diffusion start and variance

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 = σ2 (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”.
By the way, the “mean” above is      logS0 + [(mu – σ2/2)(T-t)], where mu and σ are parameters of the original GBM dynamics. 


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