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# Visual Derivation of Straddle Approximation

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Pinning this for reference

#### 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:

#### 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.

We invoke Black Scholes:
…specifically, we zoom in on d1:
We are computing the call price for the strike K = ATF

Plug back into d1:
Recall from the definition of B-S:
Plug and chug:
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Checkpoint: We established 3 identities that occur at-the-forward

Let’s plug these identities back into the B-S equation for call struck ATF:
Hmm, this looks fairly docile. Stare at it hard. The next section will feel good.

#### Visualizing the call option

We established this so far:
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:
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:

Let’s bring this all together into a picture:

#### 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

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:
For the at-the-forward strike the call and put are equal because of put-call parity!

The rest is easy: