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Your mistake is actually made at the beginning: "Introducing a new process: $d\tilde{W}_t = dW_t +\frac{\mu-r}{\sigma} dt $" This is incorrect. Rather, $d\tilde{W}_t = dW_t -\frac{\mu-r}{\sigma} dt $ Otherwise, your derivation is correct. After correcting for the sign error, your final equation becomes $\Phi(x)=e^{-\lambda x-\frac{1}{2}\lambda^2 t}$. ...


The error is in the application of Girsanov theorem. We have multivariate Black-Sholes market, however I apply one-dimensional Girsanov theorem. I should apply multi-dimensional Girsanov theorem. Then there would be now such equations, except the case for $\rho=1$. The alike task is formulated here ...


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 $\mathbb{E}\left[ S_T \right] = S_0 \, e^{rT} \,.$


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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