What is a Free-Boundary Problem and What is the Optimal-Stopping Time for an American Option?

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.

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 a final condition. We must also specify two conditions in the asset space. For example, a put option has zero value at infinite stock price and is the discounted strike at zero stock price. These are examples of boundary conditions. These three are just the right number and type of conditions for there to exist a unique solution of the Black-Scholes parabolic partial differential equation.

In the American put problem it is meaningless to specify the put's value when the stock price is zero because the option would have been exercised before the stock ever got so low. This is easy to see because the European put value falls below the payoff for sufficiently small stock. If the American option price were to satisfy the same equation and boundary conditions as the European then it would have the same solution, and this solution would permit arbitrage.

The American put should be exercised when the stock falls sufficiently low. But what is 'sufficient' here?

To determine when it is better to exercise than to hold we must abide by two principles:

• The option value must never fall below the payoff, otherwise there will be an arbitrage opportunity.

• We must exercise so as to give the option its highest value.

The second principle is not immediately obvious. The explanation is that we are valuing the option from the point of view of the writer. He must sell the option for the most it could possibly be worth, for if he undervalues the contract he may make a loss if the holder exercises at a better time. Having said that, we must also be aware that we value from the viewpoint of a delta-hedging writer. He is not exposed to direction of the stock. However the holder is probably not hedging and is therefore very exposed to stock direction. The exercise strategy that is best for the holder will probably not be what the writer thinks is best. More of this anon.

The mathematics behind finding the optimal time to exercise, the optimal-stopping problem, is rather technical. But its conclusion can be stated quite succinctly. At the stock price at which it is optimal to exercise we must have

• the option value and the payoff function must be continuous as functions of the underlying

• the delta, the sensitivity of the option value with respect to the underlying, must also be continuous as functions of the underlying.

This is called the smooth-pasting condition since it represents the smooth joining of the option value function to its payoff function. (Smooth meaning function and its first derivative are continuous.)

This is now a free-boundary problem. On a fixed, prescribed boundary we would normally impose one condition. (For example, the above case of the put's value at zero stock price.) But now we don't know where the boundary actually is. To pin it down uniquely we impose two conditions, continuity of function and continuity of gradient. Now we have enough conditions to find the unknown solution.

Free-boundary problems such as these are nonlinear. You cannot add two together to get another solution. For example, the problem for an American straddle is not the same as the sum of the American call and the American put.

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, 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 conversion of a convertible bond, callability, shout options, choosers.