Anomalous Impact in Reaction-Diffusion Financial Models

Abstract

We generalize the reaction-diffusion model A+B→0 in order to study the impact of an excess of A (or B) at the reaction front. We provide an exact solution of the model, which shows that the linear response breaks down: the average displacement of the reaction front grows as the square root of the imbalance. We argue that this model provides a highly simplified but generic framework to understand the square-root impact of large orders in financial markets.

Slow Decay of Impact in Equity Markets

“…4, 0.7]. Theoretical models allowing one to understand the mechanisms underlying this strongly concave, nonintuitive impact function, have been put forward in [6,9,11,12] (on a related topic, see also [13]). Still, whereas the description of the contemporaneous impact of orders in terms of Eq.…”

The Square-Root Impace Law Also Holds for Option Markets

“…intended orders to buy (bid) and sell (ask), diffuse on a line and disappear whenever they meet, corresponding to a transaction. The boundary between the B-rich region and the A-rich region therefore corresponds to the price p t [13,14]. This highly stylized model can be argued to capture the essential features of the liquidity dynamics in financial markets, at least on long enough time scales where the precise microscopic details of the model become irrelevant (see Refs.…”

“…This highly stylized model can be argued to capture the essential features of the liquidity dynamics in financial markets, at least on long enough time scales where the precise microscopic details of the model become irrelevant (see Refs. [13][14][15] for more insights). In fact, the mechanism leading to a V-shaped liquidity is so robust that one expects to observe an approximate square-root law in a very wide range of situations -as alluded to above.…”

“…The fact that impact in option markets is fully compatible with all previously published results [2][3][4][5][6][7][8][9][10], including Bitcoin [12], bolsters the hypothesis that the square-root impact is universal across all traded markets, and vindicates the idea of a locally linear latent liquidity profile around the traded price, as advocated in Refs. [6,[13][14][15].…”

An introduction to econophysics and quantitative finance

“…The simplest description of the latent order book, proposed in [38], is provided by a model in which particles (orders) of two types B and A (respectively bid, for buy orders, and ask, for sell orders) sit on two opposite sides of a discrete price grid of length L, whose coordinates will be indexed by x, similar to the setting considered in [7,46]. The mid-point among the rightmost bid-particle and the left-most ask particle will be defined as the price, and will be denoted by x t .…”