# 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 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 are modeled by means of the more general Itô-process: This model does depend on the unknown functions and A useful and simpler variant utilizing only two unknown real model parameters and 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: Since this condition is satisfied if for given Additionally, 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 as a solution of the stochastic differential equation where is the expected return on the stock, and the volatility. Such a process is called geometric Brownian motion because By applying Itôs lemma, which we introduce in Section 5.5, it can be shown that for a suitable Wiener process it holds bzw. Since is normally distributed, 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. 