Finding Option Strike or Volatility from Delta

Summary

The document addresses how to identify the implied volatility of a 25-delta call or negative-25-delta put, and how to infer the corresponding strike when volatility is known. It presents the Black–Scholes call delta as a function of spot, strike, time to expiry, rates, and volatility. One approach is to solve for volatility such that the call delta matches the target; put-call parity relates call and put deltas under the stated convention.

A second response describes reversing the Black–Scholes delta formula to calculate strike from a chosen delta, using inputs including spot, expiry, rate, volatility, and option right. These methods connect delta-based skew measures to strike and volatility lookup. The discussion does not spell out numerical root-finding, market conventions, or adjustments that may matter for options on futures, so implementation should use the delta definition and pricing conventions appropriate to the product.

Key ideas

• Black–Scholes delta depends on strike, volatility, expiry, spot, and rates.
• Implied volatility for a target delta can be found by solving the delta equation.
• The strike corresponding to a chosen delta can be obtained by rearranging the delta formula.
• Put and call deltas are related through put-call parity under consistent conventions.
• Options on futures may require product-specific delta conventions beyond the basic explanation.

Tags

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


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












I have calculated the implied volatility for all strikes of a certain product (options on futures) and approximated the ATM volatility. My question is how can I figure out the implied volatility for a 25 delta call and -25 delta put? I have come across a lot of information about delta, but can't quite put it together to solve this specific problem.

I am trying to implement Mixon's skew measure.

## Answer by emcor (score 3, accepted)

https://quant.stackexchange.com/a/12888

For Black-Scholes, $\Delta_C=\partial_{S} C=N(d_1)$, $d_1= \frac{\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)}{\sigma\sqrt{T - t}}$

You may fit the volatility $\sigma$ to this term by $$\Delta_C({\hat{\sigma}})=0.25$$Note that $\Delta_P=1-\Delta_C$ by Put-Call-Parity.

## Answer by Stu (score 1)

https://quant.stackexchange.com/a/12887

Found a nice source, hopefully someone can verify: http://www.elitetrader.com/vB/showthread.php?p=3482827

The trick is to back into the strike by using the delta formula (of course). Here is the R code posted at the site above:

```
BSStrikeFromDelta <- function(S0, T, r, sigma, delta, right)
{
strike <- ifelse(right=="C", 
S0 * exp(-qnorm(delta * exp((r)*T) ) * sigma * sqrt(T) + ((sigma^2)/2) * T),
S0 * exp(qnorm(delta* exp((r)*T) ) * sigma * sqrt(T) + ((sigma^2)/2) * T))
return( strike);
}
```

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.