Though I am in a quant finance program, I have decided not to pursue quant jobs as my career. With all due respect, the more I learn about finance, including quant fin, the more I believe it's not the core of the financial markets, though they are extremely crucial in derivative pricing, trading and risk management. It's all a matter of choice, but I reckon now that I've learnt a lot about Brownian motions stuffs, it seems like a waste if I don't share them with people. So bear with me and let me give you my understandings.
Intuitively you should find it easy to understand why stock price S(t) follows a geometric Brownian motion. By Ito's lemma, you will get the stochastic differential equation of the GBM S(t) to become dS(t)/S(t) = ln[S(t)] = stock return = alpha * dt + sigma * dW(t). (Note that you can ONLY get this stuff if stock price follows GBM. You wouldn't pay too many efforts to prove it.) Well, that "alpha" is nothing more than the average return of the stock (here, a stock's average return is alpha, not risk-free rate. Save risk-free rate for the purpose of pricing a stock derivative using Monte-Carlo). You can think of it as the compound return that you consistently earn from a stock (well, only fabulous stocks can give you this kind of return, lol. If you buy Huishan Dairy, then forget about any Brownian motion). The sigma part is the only uncertain part in a stock price process, because it has a Wiener process W(t), which follows N(0, t). You will find it make sense if you think about it: on average, the stock return will just be the average return alpha * dt, because E[stock return] = E[dS(t)/S(t)] = alpha * dt. However, the stock returns are volatile and moving around all over the place when it is traded; thus, those smart finance guys just said "why don't we add some random favor in it and make the stock returns to follow some normal distribution?" So wham! We have the sigma * dW(t) part, indicating that the stock is moving around the average returns. It also illustrates the so-called "mean reversion" phenomenon. You might wonder if this crazy model makes any sense in real world. Trust me, it indeed makes some sense.
There are some great stocks in the market that look exactly like a geometric Brownian motion. For example, the stock price of Tencent.
It's a pity that I can't upload any picture here. You can search for Tencent's stock price diagram and click "Max", and it will blow your mind: it looks exactly like geometric Brownian motion.
Some people argue that the model is wrong. Of course, all models are wrong, but some are useful :)
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