A theory of power-law distributions in financial market fluctuations

Abstract

Insights into the dynamics of a complex system are often gained by focusing on large fluctuations. For the financial system, huge databases now exist that facilitate the analysis of large fluctuations and the characterization of their statistical behaviour. Power laws appear to describe histograms of relevant financial fluctuations, such as fluctuations in stock price, trading volume and the number of trades. Surprisingly, the exponents that characterize these power laws are similar for different types and sizes of markets, for different market trends and even for different countries--suggesting that a generic theoretical basis may underlie these phenomena. Here we propose a model, based on a plausible set of assumptions, which provides an explanation for these empirical power laws. Our model is based on the hypothesis that large movements in stock market activity arise from the trades of large participants. Starting from an empirical characterization of the size distribution of those large market participants (mutual funds), we show that the power laws observed in financial data arise when the trading behaviour is performed in an optimal way. Our model additionally explains certain striking empirical regularities that describe the relationship between large fluctuations in prices, trading volume and the number of trades.

Patterns in high-frequency FX data: discovery of 12 empirical scaling laws

“…In finance, there is one scaling law that has been widely reported (Müller et al [1990], Mantegna and Stanley [1995], Galluccio et al [1997], Guillaume et al [1997], Ballocchi et al [1999], , Corsi et al [2001], Di Matteo et al [2005]): the size of the average absolute price change (return) is scale-invariant to the time interval of its occurrence. This scaling law has been applied to risk management and volatility modelling (see Ghashghaie et al [1996], Gabaix et al [2003], Sornette [2000], Di Matteo [2007]) even though there has been no consensus amongst researchers for why the scaling law exists (e.g., Bouchaud [2001], Barndorff-Nielsen and Prause [2001], Farmer and Lillo [2004], Lux [2006], Joulin et al [2008]). …”

The Price Impact of Order Book Events

“…However, the linear equation (2) is quite different from models of price impact that consider only the size of trades [18,20,29,43,38,39]. Instead of modeling price impact of trades as a (nonlinear) function of trade size, we show that the price impact of all events (including trades) is a linear function of their size after events are aggregated into a single imbalance variable.…”

“…al [3]). Alternatively, empirical studies on public data [16,18,20,28,29,43,38,39] have investigated the relation between the direction and sizes of trades and price changes and typically conclude that the price impact of trades is an increasing, concave ("square root") function of their size. This focus on trades leaves out the information in quotes, which provide a more detailed picture of price formation [15], and raises a natural question: is volume of trades truly the best explanatory variable for price movements in markets where many quote events can happen between two trades?…”

Diversity of scales makes an advantage: The case of the Minority Game

“…We see that in absolute numbers the MG with heterogeneous impacts remains, of course, a negative-sum game, but subjectively felt relative profit of agents averaged over the whole system may be positive. This observation could give a cause for the heterogeneity of investors' sizes, which is one of the most fundamental features of the stock market [27]. Interpreting the utility function U (I) as an investor's subjective evaluation of the capital unit I, the notion of Π {U(I)} is a tool suitable for finer analysis of this phenomena.…”

“…The calculation of (27) is valid only in the ergodic phase of the MG as it was already examined in Ref. [9].…”