Flash Crashes: The Role Of Information Processing Based Subordination And The Cauchy Distribution In Market Instability

Edgar Parker Jr.
New York Life Insurance Company

July 15, 2016

Abstract:

While a wide variety of hypotheses have been offered to explain the anomalous market phenomena known as a “Flash Crash”, there is as of yet no consensus among financial experts as to the sources of these sudden market collapses. In contrast to the behavior expected from standard financial theory, both the equity and bond markets have been thrown into freefall in the absence of any significant news event. The author posits that a combination of probability and information theory, and diffusion dynamics offers a relatively simple explanation of the causes of some of these dramatic events. This new avenue of research also suggests new policies or measures to lower the probability of occurrence and to mitigate the effects of these extreme events.

Flash Crashes: The Role Of Information Processing Based Subordination And The Cauchy Distribution In Market Instability – Introduction

Two of the key variables studied in this paper that affect the behavior of the markets are the information arrival (CCA) and processing rates (CCL) of the market (and by extension the market participants). As will be shown the value of the ratio of these rates (CCA/CCL) can determine different regimes of normal and “anomalous” behaviors for security returns. As this ratio evolves over a continuum of values, security returns can be expected to go through phase transitions between different types of behavior. These dramatic phase transitions can occur even when the underlying information generation mechanism is unchanged. Additionally when the information arrival and processing rates are assumed to fluctuate independently and normally, the resulting ratio (CCA/CCL) is shown to be Cauchy distributed and thus fat tailed.

This line of research actually suggests significantly different remedies to market instability compared to those currently utilized such as so called “circuit breakers”. The level and stability of the information processing rate CCL of the market (and market participants) turns out to be the most important variable in the model. Policies which increase this level and minimize the variance of the information processing rate will reduce the probability of occurrence and the ultimate severity of anomalous market fluctuations.

Non-normality, Subordination, and the Limits of Computation

One obvious and well documented feature of securities markets is the observed non-normality of market returns. The concept of trading time or subordinated Brownian motion as pioneered by (Mandelbrot and Taylor 1967), and (Clark 1973) provides a method of retrieving the normality assumption. The traditional calendar clock is replaced with stochastic time due to the random arrival of information at the market. Market activity in terms of the number and/or size of trades is often used as a proxy of the information arrival rate. After taking into account this stochastic changing of time or “time deformation” returns have been demonstrated to be approximately normality distributed for most return data.

Failure of the Subordination Hypothesis for Large Price Movements

The time deformation or subordination methods have been extensively studied and provide a useful explanation and remedy for a large bulk of the observed market non-normality. However, there is evidence that subordination fails to completely explain extreme price movements. (Silva 2005) determined that subordination only effectively explains the center (?85%) of price movements, and that the subordination hypothesis is rejected for returns that reach the 2 standard deviation level and above. While (Farmer et al. 2008) provides evidence that subordination does not explain large price movements in the London Stock Exchange.

This author argues that prior time deformation methods are missing an additional important factor which may affect the subordination process. The missing factor is the information processing ability of the market (and market participants) relative to the randomly arriving information. The decisive role of the rate information processing rate and its variance in the behavior of market returns is the subject the following analysis.

In the following section an alternative formulation of the standard subordination process is derived and its relation to the typical trade number and volume type time deformation process is presented. Afterwards the author shows how this alternative structure naturally generates different behaviors including the increased likelihood of extreme events. Finally the remedies suggested by this perspective to reduce the severity and probability of occurrence of these extreme events are discussed.

Flash Crashes, Market Instability

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