A share at 50 goes to 65 or to 40, and the right to buy it at 50 is worth exactly 6, because three fifths of a share against 24 borrowed pays the option in both states and costs 6 today. That bill contains no probability at all, which symbolic differentiation shows and a sweep never could, so the price may be computed under whichever beliefs are convenient and the artificial 2/5 returns the same 6. Discounting the mean payoff at the share's required 15 percent gives 9.13, and the rate that does work is the option's own 75 percent, which cannot be known before the price is.
Short a call struck at 100 with the share at 113.40 and two months left, the hedge holds 0.90028 shares; a month later at 112 it wants 0.94591, so you buy 4.56 per hundred. The reflex to sell is not a blunder, because freezing the clock and letting the same fall happen alone really does take the hedge to 0.87726, but the time effect is 2.7 times larger and points the other way. Holding the hedge constant traces the curve S(T) = X exp(d1 sigma root T minus half sigma squared T), which puts the break-even fall at 3.51 percent and, at expiry, at the strike itself.
Whether an up move followed by a down move lands where a down move followed by an up move lands decides between a quadratic node count and an exponential one, and at twenty steps the gap is a factor of 9,078.6. Both sums carry N+1 terms rather than N, because a twenty-step tree has twenty-one dates on it, and the off-by-one costs the entire final row. The article also states the recombination hypothesis exactly, which is weaker than the usual ud = 1.
A holding pays $200 if a team wins four games first, you must take a symmetric position on every game, and committing the whole hundred to game one produces the right payoffs a week too early. Backward induction on the lattice fixes the amount at half the gap between the two successor values, $31.25. The same number is 5/16 of the holding, which is the chance the other six games split three each, and no win probability appears anywhere in the derivation.
A stock at 100 goes to 130 with probability 0.8 or 70 with probability 0.2, rates are zero, and the right to buy at 110 is worth 10 rather than 16. A third of a share funded by borrowing 70/3 reproduces both payoffs and costs 10 today, which prices the option without using a probability anywhere. The general risk-neutral probability (S-d)/(u-d) is a half here only because rates are zero and 100 sits midway between the two outcomes.