ElliottAgents Market Simulation

• ElliottAgents is a multi-agent framework that simulates limit order book dynamics at microscopic time scales using stochastic, information-driven agent behavior.
• It employs a multiplicative decomposition of market sentiment, blending public metrics and private noise to drive non-utility-based trading decisions.
• The model reproduces key empirical phenomena such as fat-tailed returns, volatility clustering, and realistic order book liquidity profiles.

ElliottAgents is a multi-agent computational framework developed primarily for simulating and analyzing the microstructure and dynamical properties of limit order books (LOBs) in double auction markets at microscopic time scales. The distinguishing feature of ElliottAgents lies in its agent-based modeling paradigm, which eschews utility optimization in favor of stochastic, information-driven decision processes, and in its ability to reproduce numerous nontrivial empirical stylized facts observed in high-frequency financial data (Bartolozzi, 2010). At its core, the system embodies agents whose individual trading actions are dictated by a noise-generated “market sentiment,” representing a composite of filtered public and private informational states. The resulting collective agent behavior gives rise to realistic market microstructure, including intermittent price dynamics, volatility clustering, and return autocorrelations characteristic of real-world exchanges.

1. Agent-Based Decision-Making and Market Sentiment

ElliottAgents instantiate agents whose activity at each time step is controlled by an idiosyncratic “market sentiment” variable, $\varphi_i(t)$. This sentiment is a multiplicative mixture:

$$\varphi_i(t) = \phi_0 \cdot \kappa_i \cdot \psi^*(t) \cdot \eta(t) \cdot \varepsilon_i(t)$$

where:

• $\phi_0$ is a global activity scaling parameter;
• $\kappa_i$, uniformly sampled in $[0.25, 0.75]$, encodes agent-specific risk aversion;
• $\psi^*(t) = 1 - \psi(t)$ is a volatility risk modulator governed by a cancellation process;
• $\eta(t)$ reflects instantaneous liquidity risk as the ratio of current limit orders to agent population;
• $\varepsilon_i(t) = \nu(t) \cdot G(0,1)$ is the private Gaussian information, scaled by the estimated volatility $\nu(t)$.

The observed state of the LOB is filtered via an exponential moving average to derive public statistics, particularly volatility. The action probability for agent $i$ is passed through a sigmoidal transfer: $p_i(t) = (2/\pi)|\arctan [\varphi_i(t)]|$, and the sign of $\varphi_i(t)$ determines bid/ask direction. This construction yields a “zero-intelligence plus memory” decision rule, as agents react to stochastic signals, modulated by market-derived memory variables.

2. Mixing of Public and Private Information

The architecture of ElliottAgents decomposes market sentiment as the overlap of filtered public information with private informational noise. The system applies an exponential moving average:

$$x(t) = \frac{1}{L} x(t) + \left(1 - \frac{1}{L} \right) x(t-1)$$

to public variables such as mid-price, and perceived volatility is computed as:

$$\nu(t) = \sqrt{\text{EMA}(r^2(t))}, \quad r(t) = P_m(t) - P_m(t-1)$$

The agent’s private forecast is generated per step, as $\varepsilon_i(t) = \nu(t) \cdot G(0,1)$. The sentiment variable combines these signals multiplicatively. This approach—notably different from mean-based additive mixing—enables the emergence of complex stochastic, nonlinear dependencies, accounting for both exogenous market conditions and endogenous agent bias in every decision step.

3. Order Generation, Submission, and Execution

Agents in ElliottAgents are permitted a single open position and must liquidate before re-entering the LOB, precluding explicit market making and focusing the model on position/inventory-based microstructure. Order prices and sizes are stochastically generated:

• Buy price: $P_{\ell} = P_b - (\xi - Q_P)$
• Sell price: $P_s = P_a + (\xi - Q_P)$

where $P_b$/$P_a$ denote prevailing bid/ask, and $\xi$ is log-normally distributed (with median subtraction for symmetry). Volumes are similarly sampled and bounded. If a submitted order crosses the LOB, it is fully executed as a market order; otherwise, it queues as a limit order (FIFO at same price). The random, threshold-based order generation reflects a probabilistic impatience and allows the simulation of LOB reshaping by non-utility traders sensitive to volatility and liquidity.

