Brownian motion is clearly inadequate as a market model, not least because it would predict negative stock prices. However, by considering functions of Brownian motion we can produce a wide class of potential models. The basic model underlying the Black–Scholes pricing theory, geometric Brownian motion, arises precisely in this way. It will inherit from the Brownian motion very irregular paths. In §4.1 we shall see why a stock price model with rough paths is forced upon us by arbitrage arguments. This is not in itself sufficient to justify the geometric Brownian motion model. However in §4.7 we provide a further argument that suggests that it is at least a sensible starting point. A more detailed discussion of the shortcomings of the geometric Brownian motion model is deferred until Chapter 7.
In order to study models built in this way, we need to develop a calculus based on Brownian motion. The Itô stochastic calculus is the main topic of this chapter. In §4.2 we define the Itô stochastic integral and then in §4.3 we derive the corresponding chain rule of stochastic calculus and learn how to integrate by parts.
Just as in the discrete world, there will be two key ingredients to pricing and hedging in the Black–Scholes framework. First we need to be able to change the probability measure so that discounted asset prices are martingales. The tool for doing this is the Girsanov Theorem of §4.5.