Wellposedness of the boundary value formulation of a fixed strike Asian option

摘要

This work is the follow up to [J. Hugger, Numerical Mathematics and Advanced Applications—Enumath 2001, Springer, Italy, 2003] where a partial differential equation equivalent to the stochastic formulation for a fixed strike Asian option was derived.In the present work the differential equation is complemented with boundary value conditions that are derived from financial conditions.With the complete boundary value formulation thus recovered, wellposedness of the problem is adressed. It turns out that the problem takes the form of a degenerated parabolic boundary value problem with a second-order, linear, time-dependent PDE with non-negative characteristic form. Apart from the degeneracy in the PDE, also the boundary conditions (derived from the financial understanding) are “the wrong ones” or at least are non-standard. There are conditions on boundaries where none are expected to be needed bacause of the degeneracy and there are boundaries where conditions are expected to be needed but none can be found.