This is one the many nitty-gritty pitfalls.

Black Scholes assumes a lognormal distribution of stock price as of any given future date, including the expiration date T --

log ST ~ N( mean = ... , variance = σ

^{2}(T – t) )
This says that the log of that yet-unrealized stock price has a normal distribution. Now, as the valuation time “t” moves from ½ T to 0.99 T (approaching expiry), why would variance shrink? I thought if the “target” date is more distant from today, then variance is wider.

Well, I would say t is the so-called diffusion-start date. The price history up until time-t is known and realized. There’s no uncertainty in St. Therefore, (T - t) represents the diffusion window remaining. The longer this window, the larger the spread of diffusing "particles".

^{2}/2)(T-t)], where mu and σ are parameters of the original GBM dynamics.

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