ABSTRACT
In Theorie de la Speculation, Louis Bachelier made the first attempt to model the inherent randomness in stock prices using a continuous-time counterpart of white noise, the Brownian Motion. For many years this modelling approach was of purely academic interest as financial institutions used less mathematically demanding methods. This chapter introduces the merits of one of the most important financial derivatives, the European call option, by considering a fairly simple transaction between two companies. The Gaussian distribution may be fine for many applications, but it does not represent the unconditional distribution of returns well. Large and extreme events are much more common than predicted by the normal distribution. Returns computed over long time periods are more Gaussian than returns computed for short time periods. The return over a long period can be written as a sum of returns over shorter periods. That suggests, under some conditions, that the return over long periods should become increasingly Gaussian.
