Optimal Liquidation Behaviour Analysis for Stochastic Linear and Nonlinear Systems of Self-Exciting Model with Decay

Jiangming Ma,Xiankang Luo,Yue Song

doi:10.1155/2021/8553689

Abstract

When the market environment changes, we extend the self-exciting price impact model and further analysis of investors’ liquidation behaviour. It is assumed that the model is accompanied by an exponential decay factor when the temporary impact and its coefficient are linear and nonlinear. Using the optimal control method, we obtain that the optimal liquidation behaviours satisfy the second-order nonlinear ODEs with variable coefficients in the case of linear and nonlinear temporary impact. Next, we solve the ODEs and get the form of the investors’ optimal liquidation behaviour in four cases. Furthermore, we prove the decreasing properties of the optimal liquidation behaviour under the linear temporary impact. Through numerical simulation, we further explain the influence of the changed parameters ρ , a , b , x , and α on the investors’ liquidation strategy X t in twelve scenarios. Some interesting properties have been found.

Highlights

Bertsimas and Lo [1] proposed the optimal execution model which has a linear and discrete price impact model in a fixed time. ey got the optimal liquidation behaviours by an optimal control method in some cases
We introduce the classical exponential decay factor in the selfexciting price impact model
In eorem 1, we address the optimal liquidation behaviour when there is a self-exciting model with exponential decay and give the specific form of optimal liquidation behaviour. at is, the coefficient of temporary impact is multiplied by the exponential function

Summary

Introduction

Bertsimas and Lo [1] proposed the optimal execution model which has a linear and discrete price impact model in a fixed time. ey got the optimal liquidation behaviours by an optimal control method in some cases. Almgren [3] further extended the previous model which is assumed that the temporary impact is nonlinear and got the form of the optimal liquidation behaviour. Based on the premise of no dynamic arbitrage, Gatheral [8] studied the optimal investment behaviour under the condition of market impact with decay factors. He gave the specific form of the optimal investment behaviour from the case of fixed and temporary market impact with different decay factors. Ey considered that a large sell order creates persistent selling pressure which incurs price impact and increases the execution costs They only discussed that the temporary impact is a linear function. The objective function of investors is minimize E[C(X)]

Main Results

Numerical Simulation

Conclusion