Barrier option

Last updated March 26, 2022

A barrier option is an option whose payoff is conditional upon the underlying asset's price breaching a barrier level during the option's lifetime.

Types

Barrier options are path-dependent exotics 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 breaches a predetermined level (the barrier).

"In" options only become active in the event that a predetermined knock-in barrier price is breached:

If the barrier price is far from being breached, the knock-in option will be worth slightly more than zero.
If the barrier price is close to being breached, the knock-in option will be worth slightly less than the corresponding vanilla option.
If the barrier price has been breached, the knock-in option will trade at the exact same value as the corresponding vanilla option.

"Out" options start their lives active and become null and void in the event that a certain knock-out barrier price is breached:

If the barrier price is far from being breached, the knock-out option will be slightly less than the corresponding vanilla option.
If the barrier price is close to being breached, the knock-out option will be worth slightly more than zero.
If the barrier price has been breached, the knock-out option will trade at the exact value of zero.

Some variants of "Out" options compensate the owner for the knock-out by paying a cash fraction of the premium at the time of the breach.

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.

By in-out parity, we mean that the combination of one "in" and one "out" barrier option with the same strikes and expirations yields the price of the corresponding vanilla option: $C=C_{in}+C_{out}$ . Note that before the knock-in/out event, both options have positive value, and hence both are strictly valued below the corresponding vanilla option. After the knock-in/out event, the knock-out option is worthless and the knock-in option's value coincides with that of the corresponding vanilla option. At maturity, exactly one of the two will pay off identically to the corresponding vanilla option, which of the two that depends on whether the knock-in/out event has occurred before maturity.

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 warrant is a barrier option namely a knock out call that is initially in the money and with the barrier at the same level as the strike.

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 to use a static replicating portfolio of vanilla options (which can be valued with Black–Scholes), 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, but usually only by assuming that the Black-Scholes model is correct. This method is therefore inappropriate when there is a volatility smile. For a more general but similar approach that uses numerical methods, see Derman's "Static Options Replication."^[1]

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 partial differential equation (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 become 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 (sensitivities) using this approach is numerically unstable.
A faster approach is to use Finite difference methods for option pricing 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.
A simple approach of binomial tree option pricing also applies.

Related Research Articles

The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price given the risk of the security and its expected return. The equation and model are named after economists Fischer Black and Myron Scholes; Robert C. Merton, who first wrote an academic paper on the subject, is sometimes also credited.

In financial mathematics, put–call parity defines a relationship between the price of a European call option and European put option, both with the identical strike price and expiry, namely that a portfolio of a long call option and a short put option is equivalent to a single forward contract at this strike price and expiry. This is because if the price at expiry is above the strike price, the call will be exercised, while if it is below, the put will be exercised, and thus in either case one unit of the asset will be purchased for the strike price, exactly as in a forward contract.

In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These options—as well as others where the payoff is calculated similarly—are referred to as "vanilla options". Options where the payoff is calculated differently are categorized as "exotic options". Exotic options can pose challenging problems in valuation and hedging.

In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting.

In mathematical finance, the Greeks are the quantities representing the sensitivity of the price of derivatives such as options to a change in underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the most common of these sensitivities are denoted by Greek letters. Collectively these have also been called the risk sensitivities, risk measures or hedge parameters.

Rational pricing is the assumption in financial economics that asset prices - and hence asset pricing models - will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.

Monte Carlo methods are used in corporate finance and mathematical finance to value and analyze (complex) instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the distribution of their value over the range of resultant outcomes. This is usually done by help of stochastic asset models. The advantage of Monte Carlo methods over other techniques increases as the dimensions of the problem increase.

Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter that is needed to be modified for the Black–Scholes formula to fit market prices. In particular for a given expiration, options whose strike price differs substantially from the underlying asset's price command higher prices than what is suggested by standard option pricing models. These options are said to be either deep in-the-money or out-of-the-money.

A variance swap is an over-the-counter financial derivative that allows one to speculate on or hedge risks associated with the magnitude of movement, i.e. volatility, of some underlying product, like an exchange rate, interest rate, or stock index.

In finance, a bond option is an option to buy or sell a bond at a certain price on or before the option expiry date. These instruments are typically traded OTC.

In finance, a price (premium) is paid or received for purchasing or selling options. This article discusses the calculation of this premium in general. For further detail, see: Mathematical finance § Derivatives pricing: the Q world for discussion of the mathematics; Financial engineering for the implementation; as well as Financial modeling § Quantitative finance generally.

In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is required at "all" times before and including maturity. A continuous model, on the other hand, such as Black–Scholes, would only allow for the valuation of European options, where exercise is on the option's maturity date. For interest rate derivatives lattices are additionally useful in that they address many of the issues encountered with continuous models, such as pull to par. The method is also used for valuing certain exotic options, where because of path dependence in the payoff, Monte Carlo methods for option pricing fail to account for optimal decisions to terminate the derivative by early exercise, though methods now exist for solving this problem.

Martingale pricing is a pricing approach based on the notions of martingale and risk neutrality. The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety of derivatives contracts, e.g. options, futures, interest rate derivatives, credit derivatives, etc.

In finance, a spread option is a type of option where the payoff is based on the difference in price between two underlying assets. For example, the two assets could be crude oil and heating oil; trading such an option might be of interest to oil refineries, whose profits are a function of the difference between these two prices. Spread options are generally traded over the counter, rather than on exchange.

Option (finance) Right to buy or sell a certain thing at a later date at an agreed price

In finance, an option is a contract which conveys to its owner, the holder, the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on or before a specified date, depending on the style of the option. Options are typically acquired by purchase, as a form of compensation, or as part of a complex financial transaction. Thus, they are also a form of asset and have a valuation that may depend on a complex relationship between underlying asset value, time until expiration, market volatility, and other factors. Options may be traded between private parties in over-the-counter (OTC) transactions, or they may be exchange-traded in live, orderly markets in the form of standardized contracts.

In finance, a volatility swap is a forward contract on the future realised volatility of a given underlying asset. Volatility swaps allow investors to trade the volatility of an asset directly, much as they would trade a price index. Its payoff at expiration is equal to

In mathematical finance, the Black–Scholes equation is a partial differential equation (PDE) governing the price evolution of a European call or European put under the Black–Scholes model. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of options, or more generally, derivatives.

Finite difference methods for option pricing are numerical methods used in mathematical finance for the valuation of options. Finite difference methods were first applied to option pricing by Eduardo Schwartz in 1977.

The Vanna–Volga method is a mathematical tool used in finance. It is a technique for pricing first-generation exotic options in foreign exchange market (FX) derivatives.

In finance, a contingent claim is a derivative whose future payoff depends on the value of another “underlying” asset, or more generally, that is dependent on the realization of some uncertain future event. These are so named, since there is only a payoff under certain contingencies. Any derivative instrument that is not a contingent claim is called a forward commitment.

References

↑ Derman, Emanuel; Ergener, Deniz; Kani, Iraj (31 May 1995). "Static Options Replication" (PDF). The Journal of Derivatives. 2 (4): 78–95. doi:10.3905/jod.1995.407927.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Derman, Emanuel; Ergener, Deniz; Kani, Iraj (31 May 1995). "Static Options Replication" (PDF). The Journal of Derivatives. 2 (4): 78–95. doi:10.3905/jod.1995.407927.

[1]