Wealth dynamics in a market with information asymmetries

Financial markets are usually investigated through time series such as prices and volumes that describe the behavior of the system as a whole. Such observables emerge from microeconomic interactions between market participants. Agent-based models have been utilized to shed light on this process. The model's ability to produce statistics frequently found in empirical data is evidence of some correspondence with real markets. Here, an agent-based market model is proposed. Different trader profiles with short- and long-term motivations are considered, and limitations on the agents' skills to manipulate information are inserted into the model. According to their profile and limitations, agents are rational. A differential equation approximation is employed to find the value to which the price converges and the timescale of this process. The relationship between agents' attributes and the evolution of their wealth is explored in different scenarios. Agent's average wealth was not significantly affected by information processing accuracy, but the standard deviation was. The increased risk is the main consequence of low accuracy. The model yielded price series with multifractal behavior and heavy-tailed return distributions, which are nontrivial statistics frequently observed in empirical series.

Figure 1

Differential equation approximation. Comparison of price evolution computed by the agent-based model (solid line) with its differential equation approximation (dashed line) for $n=101$ and $c_p=0.001$. The parameters $c_r\left(a\right)=1$ and $c_s\left(a\right)=0.01$ are the same for all agents. Values of $P_r\left(a\right)$ are distributed from 9 to 11 with step 0.02; here, they do not vary in time. Values of $\mathrm{\Delta }\left(a\right)$ are distributed between $-0.1$ and 0.1 with step 0.002. The curves show good agreement.

Figure 2

Three homogeneous sets. A total of 303 agents were grouped into three sets of 101 elements, within which the agents' parameters do not vary. In set 1, agents are driven only by the reference price $[c_r\left(a\right)=5$ and $c_s\left(a\right)=0]$ and, in set 2, only by the next step price $[c_r\left(a\right)=0$ and $c_s\left(a\right)=1]$; in set 3, both prices influence the agents $[c_r\left(a\right)=5$ and $c_s\left(a\right)=1]$. The initial price is $P\left(0\right)=10$. The reference price is the same for all agents affected by it $[P_r\left(a\right)=P_r]$. It starts at $P_r=10$ and at each time step can change with a probability of 0.04. The logarithm of such variation follows a Gaussian distribution: the new value is computed by multiplying the current one by $exp(\eta /100)$, where $\eta$ is a random variable with standard normal distribution. Agents guided by the price of the next step predict it accurately $\left[\mathrm{\Delta }\right(a)=0]$. The parameter that relates excess demand to price variation is $c_p=3\times {10}^{-4}$. The price dynamics is given in (a) and the corresponding returns in (b). In (c), we present the evolution of wealth variation for a typical agent of set 1 (blue line), set 2 (red line), and set 3 (black line). Singularity spectra are displayed in (d): blue dots correspond to the original series and red dots to the shuffled ones. The dots in (e) represent the inverse cumulative distribution of normalized returns; for comparison, the dashed line gives the Gaussian profile.

Figure 3

Uncertainty in the fundamental analysis. Market dynamics for $n=101$ agents driven by both fundamental analysis $\left[P_r\left(a\right)\right]$ and short-term expectations $\left[P_s\left(a\right)\right]$. The simulation was performed with $c_r\left(a\right)=1$ and $c_s\left(a\right)=0.01$ for all agents, and $c_p=0.001$. The initial price is $P\left(0\right)=10$. The reference prices take values equally spaced between $0.9P_{rc}$ (for the agent with index $a=1$) and $1,1P_{rc}$ (for the agent with index $a=101$), according to $P_r\left(a\right)=P_{rc}[0.9+(a-1)0.002]$, where $P_{rc}$ is a central price that begins at $P_{rc}=10$ and can change at each time step with probability 0.04. The logarithm of such variation follows a Gaussian distribution: the new value is computed by multiplying the current one by $exp(\eta /100)$, where $\eta$ is a random variable with standard normal distribution. The agents accurately predict the next step price $\left[\mathrm{\Delta }\right(a)=0]$. Price evolution is given in (a) and the corresponding returns in (b). In (c), we present the wealth variation in time for each agent. In (d), dots represent the mean of the wealth variation computed over the whole simulation period; the upper (lower) line displays the mean plus (minus) the standard deviation. Singularity spectra are displayed in (e): blue dots correspond to the original series and red dots to the shuffled ones. The dots in (f) represent the inverse cumulative distribution of normalized returns; for comparison, the dashed line gives the Gaussian profile.

