Use Ito's Lemma to compute $d(\log S(t))$ and use this to find the closed form solution of S(t)

Question

I am having issues with this practice problem. If someone could help me solve it that would be greatly appreciated!

Let $S(t)$ be the stock price that satisfies the BSM model in SDE form $$dS(t) = \mu S(t) dt + \sigma S(t) dW_t$$ where $\mu > 0$ and $\sigma > 0$ are two constants. Use Ito's Lemma to compute $d \log S(t)$ and use this result to find the closed form solution of $S(t)$.

Answer 1

As the title of the question says, it's just a straightforward application of Ito's lemma: Since $S$ satisfies the given SDE, $(\log)'(x)=x^{-1}$, and $(\log)''(x)=-x^{-2}$, we have $$ \begin{split} d(\log (S_t))&= \frac{1}{S(t)}dS_t-\frac{1}{2}\frac{1}{S_t^2}\sigma^2S_t^2dt\\ &= \mu dt+\sigma dW_t-\frac{\sigma^2}{2}dt\\ &=(\mu-\frac{\sigma^2}{2})dt+\sigma dW_t \end{split} $$ And this is just the SDE of the Brownian with drift.

Answer 2

Let $S(t)$ be governed by the SDE

$$dS(t)=\mu S(t)dt+\sigma S(t)dW_t$$

Let $f(S)=\log(S)$.

Heuristically, we can write

$$\begin{align} d\log(S)&=\frac{\partial f(S)}{\partial t}\,(dt)+\frac{\partial f(S)}{\partial S}\,(dS)+\frac12\frac{\partial ^2f(S)}{\partial S^2}(dS)^2\\\\ &=\frac{\partial \log(S)}{\partial t}\,(dt)+\frac{\partial \log(S)}{\partial S}\,(dS)+\frac12\frac{\partial ^2\log(S)}{\partial S^2}(dS)^2\\\\ &=(0)\,dt+\frac1S\,(dS)-\frac{1}{2S^2}(dS)^2\\\\ &=\left(\mu-\frac12 \sigma^2\right)\,dt+\sigma dW_t \tag 1 \end{align}$$

Now, integrating both sides of $(1)$ yields

$$\begin{align} \int_0^t d\log(S(t'))&=\log(S(t)/S(0)\\\\ &=\int_0^t\left(\left(\mu-\frac12 \sigma^2\right)\,dt'+\sigma dW_t'\right)\\\\ &=\left(\mu-\frac12 \sigma^2\right)t+\sigma (W(t)-W(0))\\\\ S(t)=&S(0)e^{\left(\mu-\frac12 \sigma^2\right)t+\sigma (W(t)-W(0))} \end{align}$$