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finite differencing

Barrier option

In finance, a barrier option is a type of financial option where the option to exercise depends on the underlying crossing or reaching a given barrier level. Barrier options were created to provide the insurance value of an option without charging as much premium. For example, if you believe that IBM will go up this year, but are willing to bet that it won't go above $100, then you can buy the barrier and pay less premium than the vanilla option.

Types

Barrier options are path-dependent exotics that are similar in some ways to ordinary options. There are put and call, as well as European and American varieties. But they become activated or, on the contrary, null and void only if the underlier reaches a predetermined level (barrier).

"In" options start their lives worthless and only become active in the event a predetermined knock-in barrier price is breached. "Out" options start their lives active and become null and void in the event a certain knock-out barrier price is breached.

In either case, if the option expires inactive, then there may be a cash rebate paid out. This could be nothing, in which case the option ends up worthless, or it could be some fraction of the premium.

The four main types of barrier options are:

  • Up-and-out: spot price starts below the barrier level and has to move up for the option to be knocked out.
  • Down-and-out: spot price starts above the barrier level and has to move down for the option to become null and void.
  • Up-and-in: spot price starts below the barrier level and has to move up for the option to become activated.
  • Down-and-in: spot price starts above the barrier level and has to move down for the option to become activated.

For example, a European call option may be written on an underlying with spot price of $100, and a knockout barrier of $120. This option behaves in every way like a vanilla European call, except if the spot price ever moves above $120, the option "knocks out" and the contract is null and void. Note that the option does not reactivate if the spot price falls below $120 again. Once it is out, it's out for good.

In-out parity is the barrier option's answer to put-call parity. If we combine one "in" option and one "out" barrier option with the same strikes and expirations, we get the price of a vanilla option: C=C_{in}+C_{out}. A simple arbitrage argument—simultaneously holding the "in" and the "out" option guarantees that one and only one of the two will pay off. The argument only works for European options without rebate.

Barrier events

A barrier event occurs when the underlying crosses the barrier level. While it seems straightforward to define a barrier event as "underlying trades at or above a given level," in reality it's not so simple. What if the underlying only trades at the level for a single trade? How big would that trade have to be? Would it have to be on an exchange or could it be between private parties? When barrier options were first introduced to options markets, many banks had legal trouble resulting from a mismatched understanding with their counterparties regarding exactly what constituted a barrier event.

Variations

Barrier options are sometimes accompanied by a rebate, which is a payoff to the option holder in case of a barrier event. Rebates can either be paid at the time of the event or at expiration.

A discrete barrier is one for which the barrier event is considered at discrete times, rather than the normal continuous barrier case.

A Parisian option is a barrier option where the barrier condition applies only once the price of the underlying instrument has spent at least a given period of time on the wrong side of the barrier.

Barrier options can have either American or European exercise style.

Valuation

The valuation of barrier options can be tricky, because unlike other simpler options they are path-dependent -- that is, the value of the option at any time depends not just on the underlying at that point, but also on the path taken by the underlying (since, if it has crossed the barrier, a barrier event has occurred). Although the classical Black-Scholes approach does not directly apply, several more complex methods can be used:

  • The most simple way to value barrier options is using a static replicating portfolio of vanilla options (which can be valued with Black-Scholes), so chosen so as to mimic the value of the barrier at expiry and at selected discrete points in time along the barrier. This approach was pioneered by Peter Carr and gives closed form prices and replication strategies for all types of barrier options.
  • Another approach is to study the law of the maximum (or minimum) of the underlying. This approach gives explicite (closed form) prices to barrier options.
  • Yet another method is the PDE approach. The PDE satisfied by an out barrier options is the same one satisfied by a vanilla option under Black and Scholes assumptions, with extra boundary conditions demanding that the option becomes worthless when the underlying touches the barrier.
  • When an exact formula is difficult to obtain, barrier options can be priced with the Monte Carlo option model. However, computing the greeks (sensibilities) using this approach is numerically unstable.
  • A faster approach is to use finite-differencing techniques to diffuse the PDE backwards from the boundary condition (which is the terminal payoff at expiry, plus the condition that the value along the barrier is always 0 at any time). Both explicit finite-differencing methods and the Crank-Nicolson scheme have their advantages.

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