Binary options
Binary options, also known as digital options, have discontinuous payoffs. They can be used as building blocks to develop options with more complicated payoffs. For example, a regular European call option is equivalent to a long position in an assetornothing call and a short position in a cashornothing call, where the both options have the same strike price and the cash payoff of the cashornothing option equals the strike price. Unlike standard European style options, the payout for binary options does not depend on how much it is inthemoney but rather whether or not it is on the money. The option’s payoff is fixed at the options inception and is based on the price of the underlying asset on the expiration date. Binary options may also incorporate barriers, as is the case with binarybarrier options.
Binary option functions:
Asset Or Nothing 
In an assetornothing option, a predetermined asset value is paid if the asset is, at expiration, above for a call or below for a put some strike level, independent of the path taken. For a call (put) the terminal price is greater than (less than) the exercise price, the call (put) expires worthless. The exercise price is never paid. Instead, the value of the asset relative to the exercise price determines whether or not the option returns a payoff. The value of an assetornothing call (put) option is the present value of the asset multiplied by the probability that the terminal price will be greater than (less than) the exercise price. Assetornothing options can be priced analytically using a model introduced by Cox and Rubinstein (1985). The FinOptions function AssetOrNothing can be used to evaluate European double barrier options. 


Cash Or Nothing 
In a cashornothing option, a predetermined amount is paid if the asset is, at expiration, above for a call or below for a put some strike level, independent of the path taken. These options require no payment of an exercise price. Instead, the exercise price determines whether or not the option returns a payoff. The value of a cashornothing call (put) option is the present value of the fixed cash payoff multiplied by the probability that the terminal price will be greater than (less than) the exercise price. Cashornothing options can be priced analytically using a model introduced by Reiner and Rubinstein (1991). The FinOptions function CashOrNothing can be used to evaluate European cashornothing options. 


Twoasset Cash Or Nothing 
Twoasset cashornothing options can be useful building blocks for constructing more complex exotic options. There are four types of twoasset cashornothing options: A twoasset cashornothing call pays out a fixed cash amount if Asset 1, is above Strike 1 and Asset 2, is above Strike 2 at expiration. A twoasset cashornothing put pays out a fixed cash amount if Asset 1, is below Strike 1 and Asset 2, is below Strike 2 at expiration. A twoasset cashornothing updown pays out a fixed cash amount if Asset 1, is above Strike 1 and Asset 2, is below Strike 2 at expiration. A twoasset cashornothing downup pays out a fixed cash amount if Asset 1, is below Strike 1 and Asset 2, is above Strike 2 at expiration. Twoasset cashornothing options can be priced analytically using a model introduced by Heynen and Kat (1996). The FinOptions function CashOrNothingTA can be used to evaluate European twoasset cashornothing options. 


Gap 
Gap options are similar to plain options, except for the payoff. The payoff is a function of the exercise price. The payoff on a gap option depends on all of the factors of a plain option, but it is also affected by the gap amount, which can be either positive or negative. A gap call option is equivalent to being long an assetornothing call and short a cashornothing call. A gap put option is equivalent to being long a cashornothing put and short an assetornothing put. Gap options can be priced analytically using a model introduced by Reiner and Rubinstein (1991). The FinOptions function Gap can be used to evaluate European gap options. 


Supershare 
A supershare is a financial instrument that represents a contingent claim on a fraction of the underlying portfolio. The contingency is that the value of the portfolio must lie between a lower and an upper bound on its expiration date. If the value lies within these boundaries, the supershare is worth a proportion of the assets underlying the portfolio, else the supershare expires worthless. A supershare has a payoff that is basically like a spread of two assetornothing calls, in which the owner of a supershare purchases an assetornothing call with an strike price of LowerStrike and sells an assetornothing call with an strike price of UpperStrike. Supershare options can be priced analytically using a model introduced by Hakansson (1976). The FinOptions function Supershare can be used to evaluate European supershare options. 


Binary Barrier 
Binarybarrier options combine characteristics of both binary and barrier options. They are path dependent options with a discontinuous payoff. Similar to barrier options, the payoff depends on whether or not the asset price crosses a predetermined barrier. There are 28 different types of binary barrier options, which can be divided into two main categories: Cashornothing and Assetornothing barrier options. Cashornothing barrier options pay out a predetermined cash amount or nothing, depending on whether the asset price has hit the barrier. Assetornothing barrier options pay out the value of the asset or nothing, depending on whether the asset price has crossed the barrier. The barrier monitoring frequency can be adjusted to account for discrete monitoring using an approximation developed by Broadie, Glasserman, and Kou (1995). Binarybarrier options can be priced analytically using a model introduced by Reiner and Rubinstein (1991). The FinOptions function BinaryBarrier can be used to evaluate European binarybarrier options. 
The AssetOrNothing function calculates the theoretical price, sensitivities, the implied volatility, and the implied strike value of a European assetornothing option using the Cox and Rubinstein’s model.
The CashOrNothing function calculates the theoretical price, sensitivities, the implied volatility, and the implied strike value of a European cashornothing option using the Reiner and Rubinstein’s model.
The CashOrNothingTA function calculates the theoretical price, sensitivities, the implied volatility, the implied strike and the implied correlation value of a European twoasset cashornothing option using Heynen and Kat’s model.
The Gap function calculates the theoretical price, sensitivities, the implied volatility, and the implied strike value of a European gap option using the Reiner and Rubinstein’s model.
The Supershare function calculates the theoretical price, sensitivities, the implied volatility, and the implied strike value of a European supershare option using Hahansson’s model.
The BinaryBarrier function calculates the theoretical price, sensitivities, the implied volatility, and the implied strike value of all 28 different types of European binary barrier options using Reiner and Rubinstein’s model.