A model of optimal portfolio selection under liquidity risk and price impact

Abstract

We study a financial model with one risk-free and one risky asset subject to liquidity risk and price impact. In this market, an investor may transfer funds between the two assets at any discrete time. Each purchase or sale policy decision affects the price of the risky asset and incurs some fixed transaction cost. The objective is to maximize the expected utility from terminal liquidation value over a finite horizon and subject to a solvency constraint. This is formulated as an impulse control problem under state constraint and we prove that the value function is characterized as the unique constrained viscosity solution to the associated quasi-variational Hamilton-Jacobi-Bellman inequality.

A numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB)

“…As a result, a nonzero fixed cost c is introduced as a technical tool to distinguish these two cases and resolve the non-uniqueness problem. The nonzero fixed cost is commonly assumed in the impulse control literature [1,13,15,18,16].…”

“…There is a significant body of literature dealing with proofs of the strong comparison result for similar impulse control problems [1,12,15,18]. However, the proofs are very problem specific.…”

“…Refer to [13,15,18,16] for some applications of impulse control problems. This formulation has the advantage that it is easier to incorporate the complex features of real contracts, such as various reset provision features.…”

Smooth Fit Principle for Impulse Control of Multidimensional Diffusion Processes

“…This multi-dimensional control problem has been proposed and studied in various forms in different context of risk management, from optimal cash management [6] to inventory controls [14,13,34,33]. More recent papers in the literature of mathematical finance include those on transaction cost in portfolio management [2,19,20,9,25,28], insurance models [16,4], liquidity risk [22,3], optimal control of exchange rates [17,26,5], and finally real options [35,23].…”

Numerical methods for an optimal order execution problem

“…Models of market impact based on stylized order book dynamics were proposed in [15], [19] and [9]. There also has been several optimal control approaches to the order execution problem, using a penalizing function to model price impact: the papers [18] and [8] assume continuous-time trading, and use an Hamilton-Jacobi-Bellman approach for the mean-variance criterion, while [10], [13], and [11] consider real trading taking place in discrete-time by using an impulse control approach. This last approach combines the advantages of realistic modelling of portfolio liquidation and the tractability of continuoustime stochastic calculus.…”