Aim of the chapter

The easy access to enormous financial databases, containing thousands of asset time series, sampled at a frequency of minutes or sometimes seconds, allows one to investigate in detail the statistical features of the time evolution of financial assets. The description of any kind of data, be it of physical, biological or financial origin requires, however, an interpretation framework, needed to order and give a meaning to the observations. To describe necessarily means to simplify, and even sometimes betray: the aim of any empirical science is to approach reality progressively, through successive improved approximations.

The goal of the present and subsequent chapters is to present in this spirit the statistical properties of financial time series. We shall propose some plausible mathematical modelling, as faithful as possible (though imperfect) to the observed properties of these time series. The models we discuss are however not the only possible models; the available data is often not sufficiently accurate to distinguish, say, between a truncated Lévy distribution and a Student distribution. The choice between the two is then guided by mathematical convenience. In this respect, it is interesting to note that the word ‘modelling’ has two rather different meanings within the scientific community. The first one, often used in applied mathematics, engineering sciences and financial mathematics, means that one represents reality using appropriate mathematical formulae.