Goal-based portfolio selection with fixed transaction costs
Erhan Bayraktar, Bingyan Han, Jingjie Zhang
TL;DR
The work addresses multi-goal, finite-horizon portfolio optimization under fixed transaction costs by formulating a coupled quasi-variational inequality (QVI) system for the value-function array $\{V_k(t,x)\}$. It deploys stochastic Perron’s method to prove that this array is the unique viscosity solution, and shows the existence of an optimal impulse trading and goal-funding strategy. The analysis yields explicit constructions of upper and lower stochastic envelopes and proves a comparison principle, ensuring existence and uniqueness without requiring a dynamic programming principle a priori. Numerical results reveal intricate continuation regions and demonstrate that fixed costs drive trading decisions away from the frictionless $V$-shaped policy, with optimal funding ratios exhibiting substantial variability and dependence on goal weights and deadlines.
Abstract
We study a goal-based portfolio selection problem in which an investor aims to meet multiple financial goals, each with a specific deadline and target amount. Trading the stock incurs a strictly positive transaction cost. Using the stochastic Perron's method, we show that the value function is the unique viscosity solution to a system of quasi-variational inequalities. The existence of an optimal trading strategy and goal funding scheme is established. Numerical results reveal complex optimal trading regions and show that the optimal investment strategy differs substantially from the V-shaped strategy observed in the frictionless case.