Anomalous diffusion and price impact in the fluid-limit of an order book

ArXiv ID: 2310.06079 “View on arXiv”

Authors: Unknown

Abstract

We extend a Discrete Time Random Walk (DTRW) numerical scheme to simulate the anomalous diffusion of financial market orders in a simulated order book. Here using random walks with Sibuya waiting times to include a time-dependent stochastic forcing function with non-uniformly sampled times between order book events in the setting of fractional diffusion. This models the fluid limit of an order book by modelling the continuous arrival, cancellation and diffusion of orders in the presence of information shocks. We study the impulse response and stylised facts of orders undergoing anomalous diffusion for different forcing functions and model parameters. Concretely, we demonstrate the price impact for flash limit-orders and market orders and show how the numerical method generate kinks in the price impact. We use cubic spline interpolation to generate smoothed price impact curves. The work promotes the use of non-uniform sampling in the presence of diffusive dynamics as the preferred simulation method.

Keywords: Discrete Time Random Walk (DTRW), Sibuya waiting times, Anomalous diffusion, Order book dynamics, Fractional diffusion, Equities

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper heavily employs advanced mathematical concepts like fractional diffusion, discrete time random walks, and stochastic forcing functions, which is reflected in the dense equations and theoretical derivations. However, it primarily focuses on theoretical model development and numerical simulation methods (e.g., DTRW scheme, cubic spline interpolation) without presenting backtested results, empirical data analysis, or implementation-heavy validation against real market data.

  flowchart TD
    A["Research Goal:<br>Model Fluid Limit of Order Book<br>with Anomalous Diffusion"] --> B["Data: Sibuya Waiting Times<br>Non-uniformly Sampled Events"]
    B --> C["Method: Discrete Time Random Walk<br>DTRW Numerical Scheme"]
    C --> D["Simulation: Anomalous Diffusion<br>with Fractional Forcing"]
    D --> E["Analysis: Impulse Response &<br>Stylised Facts"]
    E --> F["Outcomes:<br>1. Price Impact Kinks<br>2. Cubic Spline Interpolation<br>3. Method Efficacy"]