Modeling Stock Prices

A common model for the evolution of the price of a financial security, such as a stock, is that it follows a brownian motion.

Arithmetic Brownian Motion

$$ dS = \mu dt + \sigma dZ\quad (ABM)$$

Let’s look at our first few ABM. Use the following controls to interact with the simulation.
Restart Increase Paths Decrease Paths Toggle Opacity Increase Drift Reduce Drift

After you have experimented with the above, let's make some additional observations. Toggle Average Path Toggle Terminal Distribution

ABM, is not appropriate for modeling stocks. For example the price might evolve to negative values. This, along with other issues motivates the introduction of GBM.

Geometric Brownian Motion

The Geometric Brownian Motion is a stochastic process where the diffusion is proportional to the current price of the process (that is, the security). Moves are in percentage terms. Click to observe the evolution of geometric brownian motions. The terminal distribution in this case becomes a lognormal and the price never becomes 0.

$$ \frac{dS}{S}= \mu dt + \sigma dZ\quad (GBM)$$

Jump Diffusion and Other Models

There are numerous other models that one can adopt for stock prices. Stay tuned for the simulation of more models, including Jump Diffusion and Local Volatility models.