Logarithmic Versus Relative Random Walks

A recurring point discussed in the book concerns the processes that are able to reproduce the stylized facts. It is found that the most accurate ones have no continuum limit, and a discrete setup has to be used. With discrete processes, the price update equation can be defined using a logarithmic return or using a relative return. The implications of both definitions are presented, in particular with respect to the use of innovations with a fat-tailed distribution. This analysis shows that the common logarithmic random walk should be abandoned because of diverging expectations and that relative returns should be used instead. For constant volatility, the very long-term properties of processes defined using these return definitions are analyzed, showing that both definitions have first a square root of time diffusion phase, followed by an exponential grow of the variance. Finally, the implications of the return definitions with respect to the skewness is explored, showing that the relative return definition is compatible with no skew.