A share at 100 that jumps to either 80 or 130 gives a call an exact price of 12, from two equations in two unknowns and no probability at all. Let the jump size be random, so 110 is also reachable, and that same hedge pays 18 where the option pays 10 while no other portfolio does better. The arbitrage-free prices then fill the whole interval from 20/3 to 12, and the obstruction turns out to be the kink in the payoff rather than the number of states.
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.