Quantitative Model of Price Diffusion and Market Friction Based on Trading as a Mechanistic Random Process

We model trading and price formation in a market under the assumption that order arrival and cancellations are Poisson random processes. This model makes testable predictions for the most basic properties of markets, such as the diffusion rate of prices (which is the standard measure of financial risk) and the spread and price impact functions (which are the main determinants of transaction cost). Guided by dimensional analysis, simulation, and mean-field theory, we find scaling relations in terms of order flow rates. We show that even under completely random order flow the need to store supply and demand to facilitate trading induces anomalous diffusion and temporal structure in prices.

Figure 1

Schematic of the order-placement process. Stored limit orders are shown “stacked” along the price axis, with bids (buy limit orders) negative and offers (sell limit orders) positive. New limit orders are visualized as “falling” randomly onto the price axis. New offers can be placed at any price greater than the best bid, and new bids can be placed at any price less than the best offer. Limit orders can be removed by random spontaneous deletion or by market orders of the opposite sign.

Figure 2

The mean depth profile versus price in nondimensional coordinates $\widehat{n}=n{p}_c/N_c=n\delta /\alpha$ vs $\widehat{p}=p/p_c=2\alpha p/\mu$. The origin $p/p_c=0$ corresponds to the midpoint. We show three different values of the nondimensional granularity parameter: $ϵ=0.2$ (solid line), $ϵ=0.02$ (dashed line), $ϵ=0.002$ (dotted line), all with tick size $dp=0$.

Figure 3

The average price impact corresponding to the results in Fig. 2. The average instantaneous movement of the nondimensional midprice, $⟨dm⟩/p_c$ caused by an order of size $N/N_c=Nϵ/\sigma$. $ϵ=0.2$ (solid line), $ϵ=0.02$ (dashed line), $ϵ=0.002$ (dotted line).