Monday, 8 April 2013

Analyzing financial interactions/markets as a complex network

Since most of the posts till now have been the characteristic trademark of the so-called 'techies' , I thought I will combine a new field with the analysis of complex networks . Financial interactions/markets in the world sure are a source of arousing curiosity of most of the tech-school students , though ephemerally .So here it goes .....
Economic research on network formation usually starts from the observation that social structure is important in a variety of interactions, including the transmission of information and the buying and selling of goods and services. Over the past decade, economists have started to investigate the interactions between network topologies and game-theoretic notions of stability and efficiency using the tools and jargon of graph theory. 
It is noteworthy that in a recent survey of economic models of network formation, the term degree distribution does not appear once, and it seems that the economics profession is only very recently becoming aware of the research by statistical physicists and computer scientists. As a consequence, it becomes very hard to evaluate whether the network structures emerging from various processes are efficient with respect to either the involved microscopic motives, or to the resulting and possibly unintended macroscopic regularities, or both .Interestingly, the particular economic (game-theoretic) setup does not seem to be crucial for such a trade-off between efficiency and stability. Chemical engineers have argued that generic network topologies, including polar cases like ‘stars,’‘circles,’ and ‘hubs,’ can be understood as the outcome of adaptive evolutionary processes that optimize the trade-off between efficiency and robustness in uncertain environments .The efficiency and robustness results were extended to show how power-law, exponential, and Poisson degree distributions can be understood as manifestations of the trade-off between efficiency and robustness within a unified theoretical framework based on the maximum entropy principle .Although degree distributions, small world effects, clustering measures, or other concepts from the statistical physics community have been ignored to a large extent by economists, I take the position that economically or behaviorally plausible mechanisms in the generation and evolution of empirically relevant network
structures should play an important role.

Financial time series are characterized by a number of statistical regularities in both their unconditional and conditional properties that are often termed ‘stylized facts.’ One property is the scaling law observed for large returns r .          
                                                        P(|r|>x)∼pow(x,−α) ,  
where α denotes the tail index .The ubiquitous nature of this power law decay has been documented in numerous financial data, covering different markets (stock markets, foreign exchange markets, commodity markets), different frequencies (from minute-to-minute to monthly data), and different countries. Moreover,empirical estimates lead to a universal value of α≈3, with a small interval of statistical variability between 2.5 and 4.
Taking into account the long tradition in the analysis of complex systems in statistical physics, scaling laws suggest to look at financial data as the result of social processes of a large ensemble of interacting sub-units.

 I start from the observation that multi-fractal models (i.e. models with a hierarchy of volatility components with different stochastic life times) describe the stylized facts of financial returns very well, and combine this with our aforementioned insight that the very large number of traders (called clients subsequently) in financial markets should be grouped into K structures, where K is moderate relative to the total number of clients. The multi-fractal model corresponds to a tree graph with a moderate K compared to the total number of agents .In social nets the average geodesic path length seems to be universal andrelatively small (the so-called “small world effect”). In our nets this distance is related to K , which determines the tail index. Thus I expect to explain the universality of the latter by that of K.The prediction of a hyperbolic decay of correlation in the multi-fractal model is only valid in a time window. Since this window changes according to with the time resolution, I hope to explain the transition from a power law for high frequency data to the complicated behavior of daily returns.The improvement of time behavior by including features of the multi-fractal model should lead to a substantially improved predictive power.

It might help to understand whether the intermittent character of financial returns as a signature of the ‘critical’ nature of market fluctuations serves to balance a lack of efficiency with the robustness to absorb large exogenous shocks (like 9/11) along the lines of the above literature. From an interdisciplinary point of view, such insights should be applicable as well to other phenomena that are characterized by qualitative differences among nodes in a network, like supply chains in industrial organization, or food webs in biological systems.

References :
(2)V. Bala and S. Goyal. A non-cooperative model of network formation.Econometrica, 68:1181–1230,2000 .
(3)E. Egenter, T. Lux, and D. Stauffer. Finite-size effects in Monte Carlo simulations of two stock market models.Physica A, 268:250–256, 1999
(4)M.I. Jordan.Learning in Graphical Models. MIT Press, Cambridge, 1998.
(5)T. Lux. Turbulence in financial markets: The surprising explanatory power of simple cascade models. Quantitative Finance, 1:632–640, 2005.

1 comment: