Everything you always wanted to know about log-periodic power laws for bubble modeling but were afraid to ask

Abstract

Sornette, Johansen, and Bouchaud ( 1996 Sornette, D. , Johansen, A. and Bouchaud, J. P. 1996. Stock market crashes, precursors and replicas. Journal de Physique I, 6(1): 167–75. (doi:10.1051/jp1:1996135) [Crossref] [Google Scholar] ), Sornette and Johansen (1997) Sornette, D. and Johansen, A. 1997. Large financial crashes. Physica A, 245(3–4): 411–22. (doi:10.1016/S0378-4371(97)00318-X) [Crossref] [Google Scholar] , Johansen, Ledoit, and Sornette ( 2000 Johansen, A. , Ledoit, O. and Sornette, D. 2000. Crashes as critical points. International Journal of Theoretical and Applied Finance, 3(2): 219–55. (doi:10.1142/S0219024900000115) [Crossref] [Google Scholar] ) and Sornette ( 2003a Sornette, D. 2003a. Why stock markets crash (critical events in complex financial systems), Princeton, NJ : PUP. [Google Scholar] ) proposed that, prior to crashes, the mean function of a stock index price time series is characterized by a power law decorated with log-periodic oscillations, leading to a critical point that describes the beginning of the market crash. This article reviews the original log-periodic power law model for financial bubble modeling and discusses early criticism and recent generalizations proposed to answer these remarks. We show how to fit these models with alternative methodologies, together with diagnostic tests and graphical tools, to diagnose financial bubbles in the making in real time. An application of this methodology to the gold bubble which burst in December 2009 is then presented.