Solvable Stochastic Dealer Models for Financial Markets

arXiv:0809.0481v2 [q-fin.TR] 20 Sep 2008 Solvable Stochastic Dealer Models for Financial Markets Kenta Yamada1,∗Hideki Takayasu2, Takatoshi Ito3, and Misako Takayasu1 1Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama 226-8502, Japan 2Sony Computer Science Laboratories, 3-14-13 Higashi-Gotanda, Shinagawa-ku, Tokyo 141-0022, Japan and 3Faculty of Economics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-0033, Japan We introduce solvable stochastic dealer models, which can reproduce basic empirical laws of financial markets such as the power law of price change. Starting from the simplest model that is almost equivalent to a Poisson random noise generator, the model becomes fairly realistic by adding only two effects, the self-modulation of transaction intervals and a forecasting tendency, which uses a moving average of the latest market price changes. Based on the present microscopic model of markets, we find a quantitative relation with market potential forces, which has recently been discovered in the study of market price modeling based on random walks. PACS numbers: 02.50.Ey Stochastic processes, 05.40.Jc Brownian motion, 89.65.Gh Economics; econo- physics, financial markets, business and management 1. INTRODUCTION Research on financial markets using methods and con- cepts developed in physics has increased considerably over the last decade. Various kinds of stylized facts or empirical laws of markets have been discovered from high precision market data of gigantic size [1][2][3][4][5]. The next goal of this econophysics study is to attempt to es- tablish the reasons for these empirical findings. Just as with the Boyle-Charles’ macroscopic law which can be derived from a simple microscopic ideal-gas model, we hope to construct a simple microscopic model of a market that can reproduce major empirical findings. By relating macroscopic market behavior to microscopic dealers’ ac- tions, we may find a pathway to control the markets, so as to avert bubbles and crashes, which occasionally cause problems in the market. The study of modeling dealers’ action is carried out with so-called agent-based models. This approach is sup- ported not only by economists but also by information scientists and physicists [6][7][8]. Agent-based models can in practice reproduce dealers’ actions in the mar- ket and they can also reflect empirical laws of markets to some extent. However, agent models generally include a huge number of parameters, and it has proved difficult to understand the relation between the parameters of the model and resulting market behavior. In order to find relationships between the parameters of dealers’ actions and market behavior, we have already introduced a kind of minimal model of an agent-based market which consists of dealers with simple determinis- tic time evolution rules [9][10][11]. With this model, we successfully reproduced most of the basic empirical laws using a minimal number of parameters, and found that ∗E-mail: yamada@smp.dis.titech.ac.jp there are only three important effects needed to repro- duce the empirical laws. The first effect is the compro- mise pricing of both buyers and sellers, who tend to allow the particular transaction price they have in mind to ap- proach the current market price in order to make a deal. From this effect, transactions occur spontaneously in the market and the price rises and falls almost randomly. The second effect is the self-modulation of transaction intervals, that is, the rate of a dealer’s clock depends on the latest moving average value of transaction intervals. When market activity becomes high, dealers accelerate their transaction rates, and by this effect we can repro- duce empirical statistical properties of transaction inter- vals which deviate from a simple Poisson process. The third is the trend-follow effect, that is, dealers forecast upcoming prices using the latest market trend which is defined by a moving average of price changes. This fore- casting effect makes the price change distribution follow a power law quite similar to that of the real market. In this paper we first introduce a stochastic version of the dealer model which is even simpler than the above (deterministic) model. In the case of the deterministic dealer model we needed at least three dealers to repro- duce market properties; however, in the present stochas- tic model we require only two. The advantages of this stochastic model are not only its simplicity but also its solvability by analytical calculation. In the usual agent- based approaches intensive numerical simulation is the only way to obtain results; in such cases exact or strict re- sults are rarely obtained. Based on this stochastic dealer model and its variants we can derive the major empirical results mentioned above, that have already been obtained by simulation of the dete

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