An example that helps you understand Brownian motion in stock market

I am going to explain how Brownian Motion works by using an example $AIFU.

I DIDN'T MEAN TO PROMOTE THIS STOCK. The reason why I chose this one is that it closely fits the math model.

When it comes to stock price like $AIFU, we often use geometric Brownian motion to model its price dynamics. The geometric Brownian motion is described by the following stochastic differential equation:

dS(t)=\mu S(t)dt+\sigma S(t)dW(t)

where:

● S(t) is the price of AIFU at time t.

● \mu is the drift parameter, which can be thought of as the average rate of return of AIFU over time. A positive \mu indicates that, on average, the price of AIFU is expected to increase over time.

● \sigma is the volatility parameter. It measures the degree of randomness or uncertainty in the price changes of AIFU. Higher values of \sigma mean that the price of AIFU can experience larger and more frequent fluctuations.

● dW(t) is the increment of the Brownian motion, representing the random shocks or noise in the market that affect the price of AIFU.

The solution to this stochastic differential equation is given by:

S(t)=S(0)\exp\left[\left(\mu-\frac{\sigma^{2}}{2}\right)t+\sigma W(t)\right]

where S(0) is the initial price of AIFU at time t = 0.

Here is how the formula suggests that this stock is going to grow.

● Positive Drift (\mu>0): Through in-depth fundamental analysis of AIFU, we can identify factors that suggest a positive long - term average return. Based on historical data and future projections, we estimate \mu = 0.1 (or 10% annual return on average). This means that, in the absence of random market fluctuations, the price of AIFU would grow exponentially at a rate of 10% per year.

● Volatility as an Opportunity: While high volatility (\sigma) is often seen as a risk factor, it can also present opportunities for growth. A non-zero \sigma implies that there is a chance for large upward price movements. For instance, if \sigma = 0.2, although the price of AIFU will experience random fluctuations, over time, the positive drift \mu combined with the random upward movements due to the Brownian motion can lead to significant price appreciation.

● Long-Term Trajectory: As time t increases in the formula S(t)=S(0)\exp\left[\left(\mu-\frac{\sigma^{2}}{2}\right)t+\sigma W(t)\right], the exponential term \left(\mu-\frac{\sigma^{2}}{2}\right)t will have a more pronounced effect on the price S(t) assuming \mu-\frac{\sigma^{2}}{2}>0. Even with the presence of the random W(t) component, the overall trend of the price of AIFU will be upward.