Worst-Case Portfolio Optimization with Proportional Transaction Costs

Abstract

We study optimal asset allocation in a crash-threatened financial market with proportional transaction costs. The market is assumed to be in either a normal state, in which the risky asset follows a geometric Brownian motion, or in a crash state, in which the price of the risky asset can suddenly drop by a certain relative amount. We only assume the maximum number and the maximum relative size of the crashes to be given and do not make any assumptions about their distributions. For every investment strategy, we identify the worst-case scenario in the sense that the expected utility of terminal wealth is minimized. The objective is then to determine the investment strategy which yields the highest expected utility in its worst-case scenario.We solve the problem for utility functions with constant relative risk aversion using a stochastic control approach. We characterize the value function as the unique viscosity solution of a second-order nonlinear partial differential equation. The optimal strategies are characterized by time-dependent free boundaries which we compute numerically. The numerical examples suggest that it is not optimal to invest any wealth in the risky asset close to the investment horizon, while a long position in the risky asset is optimal if the remaining investment period is sufficiently large.

Finite-horizon optimal investment with transaction costs: construction of the optimal strategies

“…Davis et al [24] (with some adaptations) and Belak et al [9,10] establish that the value function V is continuous and the unique viscosity solution of the HJB equation…”

On the Uniqueness of Unbounded Viscosity Solutions Arising in an Optimal Terminal Wealth Problem with Transaction Costs

“…The proof of the following proposition can be found in Davis et al [11] (in a slightly different context), or can be established along the lines of Shreve and Soner [25,Theorem 7.7]. More details can also be found in Belak et al [3] and Belak [2], who consider a more general setting. Note, however, that the value function needs to be continuous for all these lines of arguments.…”