**3.3
American Call Option: Zero-Dividend
Case
**

T |

he difference between an American option and its European counterpart is the right to early exercise. The American option can be exercised at any time; the European option can be exercised only at maturity.

This
complicates the valuation problem, because we now have to determine whether the
right to exercise early is worth something.
If it is, then the option price must reflect this additional value. Note that an American option is never worth *less* than a European option. This
is because you can always choose not to exercise early.

It
turns out that whether the early exercise possibility is valuable depends on
whether we have a put or a call option, and whether the stock pays a dividend
during the life of the option.

In
this topic, you will see that the right to early exercise for an American *call*
option, defined on a non*-*dividend-paying
security, is zero. To see why, let
us revisit the one-period binomial model.

By
applying the risk-neutral valuation principle, the present value of the call
times *r (*i.e., Cr) in the one-period model is:

where p
is the risk-neutral probability (see topic 2.7, __Risk-Neutral
Valuation Principle__ in Chapter 2), and *r*
is one plus the interest
rate. In the one-period model, *Cu*
and *Cd* are the terminal call
option values. In other words, *Cu*
= max{*Su-X,0*}, *Cd* = max{*Sd-X,0*}.

We
can demonstrate that the right to early exercise has zero value.
By substituting *Cu* and *Cd*
into the risk-neutral valuation equation, we get:

Adding
and subtracting *X* to this yields:

Dividing
both the left-hand and right-hand sides by *r*
implies:

because
*X/r < X* since *r* > 1.

This
demonstrates that the value of the call is always greater than what could be
obtained by exercising the call (i.e., *S -
X*), so the call option should never be exercised prior to maturity.

Therefore
the value of the right to the early exercise of an American call option on a
zero-dividend-paying stock is zero. You
will see that this does not hold for the American put option, so early exercise
may indeed be called for. This is
covered in the next topic __American
Put Option: Zero-Dividend Case__.