LQR and receding horizon approaches to multi-dimensional option hedging under transaction costs

In this paper we formulate the problem of dynamically hedging a basket option on multiple underlying stocks and in the presence of proportional transaction costs as a linear quadratic control problem subject to constraints. The linear structure is obtained by sampling over paths of the underlying stocks and linearly parameterizing control actions over basis functions. Two solutions are then proposed. The first involves quadratically penalizing transaction costs in the objective and allows the hedging problem to be solved as a standard unconstrained linear quadratic regulator problem. The second approach uses receding horizon control to solve a quadratic program over a specified prediction horizon, where the cost function utilizes the LQR solution from the first approach. A numerical example illustrates the methodology.