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4.2.2 Barrier options
[ SLIDE
up-and-in -
up-and-out -
down-and-in -
down-and-out ||
VIDEO
modem -
LAN -
DSL
]
More than an option on its own, a barrier is a feature that can be added
to most of the contracts, including binary options that have just been
discussed.
Remember that an ``in-barrier'' option
typically expires worthless unless the underlying crosses the barrier
once at least during the option lifetime. With these specifications,
it becomes important to keep track of the underlying asset price history
and is most conveniently implemented by ``tagging'' each of the possible
realizations in a Monte-Carlo simulation.
A fair value for the barrier option is then obtained from the average
payoff where only tagged realizations finally contribute to the sum.
The VMARKET applet below shows the result
in the case of an down-and-in barrier put option.
VMARKET applet: press Start/Stop
to study the payoff dynamics V(S,t) of a down-and-in barrier
put option set at Barrier=-0.1, i.e. an in-barrier 10%
below the price of the underlying when the simulation starts.
The option price V is displayed as a function of the underlying
value S using black / blue colors for put option with / without
a barrier and a grey color to remind the terminal payoff.
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You probably found and verified in your the experiments that ``in-'' and
``out-'' barriers are complementary: the sum of both gives the same price
as the option without a barrier.
SYLLABUS Previous: 4.2.1 Binary options
Up: 4.2 Exotic stock options
Next: 4.3 Methods for European