Free Boundary Problems

In financial derivatives, free boundary problems refer to mathematical models where the domain of the solution is not fixed but depends on the solution itself. In the context of American options, the optimal exercise boundary is a classic example of a free boundary.

The holder must decide when to exercise the option to maximize its value, creating a boundary that separates the region where holding the option is optimal from the region where exercising it is optimal. This boundary is unknown beforehand and must be solved simultaneously with the option price.

These problems are crucial for accurate pricing because they account for the early exercise feature inherent in American-style contracts. Solving them often requires advanced numerical techniques like finite difference methods or integral equations.

Understanding these boundaries is essential for managing risk and hedging positions effectively.