Figure 4

Uncertainty in the short-term forecast. Market dynamics for $n=101$ agents driven by both fundamental analysis $\left[P_r\left(a\right)\right]$ and short-term expectations $\left[P_s\left(a\right)\right]$. The simulation was performed with $c_r\left(a\right)=1$ and $c_s\left(a\right)=0.01$ for all agents and $c_p=0.001$. The initial price is $P\left(0\right)=10$. The reference price is the same for all agents $[P_r\left(a\right)=P_r]$. It starts at $P_r=10$, and at each time step can change with a probability of 0.04. The logarithm of such variation follows a Gaussian distribution: the new value is computed by multiplying the current one by $exp(\eta /100)$, where $\eta$ is a random variable with standard normal distribution. The next time step price prediction is imperfect: $\mathrm{\Delta }\left(a\right)$ varies from $-0.1$ (for the agent with index $a=1$) to 0.1 (for the agent with index $a=101$ ), according to $\mathrm{\Delta }\left(a\right)=-0,1+0,002(a-1)$. Price evolution is given in (a) and the corresponding returns in (b). In (c), we present the wealth variation in time for each agent. In (d), dots represent the mean of the wealth variation computed over the whole simulation period; the upper (lower) line displays the mean plus (minus) the standard deviation. Singularity spectra are displayed in (e): blue dots correspond to the original series and red dots to the shuffled ones. The dots in (f) represent the inverse cumulative distribution of normalized returns; for comparison, the dashed line gives the Gaussian profile.

Figure 5

Uncertainty in both fundamental analysis and short-term forecast. A total of 303 agents were grouped into three sets of 101 elements. Set 1 consists of agents from $a=1$ to $a=101$, which consider both $P_r\left(a\right)$ and $P_s\left(a\right)$ $[c_r\left(a\right)=c_s\left(a\right)=1]$. In this set, the reference prices take values equally spaced between $0.9P_{rc}$ (for the agent with index $a=1$) and $1,1P_{rc}$ (for the agent with index $a=101$), according to $P_r\left(a\right)=P_{rc}[0.9+(a-1)0.002]$, where $P_{rc}$ is a central price that begins at $P_{rc}=10$ and can change at each time step with probability 0.04. The logarithm of such variation follows a Gaussian distribution: the new value is computed by multiplying the current one by $exp(\eta /100)$, where $\eta$ is a random variable with standard normal distribution. The next time step price prediction is imperfect: $\mathrm{\Delta }\left(a\right)$ varies from $-0.1$ (for the agent with index $a=1$) to 0.1 (for the agent with index $a=101$), according to $\mathrm{\Delta }\left(a\right)=-0,1+0,002(a-1)$. Agents from $a=102$ to $a=202$ integrate set 2, where agents are only influenced by $P_r\left(a\right)$ $[c_r\left(a\right)=1$ and $c_s\left(a\right)=0]$. The values of $P_r\left(a\right)$ in this group are the same as in set 1: $P_r(a+101)=P_r\left(a\right)$, for $a=1,2,...,101$. Set 3 is composed of agents from $a=203$ to $a=303$; they are driven just by $P_s\left(a\right)$ $[c_r\left(a\right)=0$ and $c_s\left(a\right)=1]$. The values of $\mathrm{\Delta }\left(a\right)$ are distributed similarly to those of set 1: $\mathrm{\Delta }(a+202)=\mathrm{\Delta }\left(a\right)$, for $a=1,2,...,101$. Price begins at $P\left(0\right)=10$; its evolution is governed by excess demand according to $c_p=3\times {10}^{-4}$. Price evolution is given in (a) and the corresponding returns in (b). In (c), we present the wealth variation in time for each agent. In (d), dots represent the mean of the wealth variation computed over the whole simulation period; the upper (lower) line displays the mean plus (minus) the standard deviation. Singularity spectra are displayed in (e): blue dots correspond to the original series and red dots to the shuffled ones. The dots in (f) represent the inverse cumulative distribution of normalized returns; for comparison, the dashed line gives the Gaussian profile.