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 six-month at-the-money call on a 50 dollar share sheds a cent and a half a night to time decay, and its expected price tomorrow is higher anyway. The deterministic total differential gives minus 0.66 cents and predicts the opposite of the truth, while Ito's third term, half the gamma times the squared move, adds plus 1.13 and runs on variance rather than direction. Substituting the pricing equation for theta cancels that term exactly and leaves an expected return of the riskless rate plus elasticity times the premium, which is 35.54 percent a year here and turns negative below a real drift of 4.18.
A one-year at-the-money call on a 100 share with a 22 percent swing costs 8.7591 and carries 54.3795 dollars of share exposure, an elasticity of 6.2084. Multiply the share's market sensitivity of 1.10 by that and the call follows the market at 6.8292, so a share expected to earn 7.7 percent a year sits under a call expected to earn 47.8. The reflex answer, that an option price is a fair game with no drift, is true under the pricing measure and false under the one you live in, and both halves are measured here rather than asserted.
Holding the share above the strike and nothing below it reproduces a short call's obligation on every single path, and it is still not a hedge: the residual has a standard deviation of 9.07 dollars against a premium of 11.9235, and monitoring four and sixteen times as often leaves it at 9.00 and 9.02. A real delta hedge on the same paths goes 1.24, 0.63, 0.31, halving each time the interval is quartered. Tanaka's formula says why the refinement cannot help, because the residual is exactly the premium minus half the share's local time at the strike, a random quantity that never mentions the monitoring interval and is bounded above by the premium with no floor below.
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.
You own one-month calls struck at 110 with the share at 100, and you short 0.1452 shares against each one. If the share rallies to exactly 110 and stops, the calls expire worthless while the short has lost ten dollars a share, so the hedged position is down 2.074208 where the unhedged one would have lost only its 0.622212 premium. The worst case sits at the strike because the profit is piecewise linear with slopes of -0.1452 and +0.8548, and a rebalanced hedge on the same path loses 3.058738.
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.
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.
Traders treat opposite signs for theta and gamma as a law of the desk, but it is the pricing equation rearranged, and the equation names its own exceptions. At a zero rate the identity is exact and unbreakable; with a positive rate the interest on the bond leg buys the exception, and a deep in-the-money put has theta +7.0053 and gamma +0.0040317 together. The change of variables to the heat equation shows where the interest was hiding.
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.
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.
The hedge on a long call is the slope of its value, and a convex curve flattens as you slide left, so a falling share forces a smaller short and a smaller short is a purchase. No volatility, maturity or distribution enters that argument. The rebalance buys twenty shares, and the position gains 0.9824 per share on the fall.