpolation should be enough, given that 1 week is somewhat arbitrary anyway. In Scenario 1, you will want to use this for the initial guess of the minimization.
$$v_0$$ (and $$\theta$$) may also be implied from the curve of variance swap prices. See F. Guillaume, W. Schoutens (2010) "Use a reduced Heston or reduce the use of Heston?" This is also more in line with Bergomi variance curve approach.