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I have been doing reading and supposedly implied volatility of ATM options with 1-2 week expiries are reasonable vols to use as your $V_0$ when calibrating a Heston model.

Firstly, why would it be unreasonable to take implied vol of a 1 year contract, and use that as your long-run vol, $\theta$ of the Heston model?

Secondly, as a contract becomes more OTM, its implied vol is no longer useful for Heston calibration. So in general, when people discuss ATM options, what's the range around the spot vs the strike? 1%, 5%?

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A1: Volatility implied by what model? :)

A2: Check what the volatility smile is and how it affects your model.

UPD: As @frido insisted, I would add more details.

"Why not 1 year IV"

Because processes are different. CIR distribution (Heston) is different from brownian motion with constant IV (Black-Scholes). But you can try 1 year IV as an initial value for callibration.

"How far the OTM?"

From a first glance, I see 2 main reasons not to use far OTM options for calibration.

First of all, it is a matter of quality of your option board. Far OTM options (> 3*sigma) tend to be "rounded" or inadequate. As a result, you would calibrate on option prices affected by technical details.

Second, Heston model has no idea about "fat tails".

What is the minimum? Heston model has 5 parameters (kappa, theta, v0, rho, sigma). So, you need at least 5 points to optimise. :)

I would recommend using 1 or 2 sigma and then go deeper in time.

Links

The way to calibrate Heston model I would recommend: https://www.maths.univ-evry.fr/pages_perso/crepey/Equities/051111_mikh%20heston.pdf

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    $\begingroup$ You might want to add more colour to your answer. And I think it's pretty clear the OP means IVs of vanillas. $\endgroup$
    – Frido
    Aug 28 at 19:32
  • $\begingroup$ @Frido Updated. I hope, it is more clear now. $\endgroup$
    – Alex D
    Aug 29 at 4:36
  • $\begingroup$ Much better answer for the OP. +1. $\endgroup$
    – Frido
    Aug 29 at 6:28
  • $\begingroup$ @Frido thanks mate. Just for clarification, when you mean try to use options that are < 1 sigma OTM. What is that standard deviation supposed to be of? $\endgroup$
    – THAT'S MY QUANT MY QUANTITATIV
    Aug 29 at 12:44
  • $\begingroup$ @THAT'SMYQUANTMYQUANTITATIV Yes. $\endgroup$
    – Alex D
    Aug 29 at 13:37

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