American option pricing problem transformed on finite interval

Abstract

We study the American option pricing linear complementarity problem (LCP), transformed on finite interval as it is initially defined on semi-infinite real axis. We aim to validate this transformation, investigating the well-posedness of the resulting problem in weighted Sobolev spaces. The monotonic penalty method reformulates the LCP as a semi-linear partial differential equation (PDE) and our analysis of the penalized problem results in uniform convergence estimates. The resulting PDE is further discretized by a fitted finite volume method since the transformed partial differential operator degenerates on the boundary. We show solvability of the semi-discrete and fully discrete problems. The Brennan–Schwarz algorithm is also presented for comparison of the numerical experiments, given in support to our theoretical considerations.

Keywords:

• linear complementarity problem
• degenerate parabolic operator
• monotonic penalty method
• fitted finite volume method

2010 AMS Subject Classifications:

• 49J40
• 65K15
• 65M08