Finding an Option Strike from Delta Requires a Volatility Assumption
Summary
This discussion explains why delta alone cannot uniquely identify an option strike. Delta is calculated from an option pricing model and depends on inputs such as spot, maturity, rates, option type, and volatility. To invert a reported delta for strike, the calculation needs the same volatility convention that was used to produce that delta. When a data source supplies both delta and implied volatility, that implied volatility is the most relevant starting point because it is generally tied to the market option price.
If implied volatility or the option price is unavailable, historical volatility or the implied volatility of a comparable option can serve as an estimate, but the resulting strike is conditional on that assumption. The question also notes that a flat volatility assumption may not fit observed skew. The answer does not give a numerical inversion procedure, and it cautions that the missing model inputs prevent a definitive strike from being recovered from delta alone.
Key ideas
- Delta is an output of an option pricing model, not a standalone strike specification.
- Recovering strike from delta requires the volatility used to calculate that delta.
- Market implied volatility is typically appropriate when it is available alongside the reported delta.
- Historical or comparable-option volatility can be used as an estimate when implied volatility is missing.
- A volatility skew means the assumed volatility may depend on the strike being solved for.
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Full text
# Deriving strike from Delta # Deriving strike from Delta 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. ## Answer by D Stanley (score 1) https://quant.stackexchange.com/a/70613 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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