Numerical analysis of the Minimal and Two-Liquid models of the Market Microstructure

We present results of numerical analysis of several simple models for the microstructure of a double auction market without intermediaries which were introduced in [10]. These markets can be represented as a set of buyers and a set of sellers, whose numbers vary in time, and who diffuse in price space and interact through an annihilation interaction. In this paper two models suggested in [10] are studied the minimal model and the twoliquid model. We perform computer simulations of the minimal model in order to verify [email protected] [email protected] [email protected] three of the liquidity scaling laws postulated in [10]. It is found that midmarket variance, bid-offer spread, and fluctuation of the bid-offer spread scale according to D/J where D is the diffusion coefficient (trader volatility) and J is the deal rate. A logarithmic correction to the scaling law for midmarket variance is observed, but not for bid-offer spread or its fluctuation, because they are fundamentally different quantities. Scaling parameters are obtained. We show both analytically and numerically that the total number of traders in the market scales as JL/4D, where L is the width of a price space. Time to midmarket sale (τS) is found to scale as 1/J while its fluctuation goes as 0.73/J . A “reduced” time (τreduced) is also studied, and found to scale in a non-trivial way. Asymmetric fluxes are introduced to the minimal model and analytical result derived in [10], for the speed of the moving midmarket agrees with numerical results. Simulation of the two-liquid model which describes a market with both market order and limit order traders, reveals widening of the bid-offer spread when the flux of market order traders exceeds that of limit order traders. The variation of the spread with the fraction of market-order traders is investigated. The formula for asymmetric fluxes is applied to the two-liquid model and its predictions are found to agree with experiment. The critical point is approximately determined, and the ratio of the midmarkets for f = 0.0 and f = 0.5 (where f is the fraction of market-order traders) is calculated.