Numerical approximation for a superreplication problem under gamma constraints

Benjamin Bruder (1) , Olivier Bokanowski (2) , Stefania Maroso (3, 4) , Hasnaa Zidani (4, 5)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 LJLL - Laboratoire Jacques-Louis Lions
3 TOSCA - Simuler et calibrer des modèles stochastiques
4 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
5 OC - Optimisation et commande

We study a superreplication problem of European options with gamma constraints, in mathematical finance. The initially unbounded control problem is set back to a problem involving a viscosity PDE solution with a set of bounded controls. Then a numerical approach is introduced, inconditionnally stable with respect to the mesh steps. A generalized finite difference scheme is used since basic finite differences cannot work in our case. Numerical tests illustrate the validity of our approach.

Keywords: Super-replication problem; viscosity solution; numerical approximation; generalized finite difference scheme; monotone scheme; Howard's algorithm

Domains: Numerical Analysis [math.NA]

Benjamin Bruder, Olivier Bokanowski, Stefania Maroso, Hasnaa Zidani. Numerical approximation for a superreplication problem under gamma constraints. SIAM Journal on Numerical Analysis, 2009, 47 (3), pp.2289-2320. ⟨10.1137/080725222⟩. ⟨hal-00278077⟩