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 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.
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.