# Apply Girsanov theorem to derive a formula for price [duplicate]

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 $$E[S_T (I_{S_T\ge K})]$$?

• 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. – d_797 Dec 17 '20 at 17:05