Penalty Methods for Free Boundary Problems

Overview

Free boundary problems in option pricing can be recast as Linear Complementarity Problems (LCPs). The penalty method replaces the LCP with a nonlinear PDE that is easier to solve numerically while maintaining convergence guarantees.

Linear Complementarity Formulation

The American option LCP reads: find V such that

where g = max(K−S, 0) is the payoff and L is the Black–Scholes operator. The three conditions enforce: the PDE holds in the continuation region, the option value dominates the payoff, and complementarity (at most one is active).

Penalty Reformulation

The penalty method approximates the LCP by:

As ε → 0, V_ε → V (the true LCP solution). For fixed ε, this is a nonlinear PDE solvable by Picard or Newton iteration.

Convergence Theory

Under appropriate conditions, the penalty solution converges at rate O(ε) to the LCP solution. Combined with second-order FDM discretization, the overall scheme achieves second-order convergence when ε is chosen proportional to Δt.