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!
