Let
(1)
The homogeneous function of the Black-Scholes formula gives its at-the-money value at the start date as . Before that date, risk-neutral valuation and the discounted stock martingale give
(2)
After the start date the strike price is the realized value , so the complete answer is
(3)
At maturity the second line is . After entry the price depends on the historical strike price as well as the current stock, so that history cannot be discarded when giving the answer for all times.

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