## 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. The value of the payoff is not affected by the size of the difference between the underlying and the strike price, and can be in the form of a cash payment or delivery of the underlying. The options described here are path dependent, which means that the payout profile depends on the asset value during the life of the option and the value of the underlying asset when the barrier is hit or on the expiry date of the option. For a call, the payout is received if the underlying asset price is greater than the strike price, and for a put, the payout is received if the strike is greater than the underlying asset price.

## Technical Details

There are two classes of binary barrier options. The first are 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.

The payout is made either when the barrier is hit, or at option expiry. For cash payouts, this distinction will only affect the period of time over which the payment is discounted. For asset payouts, however, the distinction is more subtle. If the payout is made when the barrier is touched, then the present value of the payout is equal to the discounted barrier value – since this is the asset value when the barrier is touched. 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.

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## Analysis Supported

FINCAD binary barrier options functions can be used for the following:

- 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.
- Calculate the fair value, delta, and probability of hitting the barrier for a path dependent digital option where the payoff is made at the time the barrier is touched.

To evaluate an FINCAD product that can value Binary Barrier Options, contact a FINCAD Representative