Understanding Financial Market States Using an Artificial Double Auction Market

The ultimate value of theories describing the fundamental mechanisms behind asset prices in financial systems is reflected in the capacity of such theories to understand these systems. Although the models that explain the various states of financial markets offer substantial evidence from the fields of finance, mathematics, and even physics, previous theories that attempt to address the complexities of financial markets in full have been inadequate. We propose an artificial double auction market as an agent-based model to study the origin of complex states in financial markets by characterizing important parameters with an investment strategy that can cover the dynamics of the financial market. The investment strategies of chartist traders in response to new market information should reduce market stability based on the price fluctuations of risky assets. However, fundamentalist traders strategically submit orders based on fundamental value and, thereby stabilize the market. We construct a continuous double auction market and find that the market is controlled by the proportion of chartists, Pc. We show that mimicking the real state of financial markets, which emerges in real financial systems, is given within the range Pc = 0.40 to Pc = 0.85; however, we show that mimicking the efficient market hypothesis state can be generated with values less than Pc = 0.40. In particular, we observe that mimicking a market collapse state is created with values greater than Pc = 0.85, at which point a liquidity shortage occurs, and the phase transition behavior is described at Pc = 0.85.


Supporting Information
The switching rules of agents' opinions In this paper, we apply the modified transition rules of agents' opinions which are 1 based on Lux and Marchesi(1999) [1]. The transition probabilities introduced in Lux 2 and Marchesi(1999) [1] denoted by π A,B ∆t. π A,B is the transition rate from type B to 3 type A. 4 π +− = v 1 n c N exp (U 1 ), π −+ = v 1 n c N exp (−U 1 ), where x = (n + − n − )/n c ,N, n c , n + , n − , n f denote the number of agents, chartists, 5 optimistic agents, pessimistic agents, and fundamentalists.p denotes the current 6 market price, and p f is the current fundamental value. The subscript +, −, f denotes 7 the agent type, e.g., optimist, pessimist, or fundamentalist. The mechanism that can 8 change the agent types in transition probability consists of a herding and profit 9 strategy. If the fraction of optimistic agents (n + /N ) which is related to herding 10 strategy increases in the market, the transition probabilities from other types to 11 optimistic agents(π +− , π +f ) increase. In addition, if the profit of optimistic agents increases, the transition probabilities from other types to optimistic agents increases. 13 The other types such as pessimists and fundamentalists also have similar behaviors.
14 To avoid absorbing state(n c = 0 or n f = 0) in simulation, we apply a rule that an 15 agent with an opinion in a population less than 0.8% in the total population can not 16 change her own opinion into other opinions. 17 We assume that the investment time horizon of each agent type is different.

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Chartists have shorter investment time horizon than fundamentalists. However, in the 19 transition probabilities in Lux and Marchesi(1999) [1], the heterogeneous investment 20 time horizon of an agent is not considered. In other words, dp/dt is calculated during 21 homogeneous investment time horizons. However, in the real financial market,

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investors have heterogeneous information sets. Each investor cannot help but have a 23 different strategies due to the differences between information sets that agents have.

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With the notion of heterogeneous information sets between agents, we modify this 25 term dp/dt, which can reflect the investment time horizon of heterogeneous agent 26 types. dp/dt is calculated by the average value of ∆p/∆t during agents' own chartists.

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For more details about other parameters in transition probabilities, see Lux and 33 Marchesi(1999) [1].

Stylized facts in Artificial Double Auction Market
We analyze ADAM-generated data and observe various 'stylized facts' that were 35 observed in previous empirical studies [2][3][4][5][6][7][8][9][10][11]. Fat-tails of market microstructure 36 quantities such as absolute return, bid-ask spread and first gap, which were widely 37 observed in previous empirical studies, are observed in the ADAM (Fig. 1). We fit tails 38 of the CDF (Cumulative Distribution Function) of market microstructure quantities 39 using a power-law function, y ∼ x −α and α is estimated by maximum likelihood 40 estimation [12]. The fitted results are summarized in Table 1 (ii) In each box, the integrated time series y(i) is fitted by a polynomial function, etc.), we could apply the l-order polynomial function for the fitting. Time series y(i) is 54 detrended by subtracting the local trend y f it (i) in each box, and we calculate the For a given box size n, the root mean square(rms) fluctuation function F (n) is 57 calculated as (iii) We repeat the above computation for box sizes n to find a relationship between 59 F (n) and n, F (n) ∼ n H .

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As a result, long memory is observed in volatility, the bid-ask spread, the volume 67 and the first gap in the ADAM. In the return case, no memory is observed. These 68 memory properties of the ADAM support previous previous empirical works [3,5,9,14]. 69 Fig. 3(a),(b) depicts the rms fluctuation functions F (n) of these quantities. The 70 results of the ADAM and previous empirical results are summarized in Table 2.

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The case of the homogeneous equilibrium market     Table 1 )|  Table 2.(c)(d). F(n) is the rms fluctuation function in the homogeneous equilibrium market. The Hurst exponents measured in this case are summarized in Table 3.