6.4 The Stock Price as a Stochastic Process

Stock prices are stochastic processes in discrete time which take only discrete values due to the limited measurement scale. Nevertheless, stochastic processes in continuous time are used as models since they are analytically easier to handle than discrete models, e.g. the binomial or trinomial process. However, the latter are more intuitive and prove to be very useful in simulations.

Two features of the general Wiener process $dX_t = \mu dt + \sigma \, dW_t$ make it an unsuitable model for stock prices. First, it allows for negative stock prices, and second the local variability is higher for high stock prices. Hence, stock prices $S_t$ are modeled by means of the more general Itô-process:

$$dS_t = \mu (S_t, t) dt + \sigma (S_t, t) dW_t \, .$$

This model does depend on the unknown functions $\mu (X, t)$ and $\sigma (X,t).$ A useful and simpler variant utilizing only two unknown real model parameters $\mu$ and $\sigma$ can be justified by the following reflection: The percentage return on the invested capital should on average not depend on the stock price at which the investment is done, and of course, should not depend on the currency unit (EUR, USD, ...) in which the stock price is quoted. Furthermore, the average return should be proportional to the investment horizon, as it is the case for other investment instruments. Putting things together, we request:

$$\frac{\mathop{\text{\rm\sf E}}[dS_t]}{S_t} \, = \, \frac{\mathop{\text{\rm\sf E}}[S_{t+dt} - S_t]}{S_t} \, = \, \mu \cdot dt \, .$$

Since $\mathop{\text{\rm\sf E}}[dW_t] = 0$ this condition is satisfied if

$$\mu (S_t, t) = \mu \cdot S_t \, ,$$

for given $S_t.$ Additionally,

$$\sigma (S_t, t) = \sigma \cdot S_t \,$$

takes into consideration that the absolute size of the stock price fluctuation is proportional to the currency unit in which the stock price is quoted. In summary, we model the stock price $S_t$ as a solution of the stochastic differential equation

$$d S_t = \mu \cdot S_t \, dt + \sigma \cdot S_t \cdot dW_t \, ,$$

where $\mu$ is the expected return on the stock, and $\sigma$ the volatility. Such a process is called geometric Brownian motion because

$$\frac{dS_t}{S_t} = \mu \, dt + \sigma \, dW_t \, .$$

By applying Itôs lemma, which we introduce in Section 5.5, it can be shown that for a suitable Wiener process $\{ Y_t; \, \, t \ge 0\}$ it holds

$$S_t = e^{Y_t}$$   bzw.$$\quad Y_t = \ln S_t \, .$$

Since $Y_t$ is normally distributed, $S_t$ is lognormally distributed. As random walks can be used to approximate the general Wiener process, geometric random walks can be used to approximate geometric Brownian motion and thus this simple model for the stock price.

Fig.: Density comparison of lognormally and normally distributed random variables. SFELogNormal.xpl