Converting Black Forward Delta to an Option Strike

Summary

The document explains how to convert an option delta quoted under a Black model into the corresponding strike, given the forward price, volatility, and time to expiry. It gives separate expressions for calls and puts, using the inverse standard normal distribution to recover the model quantity needed to solve for strike. This connects a volatility smile expressed by delta to one expressed by strike or moneyness.

It also points to a numerical root-finding approach using a financial toolbox delta function, with the underlying price as an initial strike guess. The formulas and procedure assume the stated Black framework and its inputs; the note does not discuss alternative delta conventions, volatility quoting conventions, or practical edge cases. Users should ensure the delta definition and model assumptions match those used to quote the smile before applying the conversion.

Key ideas

• Under the Black framework, call delta is expressed through the standard normal cumulative distribution evaluated at d1.
• A call’s strike can be recovered from its forward delta, forward price, volatility, and time to expiry.
• The put conversion uses a different inverse-normal input from the call conversion.
• Numerical root finding can solve for a strike by matching a model delta to the target delta.
• The conversion depends on consistent model and delta conventions.

Tags

# From Delta to moneyness or strike


# From Delta to moneyness or strike












If I have volatility smile quoted with respect to the delta of an option on the forward, how can I convert this delta into the moneyness or strike of the option?

Is there any bult-in function of Matlab financial toolbox?

## Answer by NSZ (score 2)

https://quant.stackexchange.com/a/33591

The call delta in a Black framework is: $$\Delta = N(d_1)$$ with $d_1=\frac{\ln(F_t(T)/K)+(T-t)\frac{\sigma^2}{2}}{\sigma\sqrt{T-t}}$.

Then the strike of the option is: $$K=F_t(T) e^{-(N^{-1}(\Delta)+1/2) \sigma \sqrt{T-t}}$$

The same thing is done if the option is a put and we obtain:

$$K=F_t(T) e^{-(N^{-1}(\Delta+1)+1/2) \sigma \sqrt{T-t}}$$

In matlab it can be solved by doing:

```
fzero(@(Strike) blsdelta(Price,Strike,Rate,Time,Volatility,Yield)-Delta, K0)
```

where the initial guess can be `K0 = 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.