We will derive the closed form of the Geometric Brownian Motion with Itô's lemma.
The Geometric Brownian Motion (GBM) is the most popular representation of price processes in modern mathematical finance. It is most notably used to model the stock prices 🗠 of the underlying options in the Black Scholes Merton model. You can see my proof for the equation here.
The geometric brownian motion assumption for price processes is the default assumption in modern financial theory 🎓, but before that the assumption was more frequently that of a "pure" Brownian motion. That model was introduced in the early 1900s by Louis Bachelier and is still called the Bachelier model.
We introduce \(W\) the Wiener process and two parameters.
The proof is straightforward with Itô's lemma. The idea of the proof is to introduce \(df(S)=\frac{dS}{S}\). We can now directly write the SDE as follows. \[\begin{align} dS&=\mu Sdt+\sigma SdW\\ \frac{dS}{S}&=\mu dt+\sigma dW\\ df(S)&=\mu dt+\sigma dW\\ \end{align}\]
We can now apply directly Itô's lemma, which introduces quadratic variation in the derivation of the stochastic process. We recognise that \(f(S) = \ln S\). \[\begin{align} df(S)&=\frac{\partial f}{\partial S}dS + \frac{1}{2}\frac{\partial^2 f}{\partial S^2}(dS)^2\\ &=\frac{1}{S}dS - \frac{1}{2}\frac{1}{S^2}(dS)^2 \end{align}\] and we can use known properties of Bronian motion 📖, such as \(dWdt=0\) and \((dW)^2=dt\). Also, \((dt)^2=0\). \[\begin{align} (dS)^2&=(\mu Sdt+\sigma SdW)\times(\mu Sdt+\sigma SdW)\\ &=\mu^2 S^2(dt)^2+ 0 +\sigma^2 S^2(dW)^2\\ \frac{(dS)^2}{S^2}&=\sigma^2 dt\\ \end{align}\]
We can now combine with the SDE. \[\begin{align} df(S)&=\frac{1}{S}dS - \frac{1}{2}\frac{1}{S^2}(dS)^2\\ &=\mu dt+ \sigma dW - \frac{\sigma^2}{2}dt\\ d\ln S &= \left(\mu - \frac{\sigma^2}{2}\right) dt+ \sigma dW \end{align}\]
We can now solve the closed form by introducing the boundary condition \(S_0 = S(t=0)\) and knowing \(W_0 = W(t=0) = 0\). \[ \boxed{S = S_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right) t+ \sigma W(t)\right)} \]
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