Barrier option
Encyclopedia
In finance
, a barrier option is a financial derivative
which either springs into existence upon the occurrence of the event of the price of the underlying asset breaching a barrier (in the case of "up and in" and "down and in" options) or whose existence is extinguished upon the occurrence of the event of the price of the underlying asset breaching a barrier (in the case of an "up and out" or a "down and out"). Most often this contingent derivate is an option on the underlying asset whose price breaching the pre-set barrier level either springs the option into existence or extinguishes an already existing option. Barrier options are always cheaper than a similar option without barrier. 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.
s that are similar in some ways to ordinary options
. You can call
or put
, in American, Bermudan or European exercise style. But they become activated (or extinguished) only if the underlying reaches a predetermined level (the 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.
If the option expires inactive then it may be worthless, or there may be a cash rebate paid out, some fraction of the premium.
The four main types of barrier options are:
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:
. A simple arbitrage argument—simultaneously holding the "in" and the "out" option guarantees that exactly one of the two will pay off identically to a standard European option while the other will be worthless. The argument only works for European options without rebate.
Barrier options can have either American, Bermudan or European exercise style.
Finance
"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...
, a barrier option is a financial derivative
Derivative (finance)
A derivative instrument is a contract between two parties that specifies conditions—in particular, dates and the resulting values of the underlying variables—under which payments, or payoffs, are to be made between the parties.Under U.S...
which either springs into existence upon the occurrence of the event of the price of the underlying asset breaching a barrier (in the case of "up and in" and "down and in" options) or whose existence is extinguished upon the occurrence of the event of the price of the underlying asset breaching a barrier (in the case of an "up and out" or a "down and out"). Most often this contingent derivate is an option on the underlying asset whose price breaching the pre-set barrier level either springs the option into existence or extinguishes an already existing option. Barrier options are always cheaper than a similar option without barrier. 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 exoticExotic derivatives
Exotic derivatives refers to a specific type of financial asset.*Derivatives are assets whose value depends on another underlying asset.*Exotic as opposed to vanilla refers to the fact that the payoff is not standard, as is the case for a regular call option.See also Exotic option...
s that are similar in some ways to ordinary options
Option (finance)
In finance, an option is a derivative financial instrument that specifies a contract between two parties for a future transaction on an asset at a reference price. The buyer of the option gains the right, but not the obligation, to engage in that transaction, while the seller incurs the...
. You can call
Call option
A call option, often simply labeled a "call", is a financial contract between two parties, the buyer and the seller of this type of option. The buyer of the call option has the right, but not the obligation to buy an agreed quantity of a particular commodity or financial instrument from the seller...
or put
Put option
A put or put option is a contract between two parties to exchange an asset, the underlying, at a specified price, the strike, by a predetermined date, the expiry or maturity...
, in American, Bermudan or European exercise style. But they become activated (or extinguished) only if the underlying reaches a predetermined level (the 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.
If the option expires inactive then it may be worthless, or there may be a cash rebate paid out, 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:
. A simple arbitrage argument—simultaneously holding the "in" and the "out" option guarantees that exactly one of the two will pay off identically to a standard European option while the other will be worthless. 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.
- A turbo warrantTurbo warrantTurbo warrant is a kind of stock option. Specifically, it is a barrier option of the Down and Out type. It is similar to a vanilla contract, but with two additional features: It has a low vega, meaning that the option price is much less affected by the implied volatility of the stock market, and it...
is a barrier option with two additional features: it has a low vega, meaning that the option price is much less affected by the implied volatility of the stock market, and it is highly geared due to the possibility of knockout.
Barrier options can have either American, Bermudan 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 simplest way to value barrier options is using a static replicating portfolioReplicating portfolioIn the valuation of a life insurance company, the actuary considers a series of future uncertain cashflows and attempts to put a value on these cashflows...
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 explicit (closed form) prices to barrier options.
- Yet another method is the PDE (partial differential equation) 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 modelMonte Carlo option modelIn mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty or with complicated features....
. However, computing the Greeks (sensitivities) 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.
External links
- An Overview of Barrier Options (PDFPortable Document FormatPortable Document Format is an open standard for document exchange. This file format, created by Adobe Systems in 1993, is used for representing documents in a manner independent of application software, hardware, and operating systems....
), Kevin Cheng, global-derivatives.com