Delta IV to Strike and Price

Article Quant Q&A · Author: Andrej

Summary

The note explains how to price an option when implied volatility is quoted by delta rather than by strike. Given the underlying price, rate, maturity, and a volatility associated with a chosen delta, the method first uses the Black–Scholes delta relationship to infer the corresponding strike. For an unadjusted call delta, applying the inverse normal distribution gives the Black–Scholes d-one value, which can be rearranged to solve for strike. The resulting strike and volatility can then be used in the pricing formula.

This addresses the apparent circularity of needing a delta to select volatility while also needing volatility to determine option characteristics. The answer assumes the simple, unadjusted delta convention and the stated Black–Scholes inputs. Delta has multiple conventions, especially in FX markets, so the rearrangement may need modification for adjusted deltas or other quoting practices. The note gives a method, not empirical evidence or guidance for illiquid-market calibration.

Key ideas

Delta-quoted implied volatility can be mapped to a strike before applying the pricing formula.
For unadjusted call delta, the inverse normal function gives the associated d-one value.
Rearranging the Black–Scholes delta equation yields a strike from the underlying, rate, maturity, volatility, and delta.
The result depends on the delta convention, which can differ across markets.

Tags

Options Derivatives pricing Volatility

Full text

# Black Sholes option pricing with all but Delta


# Black Sholes option pricing with all but Delta












I'm trying to setup a little option pricing model in excel. I have all the information for the inputs (interest rate, IVs for different deltas, time to expiry, strike price, underlying price) but what I do not have is the actual delta for which I should be drawing the IVs from. To clarify, I have a list of IVs for each possible delta you want for the underlying security. But it's as if you're in an endless loop because you don't know what IV list to choose since it's linked to a specific delta: which you do not know.

Attaching a Delta formula in case it helps anyone visualize things.

I do not have the actual call prices since I'm dealing with illiquid cases and the end goal is to calculate those prices.

Thanks.

## Answer by Magic is in the chain (score 1, accepted)

https://quant.stackexchange.com/a/47281

If I understand the question correctly, you have the implied vol by delta, and you would like to calculate the price using the Black Scholes formula. And I assume you know the other inputs-e.g., underlying price, interest rate and maturity. Very typical problem in the FX world, so what you can do is first convert delta (using the other inputs and vol) to strike, and then you have vol by strike, which you can then plug into the BS to get the price. The delta comes in different shades, but if it is the simple unadjusted delta, then you can easily isolate strike on one side:

$\Delta=N\left(d_1\right)$

$d_1=N^{-1} \left(\Delta\right)$

Then substitute the expression for $d_1$ and solve for K.

$K=S e^{\left(r +0.5\sigma^2\right)T-\sigma \sqrt{T}N^{-1}\left(\Delta\right)}$

Now you have vol by strike, which you can use with the other inputs to calculate the price.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.

Recovering Strike from Delta-Quoted Implied Volatility for Black–Scholes Pricing

Summary

Key ideas

Tags