[Paper Review] A non-linear model of trading mechanism on a financial market

N. D. Vvedenskaya, Yuri Suhov+1 more|arXiv (Cornell University)|Jan 22, 2012

Advanced Queuing Theory Analysis10 references4 citations

TL;DR

This paper proposes a non-linear stochastic model of a limit order book using a continuous-time Markov process that models order arrivals, cancellations, price movements, and trades. Through a scaling limit, the model converges to a deterministic system of non-linear ODEs, yielding a unique equilibrium fixed point that characterizes the long-term behavior of market order flows and liquidity dynamics.

ABSTRACT

We introduce a prototype model in an attempt to capture some aspects of market dynamics simulating a trading mechanism. The model description starts with a discrete-space, continuous-time Markov process describing arrival and movement of orders with different prices. We then perform a re-scaling procedure leading to a deterministic dynamical system controlled by non-linear ordinary differential equations (ODEs). This allows us to introduce approximations for the equilibrium distribution of the model represented by fixed points of deterministic dynamics.

Motivation & Objective

• To develop a stylized, analytically tractable model of limit order book dynamics in financial markets with explicit order flow mechanisms.
• To analyze the asymptotic behavior of the system under large-market scaling, where the number of participants is large and individual actions are rare but impactful.
• To derive a deterministic approximation via scaling that captures equilibrium distributions through fixed points of non-linear ODEs.
• To explore the mathematical structure of the limit system, particularly the uniqueness and stability of the equilibrium state.
• To provide a foundation for understanding how parameters like order arrival rates, cancellation rates, and trade speed affect market liquidity and price formation.

Proposed method

• Model the market as a continuous-time, discrete-space Markov process with state variables representing buy and sell orders at N price levels.
• Define transition rates for key events: trades (at rate ρ_T), cancellations (ρ_Q), and price movements (ρ_M) for both buyers and sellers.
• Apply a diffusion-type scaling limit by rescaling time and space, assuming large market size and rare individual actions, leading to a deterministic ODE system.
• Derive the limiting system of non-linear ODEs governing the evolution of expected order volumes x_i(t) and y_i(t) for buy and sell sides.
• Use iterative fixed-point algorithms to compute the equilibrium solution (x*, y*) by solving a recursive system based on balance equations.
• Prove convergence of the finite-scale distributions to a delta measure concentrated at the unique fixed point under the scaling limit.

Experimental results

Research questions

• RQ1 How does the scaling limit of a stochastic limit order book model lead to a deterministic non-linear ODE system?
• RQ2 What conditions ensure the uniqueness of the equilibrium fixed point in the limiting deterministic system?
• RQ3 How do the parameters γ (trade speed), α_Q (cancellation rate), α_M (price movement rate), and λ_b/s (order arrival rates) affect the equilibrium order book shape?
• RQ4 What happens to market efficiency and liquidity when trade speed γ increases toward infinity?
• RQ5 How does the model capture phenomena such as overproduction crises, where excess sellers fail to trade due to lack of offsetting demand?

Key findings

• The scaling limit results in a unique fixed point (x*, y*) for the deterministic ODE system, implying convergence of the finite-scale process to a single equilibrium state.
• The equilibrium solution is computed via a convergent iterative algorithm that alternately updates buy and sell volume estimates based on local balance and trade interaction terms.
• When λ_s is large but x_i < y_i for all i, the system reaches a state where additional sellers do not increase trading volume, illustrating a market inefficiency or 'overproduction crisis'.
• As γ → ∞, trades occur instantaneously after order arrival, and the majority of trades concentrate at two adjacent price levels i₀ and i₀+1, with negligible order flow at distant levels.
• The model shows that the equilibrium distribution is insensitive to further increases in seller arrival rate beyond a certain threshold, due to the saturation of trade volume at the best bid-ask spread.
• The limiting system allows for straightforward generalizations, including state-dependent parameters, multi-level entry for new orders, and priority rules such as FCFS, enhancing its applicability to real market mechanisms.