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Let $[S_t]_{t\ge 0}$ be a geometric Brownian Motion with drift $r$ and volatility $\sigma$.

The dynamics of $S$ are:

$$dS_t=rS_tdt+\sigma S_tdW_t$$

Instead of completing the square technique, how to apply the Girsanov theorem to derive a formula for $E[S_T (I_{S_T\ge K})]$?

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    $\begingroup$ I'm not sure what completing the square would mean here, but what you'll need to do is change numeraire from a money market account to S_t. Take a look at Shreve's book for how to apply this technique. $\endgroup$ – d_797 Dec 17 '20 at 17:05