At each node a binomial-model compares two numbers: the value of continuing to hold and the value of exercising now. The set of prices where exercise wins forms a boundary that moves through time, approaching the strike as expiration nears.
Understanding the boundary is what lets a short option seller anticipate early-assignment instead of being surprised by it. If a stock crosses the boundary, assignment is not bad luck; it is the rational act of an informed counterparty, and it happens overnight without warning.
Example: for an XYZ $40 put with 60 days and 5% rates, the boundary might sit near $29. Above $29 the remaining extrinsic-value exceeds the interest you would earn on the strike proceeds. Below it, the holder should exercise, and a rational one will.
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