Optimal control of European double barrier basket options

We consider European double barrier basket call options on two underlyings with an upper and a lower knock-out barrier featuring a finite number of cash settlements at prespecified values of the underlyings between the strike and the upper barrier. The bilaterally constrained cash settlements are considered as controls that have to be chosen such that the Delta of the option is as close as possible to a predefined constant profit/loss. This leads to a control constrained optimal control problem for the two-dimensional Black-Scholes equation with Dirichlet boundary control and finite time control. Based on the variational formulation of the problem in an appropriate Sobolev space setting, we prove the existence of a unique solution and state the first order necessary optimality conditions. A semi-discretization in space by conforming P1 finite elements with respect to a simplicial triangulation of the computational domain gives rise to a semi-discrete control constrained optimal control problem for a linear system of first order ordinary differential equations. A further discretization in time by the backward Euler scheme results in a fully discrete optimization problem that is solved numerically by the projected gradient method with Armijo line search. Numerical examples for some selected test cases illustrate the benefits of hedging with European double barrier basket options in case of optimally controlled cash settlements.