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Register a new userBidirectional bond percolation model for the spread of information in financial markets
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Information is a key component in determining the price of an asset in financial markets, and the main objective of this paper is to study the spread of information in this context. The network of interactions in financial markets is modeled using a Galton-Watson tree where vertices represent the traders and where two traders are connected by an edge if one of the two traders sells the asset to the other trader. The information starts from a given vertex and spreads through the edges of the graph going independently from seller to buyer with probability $p$ and from buyer to seller with probability $q$. In particular, the set of traders who are aware of the information is a (bidirectional) bond percolation cluster on the Galton-Watson tree. Using some conditioning techniques and a partition of the cluster of open edges into subtrees, we compute explicitly the first and second moments of the cluster size, i.e., the random number of traders who learn about the information. We also prove exponential decay of the diameter of the cluster in the subcritical phase.
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A self-organized model with social percolation process is proposed to describe the propagations of information for different trading ways across a social system and the automatic formation of various groups within market traders. Based on the market structure of this model, some stylized observations of real market can be reproduced, including the slow decay of volatility correlations, and the fat tail distribution of price returns which is found to cross over to an exponential-type asymptotic decay in different dimensional systems.
The Glosten-Milgrom model describes a single asset market, where informed traders interact with a market maker, in the presence of noise traders. We derive an analogy between this financial model and a Szilard information engine by {em i)} showing that the optimal work extraction protocol in the latter coincides with the pricing strategy of the market maker in the former and {em ii)} defining a market analogue of the physical temperature from the analysis of the distribution of market orders. Then we show that the expected gain of informed traders is bounded above by the product of this market temperature with the amount of information that informed traders have, in exact analogy with the corresponding formula for the maximal expected amount of work that can be extracted from a cycle of the information engine. This suggests that recent ideas from information thermodynamics may shed light on financial markets, and lead to generalised inequalities, in the spirit of the extended second law of thermodynamics.
We study the stationary distribution of the (spread-out) $d$-dimensional contact process from the point of view of site percolation. In this process, vertices of $mathbb{Z}^d$ can be healthy (state 0) or infected (state 1). With rate one infected sites recover, and with rate $lambda$ they transmit the infection to some other vertex chosen uniformly within a ball of radius $R$. The classical phase transition result for this process states that there is a critical value $lambda_c(R)$ such that the process has a non-trivial stationary distribution if and only if $lambda > lambda_c(R)$. In configurations sampled from this stationary distribution, we study nearest-neighbor site percolation of the set of infected sites; the associated percolation threshold is denoted $lambda_p(R)$. We prove that $lambda_p(R)$ converges to $1/(1-p_c)$ as $R$ tends to infinity, where $p_c$ is the threshold for Bernoulli site percolation on $mathbb{Z}^d$. As a consequence, we prove that $lambda_p(R) > lambda_c(R)$ for large enough $R$, answering an open question of Liggett and Steif in the spread-out case.
We present results on simulations of a stock market with heterogeneous, cumulative information setup. We find a non-monotonic behaviour of traders returns as a function of their information level. Particularly, the average informed agents underperform random traders; only the most informed agents are able to beat the market. We also study the effect of a strategy updating mechanism, when traders have the possibility of using other pieces of information than the fundamental value. These results corroborate the latter ones: it is only for the most informed player that it is rewarding to stay fundamentalist. The simulations reproduce some stylized facts of tick-by-tick stock-exchange data and globally show informational efficiency.
Consider an anisotropic independent bond percolation model on the $d$-dimensional hypercubic lattice, $dgeq 2$, with parameter $p$. We show that the two point connectivity function $P_{p}({(0,dots,0)leftrightarrow (n,0,dots,0)})$ is a monotone function in $n$ when the parameter $p$ is close enough to 0. Analogously, we show that truncated connectivity function $P_{p}({(0,dots,0)leftrightarrow (n,0,dots,0), (0,dots,0) leftrightarrowinfty})$ is also a monotone function in $n$ when $p$ is close to 1.
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