Detecting Discrete Scale Invariance In Financial Markets

This thesis presents methodologies to identify periods in financial markets where the governing regime shows characteristics of discrete scale invariance. Log-periodic power laws often occur as signatures of impending criticality of hierarchical systems in the physical sciences, and it has been proposed that similar signatures may be apparent in the price evolution of financial markets as bubbles form. The features of such market bubbles have been extensively studied over the past twenty–five years, and models derived from an initial discrete scale invariance assumption have been developed and tested against the wealth of financial data with varying degrees of success. In this thesis, the equations that form the basis for the standard log-periodic power law model and its higher extensions are compared to a logistic model derived from the solution of the Schroder equation for the renormalisation group with nonlinear scaling function. Subsequently, a methodology is developed to identify change–points in financial markets where the governing regime shifts from a constant rate-of-return, i.e. from normal growth, to superexponential growth described by a power-law hazard rate. It is suggested that superexponential regimes correspond to financial bubbles and anti-bubbles driven by herding behaviour of market participants. It is from this theory that a predictive algorithm is developed that may have merit in identifying not only when a period of herding behaviour begins, and where it ends. The theory also provides tools which may facilitate profitable trading strategies across a broad spectrum of asset classes.