Chapter 2: FAQs 223
Although the fascinating mathematics of free-boundary
problems can be complicated, and difficult or impossible
to solve analytically, they can be easy to solve by finite-
difference methods. For example, if in a finite-difference
solution we find that the option value falls below the
payoff then we can just replace it with the payoff. As
long as we do this each time step before moving on to
the next time step then we should get convergence to
the correct solution.
As mentioned above, the option is valued by maximizing
the value from the point of view of the delta-hedging
writer. If the holder is not delta hedging but speculating
on direction he may well find that he wants to exit his
position at a time that the writer thinks is suboptimal.
In this situation there are three ways to exit:
- sell the option;
- delta hedge to expiration;
- exercise the option.
The first of these is to be preferred because the option
may still have market value in excess of the payoff.
The second choice is only possible if the holder can
hedge at low cost. If all else fails, he can always close
his position by exercising. This is of most relevance
in situations where the option is an exotic, over the
counter, contract with an early-exercise feature when
selling or delta hedging may not be possible.
There are many other contracts with decision features
that can be treated in a way similar to early exercise
as free-boundary problems. Obvious examples are con-
version of a convertible bond, callability, shout options,
choosers.