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.
An American call that only wakes up at 80 and dies for good at 125 has no closed form, and it cannot be simulated either, because a path runs forward while the exercise decision looks back. Every path that avoids the ceiling either visited the floor or never did, so the contract is one knock-out minus another and both come off a standard tree. The identity is exact to machine precision for European exercise at all seven grids tested, and for American exercise only in the continuous limit: the finite-tree residual falls from 0.532 percent at 45 steps to 0.043 percent at 3,394.
An option settling on the mean of a share's closes is strictly cheaper than one settling on the closing price, and the reason is convex order rather than any pricing model: for a martingale share every intermediate price is a forecast of the last one, so the average is dominated at every strike, for calls and for puts. Quantitatively the time average of a Brownian path carries variance T/3 against T, a swing ratio of 1/sqrt(3) = 0.57735, which turns 11.9235 into 6.9013 at a 30 percent volatility. A finite grid of 252 fixings sits at 0.33532 rather than 1/3, which accounts for most of the gap to the 6.918 measured by simulation on the true arithmetic average.
Three calls struck at 100: one plain, one that dies at 90, one that dies at 120. The knock-outs cost less, and their slopes can be ranked from the two ends of the picture instead of by differentiating a barrier formula. That argument only bounds an average slope, so the article also carries the exact pointwise gap, the strike times a normal tail at the reflected share price divided by the barrier, which comes to 0.138146 and turns 0.539828 into 0.677974.
The sharp statement is stronger than the usual one: on every path, missing the ceiling plus missing the floor counts the paths that miss both exactly twice, so the pair is twice the double plus the value of the one-sided survivors. That is a polynomial identity in indicators, so it holds under every pricing measure with no volatility anywhere, and one half is the tight bound. In a worked instance the pair is 10.317 against a double of 1.494, and the fastest refutation of the trap is that the pair exceeds the plain call with no barriers at all.
A ticket paying a hundred dollars if a share finishes above its strike is squeezed between two ordinary call spreads at every width, so its value is pinned by prices already quoted with no distribution assumed anywhere. The limit is minus the derivative of the call price in the strike, which equals e to the minus rT times N(d2) because two density terms cancel exactly at every strike. Here that is 53.2325, against the 62.35 a real-world drift would give.
More volatility is worth more is a theorem about convex payoffs, and a step function is not convex. An at-the-money cash-or-nothing digital falls from 46.02 to 42.07 cents when the swing doubles, and the sensitivity is positive only below X exp(-(r + sigma squared over two)(T-t)). The cap on the payoff is only half the explanation; the falling median is the other half.
A share sits at 75, the rate is zero, and a perpetual claim pays one dollar the first time the price ever touches 100. It is worth exactly 75 cents, and no volatility number is needed to say so. The reflex answer of a dollar assumes the barrier is always reached, which a price with a floor at zero never promises: a quarter of the paths fade away without paying anything.