The Degree of Stability of Price Diffusion

The distributional form of financial asset returns has important implications for the theoretical and empirical analyses in economics and finance. It is now a well-established fact that financial return distributions are empirically nonstationary, both in the weak and the strong sense. One first step to model such nonstationarity is to assume that these return distributions retain their shape, but not their localization (mean ) or size (volatility ) as the classical Gaussian distributions do. In that case, one needs also to pay attention to skewedness and kurtosis, in addition to localization and size. This modeling requires special Zolotarev parametrizations of financial distributions, with a four parameters, one for each relevant distributional moment. Recently popular stable financial distributions are the Paretian scaling distributions, which scale both in time T and frequency . For example, the volatility of the lognormal financial price distribution, derived from the geometric Brownian asset return motion and used to model Black-Scholes (1973) option pricing, scales according to T^{0.5}. More generally, the volatility of the price return distributions of Calvet and Fisher's (2002) Multifractal Model for Asset Returns (MMAR) scales according to T^{(1/(_{Z}))}, where the Zolotarev stability exponent _{Z} measures the degree of the scaling, and thus of the nonstationarity of the financial returns. Keywords: Stable distributions, price diffusion, stability exponent, Zolotarev parametrization, fractional Brownian motion, financial markets.Stable distributions, price diffusion, stability exponent, Zolotarev parametrization, fractional Brownian motion, financial markets

Fluctuation Studies at the Subnuclear Level of Matter: Evidence for Stability, Stationarity and Scaling

It is pointed out that the concepts and methods introduced by Bachelier and by Mandelbrot to Finance and Economics can be used to examine the fluctuations observed in high-energy hadron production processes. Theoretical arguments and experimental evidences are presented which show that the relative variations of hadron-numbers between successive rapidity intervals are non-Gaussian stable random variables, which exhibit stationarity and scaling. The implications of the obtained results are discussed.Comment: 34 Pages, 15 Postscript figures added references; corrected typos; changed conten

A critical look at power law modelling of the Internet

This paper takes a critical look at the usefulness of power law models of the Internet. The twin focuses of the paper are Internet traffic and topology generation. The aim of the paper is twofold. Firstly it summarises the state of the art in power law modelling particularly giving attention to existing open research questions. Secondly it provides insight into the failings of such models and where progress needs to be made for power law research to feed through to actual improvements in network performance.Comment: To appear Computer Communication

The Random Walk of High Frequency Trading

This paper builds a model of high-frequency equity returns by separately modeling the dynamics of trade-time returns and trade arrivals. Our main contributions are threefold. First, we characterize the distributional behavior of high-frequency asset returns both in ordinary clock time and in trade time. We show that when controlling for pre-scheduled market news events, trade-time returns of the highly liquid near-month E-mini S&P 500 futures contract are well characterized by a Gaussian distribution at very fine time scales. Second, we develop a structured and parsimonious model of clock-time returns by subordinating a trade-time Gaussian distribution with a trade arrival process that is associated with a modified Markov-Switching Multifractal Duration (MSMD) model. This model provides an excellent characterization of high-frequency inter-trade durations. Over-dispersion in this distribution of inter-trade durations leads to leptokurtosis and volatility clustering in clock-time returns, even when trade-time returns are Gaussian. Finally, we use our model to extrapolate the empirical relationship between trade rate and volatility in an effort to understand conditions of market failure. Our model suggests that the 1,200 km physical separation of financial markets in Chicago and New York/New Jersey provides a natural ceiling on systemic volatility and may contribute to market stability during periods of extremely heavy trading