Pinning this for reference

## 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:

#### 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:

…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:

**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: