# Binary Option

A binary option (also known as all-or-nothing option) is a financial contract that entitles its holder to a fixed payoff when the event triggering the payoff occurs or zero payoff when no such event occurs.

Possible payoff of a traditional option ranges from zero to some upper limit (or infinity) and it depends on the actual difference between the exercise price and the price of the underlying asset. Payoff of a binary option on the other hand, is just a fixed amount which is not affected by the difference between the exercise price and the price of the underlying asset. A binary option depends on the relationship between the exercise price and the price of the underlying asset only to determine whether the payoff will occur or not.

It is also called *digital option* because its payoff is just like binary signals: i.e. 0 or 1 where 1 being the maximum payoff.

## Formula

A binary call option pays 1 unit when the price of the underlying (asset) is greater than or equal to the exercise price and zero when it is otherwise. This is expressed by the following formula:

$$ Binary\ Call\ Option\ Payoff \\ =\left\{\begin{matrix}1&,&Underlying's\ Price\ \geq\ Exercise\ Price\\0&,&Exercise\ Price < Underlying's \ Price \\ \end{matrix} \right. $$

A binary option payoff is exactly the opposite of a binary call option, as expressed by the following formula:

$$ Binary\ Call\ Option\ Payoff \\ =\left\{\begin{matrix}1&,&Underlying's\ Price\ \geq\ Exercise\ Price\\0&,&Exercise\ Price < Underlying's \ Price \\ \end{matrix} \right. $$

## Example

Keita Yoshihara is a trader at Foundation Investments. On 1 June 20Y3, he bought 1,000 CBOE binary call options on S&P 500 (SPX) with exercise price of 1,650. The options carry a $100 multiplier and are due to expire on 20 July 20X3. Find per-option and total payoff if exercise-settlement value (SET) of S&P 500 index is 1,690 at the day before expiration date. What if the SET is 1,600?

### Solution

SPX is a binary call option which means it will pay $100 if the exercise-settlement value (SET) (which is the price of the underlying asset — the S&P 500 index) is equal to or greater than the exercise price and zero if the SET is lower than the exercise price.

In the first scenario since SET is higher than the exercise price (1,690 > 1,650), it will trigger the payoff which equals the option multiplier and Keita will receive $100 per option and $100 thousand in total [= 1,000 × $100].

In the second scenario where SET is 1,600, payoff will be zero because the condition required to trigger payoff is not fulfilled i.e. the SET (1,600) is not greater than or equal to the exercise price of (1,650). In this scenario Keita will have to let the options expire wothless.