Computational Algorithms for Vertical Complementarity Arising in Finance

We consider efficient computational algorithms for vertical complementarity problems. Vertical complementarity represents the equilibrium relationship among functions such that min (F1(x),...,Fm(x))=0 . This form is more general than the ordinary complementarity relationship, min (x, F(x))=0 . We consider an application in finance in terms of an option-hedging problem under transaction costs formulated as a singular stochastic control problem. This is expressed as a quasi-variational inequality. It is fully nonlinear and non-differentiable and belongs to a class of multi-dimensional free boundary problems equivalent to a vertical complementarity problem. In order to solve the quasi-variational inequality, alternative formulations are investigated. In addition, efficient numerical schemes are considered to provide a numerical solution.