A common model for the evolution of the price of a financial security, such as a stock, is that it follows a brownian motion. There are two common styles of brownian motions: the arithmetic Brownian motion (ABM), and the geometric brownian motion (GBM). Let’s talk about ABM First.

Arithmetic Brownian Motion

The most basic brownian motion is the Wiener process $Z(t)$ with $Z(0)=0$ and $Z(t)\sim \mathcal N(0,1)$. This is the analogue of the discrete unbiased random walk in continuous time and values. Based on this basic ABM, we can define variants that have some drift $\mu$ (tendency to go up) and some volatility $\sigma$ (degree of randomness or wiggleness). A Security’s price $S$ following such modified ABM is described by the stochastic differential equation (SDE):

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

SDEs such as the above, do not follow the regular calculus quite exactly, that is, we need to take care when applying certain standard rules in regular differential equations. In the above, $dZ$ indicates an infinitesimal change of this process, which has the property that $\mathbb E[dZ]=0,\mathbb V[dZ]=dt$. Pay care in the last equality: $\mathbb V[dZ]=\mathbb E[(dZ)^2]=dt$. Typically infinitesimals squared are neglected, but here on the contrary, due to the nature of the brownian motion they have singificant variability that they need to be preserved.

Let’s look at our first few ABM. Use the following controls to interact with the simulation.

After you have experimented with the above, let's make some additional observations.

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). For example we do not expect a stock at a stock price of 1000 USD, to move in absolute terms as much as a stock at a stock price of 1 USD. Instead, the moves should be in percentage terms. Click the simulation below to observe the evolution of numerous 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.