Suppose $S_t$ is the stock price and follows the dynamics $$dS_t=\mu S_tdt+\sigma S_tdW_t$$. According to Girsanov, we can apply change of measure and obtain $dS_t=r S_tdt+\sigma S_tdW_t^Q$, this implies $\ln S_T = \ln S_0+rT+\sigma W_T^Q$, and therefore $$\mathbb{E}^Q[S_T]=S_0 e^{rT+\frac{1}{2}\sigma^2T}$$, however $\mathbb{E}^Q[S_T]=S_0e^{rT}$ by Fundamental Theorem of Asset Pricing, which is contradicted. Please correct me.
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2 Answers
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It doesn't imply
$$ \ln S_T=\ln S_0+rT+σW^Q_T,$$
it implies
$$ \ln S_T=\ln S_0+(r-0.5\sigma^2)T+σW^Q_T.$$
Look up Ito's lemma.
This is covered in just about any book on financial maths including my own Concepts etc.
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If $dS_t = r S_t \, dt + \sigma S_t \, dW_t^Q$, $$S_T = S_0 \, e^{\sigma W_T^Q + \left( r - \frac{1}{2} \sigma^2\right) T}\, .$$
Hence $\mathbb{E}\left[ S_T \right] = S_0 \, e^{rT} \,.$
