In finance, a binary option is a name given to a type of option where the payoff is either some fixed amount of some asset or nothing at all. There are two main types of binary options and these are:

• the cash-or-nothing binary option and
• the asset-or-nothing binary option.

The Cash-or-Nothing Binary Option: In The cash-or-nothing binary option if the option expires in-the-money then some fixed amount of cash is paid.

The Asset-or-Nothing Binary Option: The value of the underlying security is paid by the asset-or-nothing binary option.

Thus, due to the reason that there are only two possible outcomes so the options are binary in nature. They are also referred to as all-or-nothing options, digital options (these are more common in forex/interest rate markets), and Fixed Return Options (FROs) (on the American Stock Exchange).

For instance, assume that a purchase is made of a binary cash-or-nothing call option on XYZ Corp’s stock struck at $100 with a binary payoff of $1000. Then, if the stock is trading at or above $100 at the future maturity date, then $1000 is received. And if in case its stock is trading below $100, then nothing is received.

The value of a digital option can be expressed in terms of the cumulative normal distribution function in the popular Black-Scholes model.

Interpretation of prices

In a prediction market, in order to find out a population’s best estimate of an event occurring, binary options are used. For instance, a price of 0.65 on a binary option that has been triggered by the Democratic candidate winning the next US Presidential election might be interpreted as an estimate of 65% likelihood of him winning.

If we talk about the financial markets, then the expected returns on a stock or other instrument are already priced into the stock. However, other information is provided by a binary options market. Just as the the market’s estimate of variance (volatility) i.e. second moment is revealed by regular options market, the market’s estimate of skew, i.e. the third moment is revealed by a binary options market.

American vs European

Usually binary options are of European-style the price of the underlying must be above the strike at the expiration date, for a call. American style also exist, but these are automatically exercise whenever the price “touches” the strike price, and that yields a very different behavior.

Relationship to vanilla options’ Greeks

As we know that a binary call is a mathematical derivative of a vanilla call with respect to strike, so the price of a binary call has the shape that is identical to the delta of a vanilla call, and the delta of a binary call has the shape that is identical to the gamma of a vanilla call.

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