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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.

Hans-Peter Schrei
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AntB
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    It is common to have $(r + \frac{1}{2} \sigma^2)t$ in various mathematical formulas connected to options. But what is the exact formula that you are talking about? Where did you see this? – nbbo2 Feb 21 '23 at 13:54
  • yes, what's the reason for adding 0.5σ2 in risk-free rate to find the expected growth value of a stock? where is it coming from? – AntB Feb 21 '23 at 20:40
  • IIRC for risk neutral growth rate you actually subtract 0.5σ2. It comes about because of Stochastic Calculus, hard to explain without some exposure to that. But in simple terms If you have equal chance to be up 5% or down 5% at each step, then after 2 steps you are actually down slightly i.e. down 0.25% since 1.05*0.95 = 0.9975 rather than 1. THere is a downward bias captured by the -0.5σ2 term. For Delta neutral Strike OTOH see answer below. – nbbo2 Feb 22 '23 at 08:48
  • Thanks @nbbo2 , that (downward bias) is captured in d2 of BSM, but then why annualized Vol (σ sqrt(T)) added in d2 to find d1? Are you talking about Stochastic Volatility? – AntB Feb 22 '23 at 09:59

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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.

Bob Jansen
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Newquant
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  • I understood the DNS can be found this way, but my question is more on how d1 was formulated to add (r+1/2σ2)T into it. What is the significance of adding 1/2σ2 into ATMF (At The Money Forward strike)? if d1 = d2 + σ sqrt(T) then I am interested in knowing the reason for adding the annualized volatility in d2 to get d1..... – AntB Feb 22 '23 at 09:48
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    Check out this answer I did on this same question a few months back: https://quant.stackexchange.com/a/73404/62004

    It comes from the interaction between 'e' terms in the integral to find the expected value of the call option p/l.

    This equates to the conditional expectation of the stock price given it is above the strike. Also check out this link on the lognormal distribution https://en.wikipedia.org/wiki/Log-normal_distribution#Conditional_expectation

     – Newquant Feb 22 '23 at 13:37
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    Here: http://att.newsmth.net/bbsanc.php?path=%2Fgroups%2Fsci.faq%2FFE%2Ffe%2FM.1220512433.t0&ap=253 check out page 95-99 – Newquant Feb 22 '23 at 14:04