Brownian Motion
Brownian Motion is a stochastic process where the variable’s increments follow a normal distribution with mean zero and constant variance. It’s often denoted as \( W_t \) and defined as follows:
\( W_t = W_0 + \mu t + \sigma B_t \)
Where:
- \( W_t \): The value of the variable at time \( t \).
- \( W_0 \): The initial value of the variable.
- \( \mu \): The drift or average rate of change.
- \( \sigma \): The volatility or standard deviation.
- \( B_t \): A standard Brownian Motion, which is a random process with independent and identically distributed increments.
Example:
Suppose a particle starts at position \( W_0 = 5 \). The drift \( \mu \) is 0.2, the volatility \( \sigma \) is 0.1, and the Brownian Motion \( B_t \) after 1 second is 0.8. Then, the position of the particle after 1 second is:
\( W_1 = 5 + 0.2 \cdot 1 + 0.1 \cdot 0.8 \)
Which gives:
\( W_1 = 5 + 0.2 + 0.08 = 5.28 \)
The particle is at position 5.28 after 1 second.
Geometric Brownian Motion
Geometric Brownian Motion (GBM) extends Brownian Motion by incorporating a drift term to account for potential growth or decline over time. It’s widely used to model stock price movements. The formula is:
\( dS_t = \mu S_t \, dt + \sigma S_t \, dW_t \)
Where:
- \( dS_t \): The change in the stock price over a small time interval \( dt \).
- \( S_t \): The stock price at time \( t \).
- \( \mu \): The average return or drift of the stock price.
- \( \sigma \): The volatility or standard deviation of the stock price.
- \( dW_t \): An increment of Brownian Motion over \( dt \).
Example:
Consider a stock with a current price \( S_0 = 100 \), a drift \( \mu = 0.05 \) (5% annual growth rate), volatility \( \sigma = 0.2 \) (20% annual volatility), and a time step \( dt = 1 \) year. If the Brownian Motion increment \( dW_t \) for the year is 0.3, the price after 1 year is calculated as:
\( S_t = S_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right) \)
Substitute the values:
\( S_t = 100 \cdot \exp\left(0.05 - 0.02 + 0.06\right) \)
Which gives:
\( S_t = 100 \cdot 1.0942 = 109.42 \)
Thus, the stock price after 1 year is 109.42.
Both Brownian Motion and Geometric Brownian Motion provide foundational insights for understanding the random behavior of variables in financial contexts, especially in modeling stock prices and market volatility.
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