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

TL;DR

This paper develops a numerical scheme to simulate anomalous diffusion in order books, capturing complex order dynamics and price impacts, especially under non-uniform sampling and information shocks.

Contribution

It introduces a novel DTRW-based method incorporating Sibuya waiting times for modeling fractional diffusion in order books, enhancing simulation accuracy for anomalous diffusion.

Findings

Demonstrates price impact of flash and market orders with kinks in impact curves.

Shows the effectiveness of non-uniform sampling in simulating diffusive order book dynamics.

Provides insights into the stylized facts of anomalous diffusion in financial markets.

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.