# Binary and barrier options

On the other hand, if the payout is made at option expiry, then the present value of the payout is equal to whatever the asset value happens to be at the expiry date, discounted back to the valuation date.

The second class includes options where a payout of cash or the asset is made if the barrier is hit or not hit during the life of the option and if the option is in-the-money at expiry. These are types of knock-in and knock-out binary barrier options. There are other types of digital options available within the FINCAD library, including various flavors of double barrier binary options.

Introduction A Binary Barrier Option is a type of digital option for which an option's payout depends on whether or not the asset touched a barrier level at some time during the life of the option. Technical Details There are two classes of binary barrier options.

Calculate the fair value, risk statistics and probability of hitting the barrier for a binary barrier option with a payoff equal to the asset value if the barrier is touched, or nothing if the barrier is never touched. Calculate the fair value, risk statistics and probability of hitting the barrier for a binary barrier option with a payoff of a fixed amount of cash if the barrier is touched, or nothing if the barrier is never touched.

Calculate the fair value, risk statistics and probability of hitting the barrier for a knock-in binary barrier call or put option with a payoff equal to the value of the asset if the barrier is touched and the option is in the money. Calculate the fair value, risk statistics and probability of hitting the barrier for a knock-in binary barrier call or put option with a payoff of a fixed amount of cash if the barrier is touched and the option is in-the-money.

Calculate the fair value, risk statistics and probability of hitting the barrier for a binary barrier option with a payoff equal to the value of the asset if the barrier is not touched, or nothing if the barrier is touched.

Calculate the fair value, risk statistics and probability of hitting the barrier for a binary barrier option with a payoff of a fixed amount of cash if the barrier is not touched, or nothing if the barrier is touched. Calculate the fair value, risk statistics and probability of hitting the barrier for a knock-out binary barrier call or put option with a payoff equal to the value of the asset if the barrier is not touched and the option is in the money at expiry, or nothing if the barrier is touched.

Calculate the fair value, risk statistics and probability of hitting the barrier for a knock-out binary barrier call or put option with a payoff of a fixed amount of cash if the barrier is not touched and the option is in the money at expiry, or nothing if the barrier is touched.

Calculate the fair value, delta, and probability of hitting the barrier for a path dependent digital option where the payoff is on the expiration date. If the option expires inactive, then it may be worthless, or there may be a cash rebate paid out as a fraction of the premium. Once it is out, it's out for good. Also note that once it's in, it's in 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 argumentâ€”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. 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. Barrier options are sometimes accompanied by a rebate , which is a payoff to the option holder in case of a barrier event.