A call whose payoff is the square of the stock minus 100 does not start paying at 100, it starts paying at 10, because the square clears the strike exactly when the stock clears its square root. Placing the kink at the written strike prices the option at essentially zero on a stock at 12, when its real value is 53.8168. The closed form is ordinary Black-Scholes on the transformed asset with a growth term of 4 percent and a strike leg that still discounts at the riskless rate, and the value curve sits above intrinsic everywhere while being shallower than it at the spot in question.
A share at 100, a one-year call struck at 100, a zero rate and a twenty percent swing return 0.539828 and 0.460172. The first is the number of shares in the replicating portfolio, and reading it as the chance of finishing in the money hands you the complement of the right answer, since at the money with a zero rate the two numbers sum to one. The article carries the density identity that makes the share count exact, the measure under which the first number is a probability after all, and the four cents the straight line misses over a two-dollar move.
At the money the shorter maturity always wins, and a single negative derivative settles it for every volatility and every maturity: curvature runs 0.06907 against 0.02814 for one month against six. Ten percent out of the money the order reverses, 0.01854 against 0.02352, and the two curves cross 8.845 percent above the strike. What forces a crossover to exist is a conservation law, since every option in the family carries exactly the same total curvature and can only choose how to spread it.
Moving the average inside the exponential returns 1, and 1 happens to be the exact median and the exact geometric mean of e^X, which is why the mistake survives every re-check of the arithmetic. Completing the square in the exponent gives the true value e^(sigma squared over two), or 1.6487 at unit spread, because multiplying a Gaussian density by e^x slides its centre and scales its mass. Convexity settles the direction before any integral is set up, and on a heavy-tailed variable the quantity stops being finite at all.
The put reaches its strike more often, 0.3348 against 0.2821, and the call is still worth more, 4.2920 against 3.5891. The mechanism is not the unbounded-upside story, which would predict a gap at the money where put-call parity provably gives none; at a zero rate the 110 call equals 1.1 times a put struck at 90.909, and the put on offer is struck lower than that. The article also records two circulating claims that fail at these strikes, since the in-the-money chances at r = sigma^2/2 are 0.3168 and 0.2992 rather than equal, and the price ratio is 1.63 rather than 2.
An at-the-money call has no ceiling on its payoff and an at-the-money put is capped at the strike, yet at a zero interest rate the two cost exactly the same. The reason is put-call parity and it uses no model at all: the difference of the two payoffs is a straight line, so pricing it needs only the risk-neutral mean. The equality was checked on five terminal distributions with mean at the strike, and on a sixth whose mean is 120, where the gap is exactly 20.