What does the ATM straddle tell us?

[TheBigShort]

This is a bit more mathematical, but I know there are some smart guys on here.

ATM straddle price = sqrt(2/pi) * S * vol * sqrt(t/T).

where t = time to maturity
and T = 1year in days
and S = stock price

This formula actually gives us the expected value of the trade. The expected value is = to the mean of a distribution.

So why do people say a straddle tells us the expected standard deviation? Am I missing something? Thanks in advance.

 

[Kevin Schmit]

price = sqrt(2/pi) * S * vol * sqrt(t/T).
So why do people say a straddle tells us the expected standard deviation? Am I missing something? Thanks in advance.

Your formula, rearranged:

vol * sqrt(t/T) = price / (S * sqrt)2/pi))

vol*sqrt(t/T) is the standard deviation of the expected terminal distribution of the stock price. vol*sqrt(t/T)*0.6028 is the standard deviation of the expected terminal distribution of the ATM straddle value.

Above is approximate, does not include rate, divs...

 

[TheBigShort]

.6028? Where is this # from?
Thanks for the intuition there. However if I want to find the expected value of the stock price distribution is it not the straddle price? If it was more you could always sell and if it was less you would always buy? Thanks Kev

 

[Kevin Schmit]

.6028? Where is this # from?

From the equation for standard deviation of a truncated normal distribution with a=0, b=+Inf, mu=0, and sigma=1.0. It is complex but in this situation simplifies to half normal (both equations are on Wikipedia). In R:

> require(truncnorm)
> sqrt(vtruncnorm(a=0))
[1] 0.6028103

However if I want to find the expected value of the stock price distribution is it not the straddle price?

No, the expected value of the stock price under the risk-neutral measure is the current stock price (more correctly the current forward price), this is the mean (expectation) of the expected terminal distribution.

The straddle price is the net present value (NPV) of the expected terminal value of the straddle under the risk neutral measure.

The expectation of the ATMF call is F * 0.7979 * root-time-vol * probability-of-being-ITM

probaility of being ITM is ~ 0.5 (1-pnorm(0) or 1-plnorm(1) in R)

Expectation of ATMF put is roughly the same. Add the call and put,
or twice the call, and you get the straddle price given by your approximatiion equation in your previous post.

 

[TheBigShort]

From the equation for standard deviation of a truncated normal distribution with a=0, b=+Inf, mu=0, and sigma=1.0. It is complex but in this situation simplifies to half normal (both equations are on Wikipedia). In R:

> require(truncnorm)
> sqrt(vtruncnorm(a=0))
[1] 0.6028103


No, the expected value of the stock price under the risk-neutral measure is the current stock price (more correctly the current forward price), this is the mean (expectation) of the expected terminal distribution.

The straddle price is the net present value (NPV) of the expected terminal value of the straddle under the risk neutral measure.

The expectation of the ATMF call is F * 0.7979 * root-time-vol * probability-of-being-ITM

probaility of being ITM is ~ 0.5 (1-pnorm(0) or 1-plnorm(1) in R)

Expectation of ATMF put is roughly the same. Add the call and put,
or twice the call, and you get the straddle price given by your approximatiion equation in your previous post.

Kev whats your background? This math is a little over my head and I will need a day to work this out. I cant find good articles on this stuff. Are you on another forum that deals with the mathematics of options?