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4.2.2 Barrier options

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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.

Virtual market experiments: exotic barrier options
  1. Move the in-barrier from below to Barrier=0.1 or 10% above the initial asset value. Can you see any difference in the option prices? Try to find a reason why an investor may want to buy such an option.
  2. Move the Barrier=-0.3 or 30% below the initial asset value and check how the price becomes ill defined since few realizations ever make it to the barrier. Knowing that the relative precision of a Monte-Carlo calculation is proportional to $ 1/\sqrt{N}$ , estimate the number of random walkers $ N$ that are needed to achieve a 10% precision.
  3. Reload the initial applet parameters and switch from inBarrier* to outBarrier* to experiment with the payoff of an down-and-out barrier option. How are the in-  and out-barriers related?
  4. Change the terminal option payoff and study the evolution of prices in the case of a vertical spread call featuring an in-barrier.

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