The Study On A Global Algorithm For Two-Period Financial Derivatives Liquidation Problem

The portfolio liquidation problem focuses on how to develop the optimal liquidation strategy that result in acceptable levels of leverage or generate sufficient cash flow with minimal transaction costs.According to the theory of market microstructure,investors need to consider price impacts when liquidating investment portfolios,which has prompted many scholars to study single period and two-period portfolio liquidation problems with market price impacts.Recently,the literature has proved that under the convex assumption that the temporary price influence dominates,the optimization model of the two-period financial derivatives robust liquidation problem is equivalent to a convex quadratic programming problem with semi-definite constraints.However,empirical research shows that this convex assumption does not always hold.This thesis aims to study the two-period financial derivatives liquidation problem without restricting the relationship between the magnitude of the temporary and permanent price impact parameters,whose optimization model is a quadratically constrained quadratic programming(QCQP)problem with linear and single nonconvex quadratic constraints.In this thesis,we combine successive convex optimization(SCO)approach and convex relaxation techniques to investigate the global optimization algorithm for the two-period financial derivatives liquidation problem and its numerical implementation.The following are the main results of this thesis:(1)We investigate the semi-definite programming relaxation method for the two-period financial derivatives liquidation problem.We first present a semi-definite programming(SDP)relaxation with secant cut for this nonconvex QCQP problem and estimate the gap between it and the original problem.The numerical results of the stochastic example show that this SDP relaxation can obtaion a tight upper bound to the original problem,thus providing a method for finding a good approximate solution to the original problem.(2)We investigate the successive convex optimization algorithm for the two-period financial derivatives liquidation problem and its global convergence.First,by means of the special structure of Hessian matrix in the objective and constraint functions of the problem,we transform the original problem into an equivalent DC(Difference of Convex function,DC)programming problem.We then obtain a quadratic convex approximation for the transformed problem by linearizing the concave quadratic terms in the objective and quadratic constraint functions.Second,based on the quadratic convex approximation problem,we propose a successive convex optimization algorithm for solving the transformed problem,and show that the solution sequence generated by the algorithm converges to a KKT point of the transformed problem.Finally,numerical experiments are conducted to verify the effectiveness of the SCO algorithm.(3)We develop a global optimization algorithm for the two-period financial derivatives liquidation problem and investigate its convergence.First,we present a quadratic convex relaxation for the equivalent transformed problem by replacing the concave quadratic term in the transformed problem with its convex envelope,and estimate the gap between the relaxation and the transformed problem.Second,we propose a global optimization algorithm for solving the transformed problem by combining the quadratic convex relaxation,the SCO method,and the branch-and-bound framework,which finds the global optimal solution of the transformed problem within a pre-specified -tolerance.We also establishes the global convergence of the algorithm and its complexity estimation.Finally,numerical results show that the global optimization algorithm can effectively find the global optimal solution of the two-period financial derivatives liquidation problem.