Derivative Securities – Fall 2007 – Section 8
Notes by Robert V. Kohn, extended and improved by Steve Allen.
Courant Institute of Mathematical Sciences.
American and exotic options.
We have thus far focused on European options.
This
week’s topic is the valuation and hedging of American and exotic options.
This short document (4 pages) discusses only American options.
Please also read Steve
Allen’s Section 8 notes
(posted on Blackboard); they focus mainly on (a) the numerical
valuation of pathdependent options, and (b) creation of a binomial tree that’s consistent
with an observed volatility skew/smile.
************************
American options
. American options are different in that they permit early exercise: the
holder of an American option can exercise it at any time up to the maturity
T
.
Of the
options actually traded in the market, the majority are American rather than European.
Clearly an American option is at least as valuable as the analogous European option, since
the holder has the option to keep it to maturity.
Fact:
An American call written on a stock that earns no dividend has the same value as
a European call; early exercise is never optimal. To see why, suppose the strike price is
K
and consider the value of the American option “now,” at some time
t < T
. Exercising the
option now achieves a value at time
t
of
s
t

K
. Holding the option to maturity achieves a
value at time
t
equal to that of a European call,
c
[
s
t
, K, T

t
]. Without using the Black
Scholes formula (thus without assuming lognormal stock dynamics) we know the value of a
European call is at least that of a forward with the same strike and maturity. Thus holding
the option to maturity achieves a value at time
t
of at least
s
t

e

r
(
T

t
)
K
. If
r >
0 this is
larger than
s
t

K
. So early exercise is suboptimal, as asserted.
The preceding is in some sense a fluke. When the underlying asset pays a dividend early
exercise of a call can be optimal. But the simplest example where early exercise occurs is
that of a put on a nondividendpaying stock:
Fact:
An American put written on a stock that earns no dividend can have a value greater
than that of the associated European put; early exercise can be optimal.
To see why,
consider once again the value of the American option “now,” at some time
t < T
. Exercising
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 Fall '11
 Bayou
 Finance, Options, Real options analysis, Mathematical finance, Myron Scholes

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