A binary option (also called a digital option) is a cash settled option that has a discontinuous payoff. Binary options come in many forms, but the two most basic are: cash-or-nothing and asset-or-nothing. Each can be European or American and can be structured as a put or call.
A European cash-or-nothing binary option pays a fixed amount of money if it expires in the money and nothing otherwise. For example, a European cash-or-nothing call makes a fixed payment if the option expires with the underlier above the strike price. It pays nothing if it expires with the underlier equal to or less than the strike price. Exhibit 1 compares the payoff of a European vanilla call with that of a European cash-or-nothing binary call:
Exhibit 1: Expiration values for a European vanilla call and a European binary cash-or-nothing call. The cash-or-nothing call makes a fixed payment if it expires in the money. It pays nothing if it expires at the money or out of the money.
An American cash-or-nothing binary option is issued out-of-the-money and makes a fixed payment if the underlier value ever reaches the strike. The payment can be made immediately or deferred until the option’s expiration date.
A European asset-or-nothing binary option pays the value of the underlier (at expiration) if it expires in the money. It pays nothing otherwise. For example, a European asset-or-nothing call pays the value of the underlier at expiration if it exceeds the strike price. A European asset-or-nothing put pays the value of the underlier at expiration if it is less than the strike price. Exhibit 2 compares the expiration values of European asset-or-nothing put and calls:
Exhibit 2: Expiration values for European asset-or-nothing binary put and calls.
An asset-or-nothing binary option might be structured as an American option with deferred payment, but this structure is not common.
Issuers of asset-or-nothing options can construct the instruments by combining a cash-or-nothing binary with a vanilla put or call. A cash-or-nothing binary can be dynamically hedged, but issuers sometimes hedge with a call spread instead. Either approach becomes problematic if the binary is at-the-money as it approaches expiration.