4. Limit Order Book Microstructure and Dynamics

The LOB is simulated as a continuous double auction. Order matching strictly follows FIFO at each price level, with timeouts and volatility-sensitive cancellations implemented. The dynamic cancellation probability:

$$\psi(t) = 1 - e^{-\gamma \cdot \nu(t)}, \quad \gamma = 0.02$$

mirrors increased strategic order withdrawal under high perceived volatility. Quantities tracked include spread time series, volume imbalance:

$$\Delta V(t) = \sum_{i \in b} V^b_i(t) - \sum_{j \in a} V^a_j(t)$$

and the distribution of book volume as a function of ticks from the mid-price. These dynamical measurements capture liquidity barriers, order clustering, and price impact asymmetry, central to LOB theory.

5. Empirical Stylized Facts and Model Validation

The model reproduces several empirical microstructural signatures:

• One-step returns are leptokurtic with intermittent bursts, exhibiting “fat tails” (decay between exponential and Gaussian).
• Strong negative 1-lag autocorrelation (bid-ask bounce), confirmed by both autocorrelation and detrended fluctuation analysis (DFA); Hurst exponent for returns is approximately 0.495–0.5.
• Pronounced volatility clustering; absolute returns persist over hundreds of steps (DFA $H \approx 0.61$).
• Traded volume also exhibits clustering and long-memory (DFA $H \approx 0.70$).
• The impact function is linear for small volumes, but nonlinearity arises for large trades due to emergent volume barriers.
• Simulated average LOB shape resembles empirical order book profiles—high volume at a few ticks from the mid then a power-law or exponential decay.
• Spread and imbalance series display both persistence and mean-reverting structure.

Numerical validation relies on kurtosis, autocorrelation computation, and DFA for Hurst exponent estimation. Simulations with $N = 10^4$ agents over $10^5$ time steps demonstrate reproducibility of these facts in the absence of explicit agent rationality.

6. Conceptual and Practical Implications

The ElliottAgents methodology demonstrates that a high-fidelity reproduction of complex LOB dynamics does not require optimizing agents; rather, noise-driven, memory-modulated decision rules are sufficient for emergent market structure and volatility dynamics. Liquidity and volatility risk, filtered through agents’ internal stochastic processing, endogenously drive microstructure phenomena previously attributed to higher-level strategic reasoning.

This approach substantiates theories that key characteristics of market microstructure—return fat-tails, volatility and volume clustering, order book shape, non-linear price impact—can emerge from elementary stochastic trader behavior moderated by filtered public signals, as opposed to strategic equilibrium. The explicit modeling of cancellation as a volatility-driven stochastic process provides a mechanism for decoupling liquidity from order flow, crucial for understanding market resilience.

7. Relation to Other Agent-Based and Economic Frameworks

ElliottAgents’ framework aligns with modern probabilistic agent-based economic models that view agents as hierarchically structured, path-dependent stochastic processes embedded in evolving information networks (Theodosopoulos, 2013). It supports the thesis that long-term market equilibria and observable properties may reflect history-dependent, non-ergodic collective dynamics rather than static utility maximization. Such agent architectures underlie much of contemporary econophysics and microstructure modeling, positioning ElliottAgents as both a practical simulator and a theoretical tool to explore path-dependent equilibria in financial markets.

Conclusion

ElliottAgents is a multi-agent market simulation framework that bridges stochastic agent modeling and empirical market microstructure research. By operationalizing market sentiment as a multiplicative interaction of filtered public information and individual random forecast, and by representing agent actions with noise-driven statistical rules, the system furnishes a synthetic experimental platform that matches salient statistical properties of real high-frequency market data. The model’s validation through return, volatility, and volume statistics, along with realistic LOB liquidity profiles, underpins its utility for both theoretical microstructure analysis and design of robust agent-based market simulators.