220 Frequently Asked Questions In Quantitative Finance
What is a Free-boundary Problem and
What is the Optimal-Stopping Time
foranAmericanOption?
Short Answer
A boundary-value problem is typically a differential
equation with specified solution on some domain. A
free-boundary problem is one for which that boundary
is also to be found as part of the solution. When to
exercise an American option is an example of a free-
boundary problem, the boundary representing the time
and place at which to exercise. This is also called an
optimal-stopping problem, the ‘stopping’ here referring
to exercise.
Example
Allow a box of ice cubes to melt. As they do there
will appear a boundary between the water and the
ice, the free boundary. As the ice continues to melt
so the amount of water increases and the amount of ice
decreases.
Waves on a pond is another example of a free boundary.
Long Answer
In a boundary-value problem the specification of the
behaviour of the solution on some domain is to pin
down the problem so that is has a unique solution.
Depending on the type of equation being solved we
must specify just the right type of conditions. Too few
conditions and the solution won’t be unique. Too many
and there may not be any solution. In the diffusion
equations found in derivatives valuation we must specify
a boundary condition in time. This would be the final
payoff, and it is an example of afinal condition.We
