Chapter 2: FAQs 221
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 ofboundary 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 under-
values 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