222 Frequently Asked Questions In Quantitative Finance
the stock. However the holder is probably not hedg-
ing 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 tech-
nical. 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 thesmooth-pasting conditionsince it rep-
resents the smooth joining of the option value function
to its payoff function. (Smooth meaning function and its
first derivative being continuous.)
This is now a free-boundary problem. On a fixed, pre-
scribed boundary we would normally impose one condi-
tion. (For example, the above example 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
twoconditions, continuity of function and continuity of
gradient. Now we have enough conditions to find the
unknown solution.
Free-boundary problems such as these are non linear.
You cannot add two together to get another solution.
For example, the problem for an American straddle is
notthe same as the sum of the American call and the
American put.
