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According to the following thread:

How can I calculate the strike price or implied volatility from a given delta?

To back out some strike given some Delta, you simply use realized vol (plus a few other inputs) and within a few steps, you can convert any Delta to a specific strike corresponding to some value of spot @ some point in history. However, the responses seem to assume a flat skew, which is not consistent with what we see IRL.

According to the following thread:

What vol to use when implying strike from delta?

User "Hui" advises that you need the IMPLIED vol corresponding to the strike you are trying to solve for. For me this is more intuitive since BSM values observed on the open market are driven by supply/demand for vols, not statistical estimations of it. However, while BSM option prices IRL are a function implied vols, I'm not sure if the derivation of Delta is one that uses implied vol or statvol in the procedure.

https://quantpie.co.uk/bsm_formula/bs_delta.php

Can someone clarify? My question is regarding SPX options not FX btw, if that makes a difference.

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    $\begingroup$ Delta uses implied vols. If you want to derive strike from Delta; given a (dense/interpolated) IVOL surface $\sigma(K)$ as a function of strike $K$ (assuming time-to-maturity is fixed), you are left with a univariate root finding exercise: $v: N(d1(K,\sigma(K)|S,r,q,\tau)=k$ $\endgroup$
    – Kermittfrog
    Apr 22, 2022 at 6:37
  • $\begingroup$ Addendum: Note that No-Arb does not dictate strict monotonicity, hence there may exist multiple roots. $\endgroup$
    – Kermittfrog
    Apr 22, 2022 at 10:40

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You should use whatever volatility was used to calculate that delta. However, you probably don't know that since delta is an output, not an input, to option pricing models.

If you are getting Delta from some data source and they also have implied vol, most likely the implied vol was calculated from the market price and used to calculate Delta, and you should use that.

If you don't have implied vol (or can't calculate it because you don't have price, for example), then you have to guess - you could use historical vol as an approximation, or the implied vol of another similar option, but there is no way to definitively find the strike that the delta corresponds to without all of the other inputs of the pricing model, including volatility.

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