A share swinging twenty dollars a year gives an at-the-money call that looks like it should cost ten, half the swing collected half the time. It costs 7.98, because the upper half of a bell curve averages 0.798 of a standard deviation rather than a whole one. The article derives the general arithmetic-Brownian price, checks both limits, and quantifies the negative-price defect that got the model retired and then rehabilitated.
At the money the log term in d1 vanishes and what remains is strictly positive for every non-negative rate and every volatility, so the delta always beats 0.5 and is 0.6554 at twenty percent. A square rather than a derivative gives the sharp floor: at six percent over a year the delta can never fall below 0.6355. The article also kills the sentence that sounds like a restatement of the answer, since the chance of finishing in the money falls to 0.4801 at forty percent volatility.
Six months of a sixty dollar year carries 60 over root two, which is 42.43 rather than 30, because variances add over disjoint intervals and standard deviations do not, so the digital is worth exactly $239,750. The figure of $250,000 in circulation comes from rounding the z score 0.7071 up to 0.75 and then reading the tail at 0.75 as 0.25, but Phi(0.75) = 0.773373, so even the rounded chain gives 0.2266. Rounding z upward has to make the tail smaller, and 0.25 is larger, which is the tell that a symbol changed meaning mid-calculation.