Statistical modelling of financial crashes.

As the stock market came to the attention of increasing numbers of physicists, an idea
that has recently emerged is that it might be possible to develop a mathematical theory
of stock market crashes. This thesis is primarily concerned with statistical aspects of such
a theory.
Chapters 1-5 discuss simple models for bubbles. Chapter 1 is an introduction. Chapter 2
describes a skeleton exploratory analysis, before discussing some economic interpretations
of crashes and a rational expectations model of financial crashes - a slightly simplified
version of that in Johansen et aZ. (2000). This model assumes that economic variables
undergo a phase transition prior to a crash, and we give some empirical support of this
idea in Chapters 4 and 5.
Chapter 3 discusses SDE models for bubbles. We describe maximum likelihood estimation
of the Bornette and Andersen (2002) model and refine previous estimation of this model
in Andersen and Bornette (2004). Further, we extend this model using a heavy-tailed
hyperbolic process, Eberlein and Keller (1995), to provide a robust statistical test for
bubbles. In Chapter 4 we examine a range of volatility and liquidity precursors. We have
some evidence that crashes occur on volatile illiquid markets and economic interpretation
of our results appears interesting. Chapter 5 synthesises Chapters 2-4.
In Chapter 6 we develop calculations in Johansen and Bornette (2001), to derive a
generalised Pareto distribution for drawdowns. In addition, we review the Bornette et aZ.
(2004) method of using power-laws to distinguish between endogenous and exogenous
origins of crises. Despite some evidence to support the original approach, it appears that
a better model is a stochastic volatility model where the log volatility is fractional Gaussian
noise.
Arias (2003) makes a distinction between insurance crisis and illiquidity crisis models.
In Chapter 7, focusing upon illiquidity crises, we apply the method of Malevergne and
Barnette (2005) to evaluate contagion in economics. Chapter 8 summarises the main
findings and gives suggestions for further work.