Visual Derivation of Straddle Approximation

Pinning this for reference
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Derivation of straddle approximation


The mean of the distribution

We want to estimate the straddle. The mean of the underlying stock distribution is centered around the forward price not the at-the-money price.
We will estimate the at-the-forward (ATF) straddle.
This means we are estimating the straddle struck at the ATF strike.
The ATF strike occurs at the ATF price:
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Approximating the ATF call option

This is the meat of the work.
[It requires no more than pre-algebra. I know this because my 5th grader is taking the Art of Problem Solving online course in it now. I’m not proud to say I’m quite rusty.]
Let’s go.
While we want the straddle, let's start with the ATF call option.
We invoke Black Scholes:
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…specifically, we zoom in on d1:
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We are computing the call price for the strike K = ATF
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Plug back into d1:
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Recall from the definition of B-S:
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Plug and chug:
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Checkpoint: We established 3 identities that occur at-the-forward
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Let’s plug these identities back into the B-S equation for call struck ATF:
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Hmm, this looks fairly docile. Stare at it hard. The next section will feel good.

Visualizing the call option

We established this so far:
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The underlying distributions for B-S is that stock prices are lognormal. The prices are lognromal but logreturns are normally distributed.
This is handy because normal distributions are familiar to work with.
d1 and d2 are like Z-scores on a Gaussian (bell) curve of logreturns!
The probability density function (PDF) for a bell curve:
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The center of our distribution is an expected logreturn of 0 corresponding to the forward Seʳᵗ
The peak of a bell-curve at that forward price corresponding to a logreturn of 0. For the standard normal curve we can assume σ = 1
Plug 0 into x of the PDF:
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Let’s bring this all together into a picture:
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Understanding the picture

The value of the ATF call is the integral of the PDF between d1 and d2 but we can estimate it!
height x base x forward price
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Note: This will slightly overestimate the value of the call (see overestimated region in the picture)

From call price to the straddle

The call estimate is:
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For the at-the-forward strike the call and put are equal because of put-call parity!
The rest is easy:
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