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In-Depth Information
Chapter 6
Exotic Options
Options with more sophisticated rules than those for plain vanillas are called
exotic
options
. There are different types.
Path dependent
options depend on the whole
history of the underlying and not just on the realization at maturity. In particular, we
consider
barrier options
which depend on price levels being attained over a period
and
Asian options
which depend on the average price of the option's underlying
over a period. Furthermore, we look at options which have different exercise styles
like
compound options
which are options on options and
swing options
which have
multiple exercise rights. We assume that the dynamics of the stock price is modeled
by a geometric Brownian motion.
6.1 Barrier Options
Barrier options differ from vanillas in the sense that the option contract is triggered
if the price of the underlying hits some barrier
B>
0. To be more specific, consider
a contract which pays a specified amount at maturity
T
provided during 0
T
the price
S
does not cross a specified barrier
B
either from above, the so-called
down-and-out
barrier option, or from below, called
up-and-out
barrier option. If the
barrier is crossed before
T
, the option expires worthless. A
knock-in
barrier option
becomes the corresponding European vanilla when the barrier
B
is crossed at time
0
≤
t
≤
≤
t
≤
T
, e.g. the up-and-in call becomes the European call when
B
is crossed
from below. We assume here for convenience that
B
is constant.
A European plain vanilla option pays the same at
T
as a down-and-out plus a
down-and-in with the same barrier
B
and of the same type (call/put) as the plain
vanilla. The standard no arbitrage consideration shows that pricing a knock-in bar-
rier reduces to pricing the corresponding knock-out barrier contract and a plain Eu-
ropean vanilla. If we denote the value of a down-and-out barrier option by
V
do
,the
value of a up-and-out barrier option by
V
uo
and correspondingly
V
di
,
V
ui
for the
knock-in barrier option, we have
V
di
(t, s)
=
V(t,s)
−
V
do
(t, s),
V
ui
(t, s)
=
V(t,s)
−
V
uo
(t, s).
(6.1)
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