Optimal Portfolio Liquidation in Target Zone Models and Catalytic Superprocesses

TL;DR

This paper develops a mathematical framework for optimal portfolio liquidation in target zone models, using catalytic superprocesses to solve the associated singular stochastic control problem with practical financial justifications.

Contribution

It introduces a novel approach employing catalytic superprocesses to solve portfolio liquidation problems in reflected diffusion models with barriers.

Findings

Solution of the control problem via catalytic superprocesses.

Application to reflected arithmetic and geometric Brownian motions.

Financial interpretation through discrete-time approximations.

Abstract

We study optimal buying and selling strategies in target zone models. In these models the price is modeled by a diffusion process which is reflected at one or more barriers. Such models arise for example when a currency exchange rate is kept above a certain threshold due to central bank intervention. We consider the optimal portfolio liquidation problem for an investor for whom prices are optimal at the barrier and who creates temporary price impact. This problem will be formulated as the minimization of a cost-risk functional over strategies that only trade when the price process is located at the barrier. We solve the corresponding singular stochastic control problem by means of a scaling limit of critical branching particle systems, which is known as a catalytic superprocess. In this setting the catalyst is a set of points which is given by the barriers of the price process. For the cases in which the unaffected price process is a reflected arithmetic or geometric Brownian motion with drift, we moreover give a detailed financial justification of our cost functional by means of an approximation with discrete-time models.