Apologies upfront if my Finance is better than my grasp on the finer points of advanced calculus. I know the argument(s) he's making; and just hope someone more Quant than I can land the point home.
He's arguing that a "simple" bet and a binary option, which is an "exotic" option, are one and the same thing! Which is true. Both end up worth 0 or 1, versus a prior cost = expected probability of p.
The conceit here is that instead of looking at the risk-reward of expected p vs p priced (classic Kelly stuff), he's looking at it in option terms, ie sigma, essentially in vol of price = vol of forecast terms. And making the not-too-exotic point if p is bounded [0,1], then it cannot roof it to [-inf,+inf]. In which case, if IV>RP, selling vol generates returns, irrespective of actual p.
Shock horror, all that before; albeit maybe put in this precise context dressed up exactly this way ;-)
Where I run short on answers is the precise formula. The exposition he's using is the European cash-or-nothing call segment of Black Scholes. Which is N(d2) bit of the classic formulae. My guess is that the integral of that ends up in the formula above.
Which leads to the oh-so revolutionary concept of selling vol when RV>IV. Never heard that one before ;- Even if the basis for the fair value of sigma is gloriously unclear. Even if the assumption that forecast probabilities are normally distributed is news to me, and evidently flawed. How can a move from a 49.5% to a 50.5% expectation be equivalent to 1%/99% going to 0%/100%. Sorry, but that's just mega-bonkers as models go. Even if all of Taleb's prior work screams arrogance on a level I'd struggle to match with anyone historically. We've had warrior-poets; philosopher-kings; warrior-poet-kings etc. The closest I can come to a warrior-poet-philosopher-king to match NNT is Marcus Aurelius. Trouble is, the basis of his philosophy was modesty and how little we could know... Taleb wins ;-)