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Under GBM $$\frac {dS_t}{S_t} = \mu dt + \sigma dW_t$$ we get $$S_T = S_0 e^{(\mu - \frac{1}{2}\sigma^2)T + \sigma W_T}$$ suggesting that $$S_T \sim \text{ln}\mathcal {N} ( \tilde {\mu}, \tilde {\sigma})$$ where \begin{align} \tilde {\mu} &= \ln S_0 + (\mu - \frac{1}{2}\sigma^2)T \\ \tilde {\sigma} &= \sigma \sqrt {T} \end{align} Now if $X \... 1 Here's a try/start: Let$A,B$, and$C$be three possible events, and let$U(event)$be the utility derived from each event. For example, if event$A$corresponds to the event of winning the lottery, then$U(A)$will presumably be a very large value. By contrast, if event$C$corresponds to the event of falling off a ladder and breaking an arm,$U(C)\$ will ...