0
$\begingroup$

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?

  • 0
    The integral $\int_0^T\frac{dS}{S}$ is improper and diverges; (you should not get a $+C$ if you are doing a *definnite* integral!) But since $S$ is a function of $T$, that integral should be from $S(0)$ to $S(T)$, not from $0$ to $T$. Likewise, when you integral $\int_0^T\mu\,dt$ you should not get a "$+C$"; that should jsut be $\mu T$.2012-05-06

1 Answers 1

3

$S$ 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: $\begin{align*} \frac{dS}{S} &= \mu \\ \int_{S_0}^{S_T}\frac{dS}{S} &= \int_0^T\mu \,dt\\ \ln|S|\Bigm|_{S_0}^{S_T} &= \mu t\Bigm|_0^T\\ \ln|S_T|-\ln|S_0| &= \mu T\\ \ln\left|\frac{S_T}{S_0}\right| &= \mu T\\ \left|\frac{S_T}{S_0}\right| & = e^{\mu T}\\ \frac{S_T}{S_0} &= e^{\mu T} \quad\text{(since }S_T,S_0\text{ are positive)}\\ S_T &= S_0e^{\mu T}. \end{align*}$