For a stock price $S_t$, its differential $dS_t$ can be modeled as an [[Itô Processes and Itô's Lemma#Itô Process|Itô Process]]. This specific form is called Geometric Brownian Motion ("GBM").
$ dS_t = \overbrace{\mu S_t}^{a} \, dt+ \overbrace{\sigma S_t}^{b} \, dB_t $
where:
- $\mu$ is the drift (expected return)
- $\sigma$ is the volatility (standard deviation of returns)
- $dB_t$ is the increment of standard [[Brownian Motion]]
To integrate this [[Stochastic Differential Equation]], we need to bring it into "standard" SDE form:
$ dX_t =a \, dt + b \, dB_t$
The important aspect here is that $a$ and $b$ are constants or deterministic functions, that do not depend on $X_t$. Currently both terms involve $S_t$, which has a stochastic component ($dB_t$).
**Side Note:**
We could divide both sides of the equation by $S_t$, which would make the right hand side fit to standard SDE form, but then $\frac{dS_t}{S_t}$ is not an ordinary derivative you can integrate with classical calculus rules anymore.
$ \frac{dS_t}{S_t}= \mu \, dt + \sigma \, dB_t$
## Application of Itô's Lemma
Since classical integration is not feasible, we apply [[Itô Processes and Itô's Lemma#Itô's Lemma for $F(t, X_t)$|Itô's Lemma]] by transforming the process via a differentiable function $F$. Here we choose
$ F(S_t) = \log S_t$
For a function $F(S_t)$, Itô's Lemma states:
$ dF(S_t)
=\frac{\partial F}{\partial t}\, dt+
\frac{\partial F}{\partial S}\, dS_t+
\frac{1}{2} \frac{\partial^2F}{\partial S^2} \, b^2\, dt$
For $F(S_t)=\log(S_t)$ we compute the [[Partial Derivative]] of each term:
$ \frac{\partial F}{\partial t}=0, \quad \frac{\partial F}{\partial S_t}=\frac{1}{S_t}, \quad \frac{\partial^2 F}{\partial S_t^2}=-\frac{1}{S_t^2}$
Substituting into Itô's Lemma:
$
\begin{align}
dF(S_t)= d \log(S_t)&= 0+\frac{1}{S_t} dS_t+ \frac{1}{2} *-\frac{1}{S_t^2} \, b^2\, dt\\[4pt]
&=\frac{\mu S_t\, dt + \sigma S_t\, dB_t}{S_t}-\frac{1}{2S_t^2}*\sigma^2S_t^2 \, dt\\[4pt]
&=\mu\, dt + \sigma \, dB_t-\frac{\sigma^2}{2} \, dt\\[4pt]
&=\left(\mu -\frac{\sigma^2}{2}\right) \, dt + \sigma \, dB_t
\end{align}
$
Thus we have recast the process into standard SDE form:
$ dX_t = a \, dt + b \, B_t \quad \text{where}
\begin{cases}
X_t &= \log(S_t) \\
a&= \mu - \frac{\sigma^2}{2} \\
b&= \sigma
\end{cases}$
## Integration of the Log-Transformed Process
We can integrate both sides of the equation from $0$ to $T$:
$
\begin{align}
\int_0^T d\log (S_t) &= \int_0^T (\mu - \frac{\sigma^2}{2})dt+ \int_0^T\sigma dB_t \tag{1}\\[2pt]
\log (S_t)- \log(S_0) &= \left(\mu - \frac{\sigma^2}{2} \right) \,T+\sigma \,(B_t-B_0) \tag{2}\\[2pt]
\log \left(\frac{S_t}{S_0}\right) &= \left(\mu - \frac{\sigma^2}{2} \right) \,T+\sigma B_t \tag{3}\\[2pt]
\frac{S_t}{S_0} &= \exp\left((\mu - \frac{\sigma^2}{2}) \,T+\sigma B_t\right) \tag{4}
\end{align}
$
where:
- (1) Integrals of linear combinations can be split up
- (3) On the left hand side, the difference of logarithms is the log of their fraction. On the right hand side, by assumption a [[Brownian Motion]] process start at $B_0=0$ almost surely.
- (4) Exponentiating both sides.
Finally the stock price $S_t$ can be written as:
$ \boxed{S_t = S_0*\exp\left((\mu - \frac{\sigma^2}{2}) \,T+\sigma B_t\right)} $
## Properties of Geometric Brownian Motion
- *Positivity:* The exponential function ensures that $S_t$ is always positive. This is a desirable feature in financial modeling because stock prices cannot be negative.
- *Limitation:* This model does not allow for the possibility of a stock price reaching zero. In reality, a company can go bankrupt, driving its stock price to zero.
>[!note:]
>It is called "geometric", because each increment $dS_t$ is proportional to the current state $S_t$, making the process evolve in a multiplicative (instead of additive) way.