Integrating $dS/S = \mu dt$ between time $0$ and time $T$ we get $S_T = S_0e^{\mu T}$, for $S_0$ and $S_T$ stock prices at time $0$ and $T$.
I'm trying to figure out how they came to this conclusion.
So if I do $\int_o^{T} dS/S$ I get $\ln (T) + C$ and integrating $\int_0^{T}\mu dt$ I get $\mu T + C$. Setting them equal to each other and raising $e$ to both sides I get $T = e^{\mu T}$. Finance notation notwithstanding, did I do something wrong in my calculations?
1 Answer
$\begingroup$$S$ does not change from $S=0$ to $S=T$, it changes from $S_0$ to $S_T$. And definite integrals don't have constants of integration. So you should have: $$\begin{align*} \frac{dS}{S} &= \mu \\ \int_{S_0}^{S_T}\frac{dS}{S} &= \int_0^T\mu \,dt\\ \ln|S|\Bigm|_{S_0}^{S_T} &= \mu t\Bigm|_0^T\\ \ln|S_T|-\ln|S_0| &= \mu T\\ \ln\left|\frac{S_T}{S_0}\right| &= \mu T\\ \left|\frac{S_T}{S_0}\right| & = e^{\mu T}\\ \frac{S_T}{S_0} &= e^{\mu T} \quad\text{(since }S_T,S_0\text{ are positive)}\\ S_T &= S_0e^{\mu T}. \end{align*}$$
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