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[ edit ] 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?
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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[ edit ] 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.
Care sunt opțiunile de barieră 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 AmericanBermudan or European exercise style.
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Valuation[ edit ] 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 care sunt opțiunile de barieră 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—Scholeschosen 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.
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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.
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.
Exemplu de opțiune de barieră. Opțiuni de barieră
Both explicit finite-differencing methods and the Crank—Nicolson scheme have their advantages. A simple approach of binomial tree option pricing also applies. The Journal of Derivatives.
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