# Can Binary Call Options Be Priced Analytically

Binary options

Binary options, also known as digital options, have discontinuous payoffs.

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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 asset-or-nothing call and a short position in a cash-or-nothing call, where the both options have the same strike price and the cash payoff of the cash-or-nothing option equals the strike price.

Unlike standard European style options, the payout for binary options does not depend on how much it is in-the-money 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 binary-barrier options.

Binary option functions:

Asset Or Nothing

In an asset-or-nothing 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.

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The value of an asset-or-nothing 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. Asset-or-nothing 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 cash-or-nothing 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 cash-or-nothing 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.

Cash-or-nothing options can be priced analytically using a model introduced by Reiner and Rubinstein (1991).

The FinOptions function CashOrNothing can be used to evaluate European cash-or-nothing options.

Two-asset Cash Or Nothing

Two-asset cash-or-nothing options can be useful building blocks for constructing more complex exotic options.

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There are four types of two-asset cash-or-nothing options:

A two-asset cash-or-nothing call pays out a fixed cash amount if Asset 1, is above Strike 1 and Asset 2, is above Strike 2 at expiration.

A two-asset cash-or-nothing put pays out a fixed cash amount if Asset 1, is below Strike 1 and Asset 2, is below Strike 2 at expiration.

A two-asset cash-or-nothing up-down pays out a fixed cash amount if Asset 1, is above Strike 1 and Asset 2, is below Strike 2 at expiration.

A two-asset cash-or-nothing down-up pays out a fixed cash amount if Asset 1, is below Strike 1 and Asset 2, is above Strike 2 at expiration.

Two-asset cash-or-nothing options can be priced analytically using a model introduced by Heynen and Kat (1996).

The FinOptions function CashOrNothingTA can be used to evaluate European two-asset cash-or-nothing 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 asset-or-nothing call and short a cash-or-nothing call.

## Description

A gap put option is equivalent to being long a cash-or-nothing put and short an asset-or-nothing 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 asset-or-nothing calls, in which the owner of a supershare purchases an asset-or-nothing call with an strike price of LowerStrike and sells an asset-or-nothing 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

Binary-barrier options combine characteristics of both binary and barrier options.

They are path dependent options with a discontinuous payoff.

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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: Cash-or-nothing and Asset-or-nothing barrier options.

Cash-or-nothing barrier options pay out a predetermined cash amount or nothing, depending on whether the asset price has hit the barrier. Asset-or-nothing 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).

## American Binary Option Pricing: 3 Period Binomial Tree Model

Binary-barrier options can be priced analytically using a model introduced by Reiner and Rubinstein (1991).

The FinOptions function BinaryBarrier can be used to evaluate European binary-barrier options.

## FinOptions XL Functions:

The AssetOrNothing function calculates the theoretical price, sensitivities, the implied volatility, and the implied strike value of a European asset-or-nothing 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 cash-or-nothing 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 two-asset cash-or-nothing 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.