An at-the-money call with a zero interest rate is worth one over the root of two pi, which is 0.39894, times the absolute swing of the terminal price, so a 10 dollar standard deviation prices it at 3.9894 and the closest of the offered 1, 5 and 10 is 5. The two-step estimate is exact under a symmetric terminal law, where the option finishes above the strike exactly half the time and the average gain when it does is 7.979. Calibrate a lognormal to the same 10 dollar swing and those two factors become 0.4801 and 8.285, whose product is still 3.98, which is why the qualifier about half cannot be cut.
Two stocks each swinging 20 percent a year with a correlation of one half give their product a volatility of 20 root 3, or 34.641 percent, rather than 40. Adding the two numbers is correct at exactly one correlation, namely 1, because the composite volatility is the law of cosines with the correlation as the cosine of the angle between two arrows. Pricing the call at 40 percent overstates it by 14.2 percent and ignoring the correlation understates it by 16.9 percent, and the other diagonal of the same parallelogram prices the ratio of the two stocks.
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.
Two weights that add to 0.98 give the variance forecast a half-life of 34.31 days; delete the second one and the half-life is 0.2744 days, gone before the next open. The same recursion turns strictly normal daily draws into a year with kurtosis exactly 297/67, and one shuffle of those same numbers separates the fat tail from the clustering. It also has a condition nobody quotes: stationarity is not enough for that kurtosis to be finite.
The standard deviation of a sum is not the sum of the standard deviations, so quadrupling the horizon only doubles the risk. The article carries the general square-root law, the ratio that diagnoses the mistake, and the controls showing a bell curve does none of the work: a two-point yearly return lands on 0.20026 and a uniform one on 0.20008. It also carries what actually breaks the rule, which is dependence rather than fat tails.
A three-month at-the-money call on a stock at 100 with 40% volatility is worth about eight dollars, and you can get there in two multiplications. The constant four tenths turns out to be the height of the normal bell at its peak, and the whole error of the mental rule is one rounding plus one cubic term. Scaling volatility linearly with time instead of with its square root gives ten dollars, which is 25.5% too high.