so the brain teaser is; what are the odds of a put strike randomly sold 1 sigma away being hit? and...what are the odds of either side of the strikes of a strangle being hit when sold 1 sigma on either side?
Not the right question. In order to answer what you've asked, you'd have to correctly guess the volatility of the underlying GOING FORWARD. If you can figure out how to do that, there's a $10B hedge fund with your name on it. So the right question is, what is the correct strike to sell so that I only have a 1/3 chance of it touching? In order to answer it, you make a prediction of what volatility and returns are going to look like, and figure out what option you want to trade. (I smell a Bayesian discussion looming.)
The formula is quite complicated, I'll see if I can dig something out. But my understanding is that the probability of the underlying touching a strike 1 StDev out is roughly twice that of it expiring at or beyond the strike.
the probability of just a 1 sigma call or just a 1sigma put being hit in the spx over the last 26 years is about 14 percent, more for the calls, less for the puts. the probability of either side of a 1 sigma strangle being hit is just over 30 percent.
You're not catching what I'm throwing. You can't say that without defining a standard deviation for the appropriate time. You need a volatility input to do that. You can only know that volatility input after the fact. So your statement is a tautology. BTW, it is true that closing at or beyond a certain deviation is twice that of touching the same strike. This is just a random walk issue. Once you reach the strike, there's a 50% chance of being up or down from there.