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.
Give a pulled-back log price the same 20 percent instantaneous swing as a free-wandering one and its horizon variance stops being sigma squared times T: at one reversion time only 0.432332 of it survives, the volatility that prices a one-year call is 13.1504 percent, and the call falls from 7.9656 to 5.2425. The same pull makes consecutive returns fight each other, with a first-order autocorrelation of exactly minus half of one minus phi, and that is the independence the pricing model rests on. The formula still returns the right European price and has lost the hedging argument that justified it.
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.
Shade the region between a diffusing particle and the time axis over one second. The box is one wide and about one tall, so the eye guesses a variance of one, and the answer is one third because each increment counts only for the time remaining after it. No stochastic integration is needed to define the object, only continuity of the path, and the constant is pinned twice over: once by the weight (T minus t) and once by integrating the covariance min(s,t) across the square.
The integral of x dx is x squared over two, so the integral of W dW ought to be W(T) squared over two, and the only false step in that chain is the conclusion. A dissected square turns the Riemann sum into an identity exact at every partition, and the term that refuses to vanish is the total of the squared steps, which equals T rather than zero. The reflex answer is the exact value of the midpoint sum over the same partition, which is why it feels so solid.
A contract struck at the current share price is worth roughly 0.3989 S sigma root T, so doubling the time to expiry multiplies the price by root two and takes 100 dollars to about 141 rather than 200. The exact ratio erf(s/2) over erf(s/(2 root 2)) is always strictly below root two because erf is concave, so 141.42 is a ceiling never reached. Strip out the strike condition and the rule collapses: the same doubling multiplies a strike 30 percent above spot by 4.19 and one 30 percent below by 1.01.
Six months of a sixty dollar year carries 60 over root two, which is 42.43 rather than 30, because variances add over disjoint intervals and standard deviations do not, so the digital is worth exactly $239,750. The figure of $250,000 in circulation comes from rounding the z score 0.7071 up to 0.75 and then reading the tail at 0.75 as 0.25, but Phi(0.75) = 0.773373, so even the rounded chain gives 0.2266. Rounding z upward has to make the tail smaller, and 0.25 is larger, which is the tell that a symbol changed meaning mid-calculation.
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.