Hedging with European Double Barrier Basket Options as a Control Constrained Optimal Control Problem

In finance, hedging strategies are used to safeguard portfolios against risk associated with financial derivatives such as options. For an option with an underlying asset, the risk can be measured in terms of the so-called Greeks. In particular, the derivative of the option price with respect to the value of the asset is referred to as the Delta. An alternative to optimize hedges for options is to optimize options for hedging. Here, we are concerned with European double barrier basket options with multiple cash settlements. The cash settlements are considered as controls and the objective is to choose the controls such that the Delta is as close to a constant as possible. This amounts to the solution of a control constrained optimal control problem for the multidimensional Black Scholes equation featuring Dirichlet boundary control and final time control. We prove existence and uniqueness of the optimal control and derive the first order necessary optimality conditions in terms of the state, the adjoint state, and the control. The numerical solution is based on a discretization in space by P1 conforming finite elements with respect to a simplicial triangulation of the spatial domain and a further discretization in time by the implicit Euler scheme with respect to a partition of the time interval. The fully discretized optimal control problem is then solved by a projected gradient method with Armijo line search. Numerical results are given to illustrate the performance of the suggested approach.