The Hedge That Lands on the Right Answer and Still Swings Nine Dollars
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.