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Deriving the Delta-Neutral Straddle Strike from Black–Scholes d1

Article Quant Q&A · Author: AntB
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Summary

The document explains why an at-the-money delta-neutral straddle strike includes half the variance term in its exponential adjustment. In the Black–Scholes framework, the straddle is delta neutral when the call’s delta is one half, which corresponds to setting d1 equal to zero. Solving the expression for d1 for the strike gives a strike based on the spot price and an exponent containing the risk-free rate plus half the volatility variance over the option’s life.

This distinguishes the delta-neutral strike from the at-the-money-forward strike, whose adjustment uses the rate over time without the half-variance term. The explanation provides the key equation and the inversion step, but does not derive the Black–Scholes delta itself or discuss dividends, alternative pricing models, volatility conventions, or practical approximations. Its result applies within the stated model assumptions.

Key ideas

  • A delta-neutral straddle has call delta equal to one half in the stated framework.
  • That condition corresponds to setting the Black–Scholes d1 quantity to zero.
  • Solving d1 equals zero for the strike introduces half the variance alongside the rate.
  • The at-the-money-forward strike uses the rate adjustment without this half-variance term.

Tags

OptionsDerivatives pricingVolatility
Full text
# What is the reason for adding 0.5 variance when calculating the ATM DNS of an option?


# What is the reason for adding 0.5 variance when calculating the ATM DNS of an option?












Why is an Option ATM DNS (Delta Neutral Straddle) strike calculated using exponential value of (rate + 0.5 variance) * t. For ATMF (At the money forward), the rate time is used as the value in exponential growth.

## Answer by Newquant (score 3)

https://quant.stackexchange.com/a/74719

The delta neutral strike occurs when $N(d_1) = 0.5$, or when $d_1 = 0$. Now invert $$d_1 =\frac{\ln(S/K)+(r+\frac{1}{2}\sigma^2)T}{\sigma \sqrt{T}}$$ to solve for the strike $K$. You will have the answer.

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.