This chapter starts with introducing the notion of return. Then, I describe the efficient market hypothesis and its relationship with the random walk. In particular, I define three types of the random walk and address the problem of predictability of returns. I also offer an overview of recent empirical findings and models describing distributions of returns. Finally, I introduce the concept of fractals and its applications in finance.
Let's start with the basic definitions. The logarithm of price P denoted further as p = log(P) is widely used in quantitative finance. One practical reason for this is that simulation of random price variations can move price into the negative region, which does not make sense. On the other hand, the negative logarithm of price is perfectly acceptable. Log price is closely related to return, which is a measure of investment efficiency. Its advantage is that some statistical properties, such as stationarity and ergodicity1 are better applicable to returns than to prices (Campbell et al. 1997). The single-period return (or simple return) R(t) between two subsequent moments t and t − 1 is defined as2