Gaussian and non-Gaussian models for financial bubbles via econophysics

Fry, J. M. (2010): Gaussian and non-Gaussian models for financial bubbles via econophysics.

Abstract

We develop a rational expectations model of financial bubbles and study how the risk-return interplay is incorporated into prices. We retain the interpretation of the leading Johansen-Ledoit-Sornette model: namely, that the price must rise prior to a crash in order to compensate a representative investor for the level of risk. This is accompanied, in our stochastic model, by an illusion of certainty as described by a decreasing volatility function. As the volatility function decreases crashes can be seen to represent a phase transition from stochastic to deterministic behaviour in prices. Our approach is first illustrated by a benchmark Gaussian model - subsequently extended to a heavy-tailed model based on the Normal Inverse Gaussian distribution. Our model is illustrated by an empirical application to the London Stock Exchange. Results suggest that the aftermath of the Bank of England's process of quantitative easing has coincided with a bubble in the FTSE 100.

• Item Type: MPRA Paper
• Language: English
• Keywords: financial crashes; super-exponential growth; illusion of certainty; bubbles; heavy tails
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C10 - General; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods ; Simulation Methods; C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods

References

Sornette, D. and Malevergne, Y. (2001) From rational bubbles to crashes. Physica A 299 40-59.

Sornette, D. and Johansen, A., (1997) Large financial crashes. Physica A 245 411-422.

Johansen, A., Ledoit, O. and Sornette, D. (2000) Crashes as critical points. International Journal of Theoretical and Applied Finance 3 219-255.

Johansen, A. (2004) Origins of crashes in 3 US stock markets: shocks and bubbles. Physica A 338 135-142.

Laloux, L., Potters, M., Cont, R., Aguilar, J.-P and Bouchaud, J.-P. (1999) Are financial crashes predictable? Europhysics Letters 45 1-5.

Johansen, A. (2002) Comment on ``Are financial crashes predictable?" Europhysics Letters 60 809-810.

Feigenbaum, J. A. and Freund, P. G. O. (1996) Discrete scale invariance in stock markets before crashes International Journal of Modern Physics B 10 346-360.

Feigenbaum, J. A. (2001a) A statistical analysis of log-periodic precursors to financial crashes Quantitative Finance 1 346-360.

Feigenbaum, J. (2001b) More on a statistical analysis of log-periodic precursors to financial crashes Quantitative Finance 1 527-532.

Chang, G., and Feigenbaum, J. (2006) A Bayesian analysis of log-periodic precursors to financial crashes Quantitative Finance 6 15-36.

Chang, G., and Feigenbaum, J. (2008)Detecting log-periodicity in a regime-switching model of stock returns. Quantitative Finance 8 723-738.

Sornette, D. (1998) Discrete scale invariance and complex dimensions. Physics Reports 297 239-270.

Zhou, W-X. and Sornette, D. (2006) Fundamental factors versus herding in the 2002-2005 US stock market and prediction. Physica A 360 459-482.

Cont, R. and Tankov, P. (2004) Financial modelling with jump processes. Chapman and Hall/CRC, Boca Raton London Hiedelberg Washington DC

Sornette, D. and Andersen, J-V. (2002) A nonlinear super-exponential rational model of speculative financial bubbles. International Journal of Modern Physics C 17 171-188.

Andersen, J-V., and Sornette, D. (2004) Fearless versus fearful speculative financial bubbles. Physica A 337 565-585.

Zhou, W-X. and Sornette, D. (2003) 2000-2003 real estate bubble in the UK but not in the USA Physica A 329 249-262.

Fry, J. M. (2008) Statistical modelling of financial crashes. Unpublished PhD thesis, Department of Probability and Statistics, University of Sheffield.

Yeomans, J. M. (1992) Statistical mechanics of phase transitions. Oxford University Press.

Cox, D. R. and Oakes, D. (1984) Analysis of survival data. Chapman and Hall/CRC, Boca Raton London New York Washington, D. C.

Lillo, F., and Mantegna, R. (2003) Power law relaxation in a complex system: Omori law after a financial market crash. Physical Review E 68 016119.

Campbell, J. Y., Lo., A., and MacKinlay, J. A. C. (1997) The econometrics of financial markets. Princeton University Press, Princeton.

Oksendal, B. K. (2003) Stochastic differential equations: an introduction with applications. Springer-Verlag, Berlin Heidelberg.

Barndorff-Nielsen, O. E. (1998a) Processes of normal inverse Gaussian type. Finance and Stochastics 2 41-68.

Barndorff-Nielsen, O. E. (1998b) Normal inverse Gaussian distributions and stochastic volatility modelling. Scandinavian Journal of Statistics 24 1-13.

Fry, J. M. (2010) Bubbles and crashes in finance. A phase transition from random to deterministic behaviour in prices. Journal of Applied Research in Finance 2 (forthcoming)