A Stable and Robust Calibration Scheme of the Log-Periodic Power Law Model

Abstract

We present a simple transformation of the formulation of the log-periodic power law formula of the Johansen-Ledoit-Sornette model of financial bubbles that reduces it to a function of only three nonlinear parameters. The transformation significantly decreases the complexity of the fitting procedure and improves its stability tremendously because the modified cost function is now characterized by good smooth properties with in general a single minimum in the case where the model is appropriate to the empirical data. We complement the approach with an additional subordination procedure that slaves two of the nonlinear parameters to what can be considered to be the most crucial nonlinear parameter, the critical time $t_c$ defined as the end of the bubble and the most probably time for a crash to occur. This further decreases the complexity of the search and provides an intuitive representation of the results of the calibration. With our proposed methodology, metaheuristic searches are not longer necessary and one can resort solely to rigorous controlled local search algorithms, leading to dramatic increase in efficiency. Empirical tests on the Shanghai Composite index (SSE) from January 2007 to March 2008 illustrate our findings.