1) Does there have to be an underlying?
Yes. The underlying provides the variability needed for the agreement to be behave like a derivative. Variability is one of the key characteristics of a derivative. Without an underlying, there is no mechanism for triggering a change in the economic value of the contract to parties.
2) Does there have to be a notional amount?
No. The notional merely serves as a factor in computing the settlement amount. The payment amount can, however, be a fixed amount or an amount determined without reference to a notional. The ASC definition states that there must be EITHER a notional OR a payment amount. If the contract has neither, then it does not meet the accounting definition of a derivative.
3) Is the underlying always a market index or some other financial measurement?
No. The underlying can be anything that is observable or objectively verifiable. Market indexes, such as stock market averages, commodity prices and interest rates, are the most common form of underlying. However, if it can be observed or verified, it can be an underlying. The accounting standards mention credit ratings and credit indexes, insurance indexes, loss indexes, and climatic and geological conditions (earthquakes, hurricanes, snowfall, rainfall, etc.). An underlying can also be the occurrence of nonoccurrence of an event specified in the contract such as a scheduled payment, or contractual obligation or covenant. This last type of underlying is known as an ‘on-off’ switched and usually triggers a payment. The most common example of this is a loan default provision that triggers either a fixed payment (e.g., penalty) or an increase in contract interest rate. As you can see, the term underlying is used very broadly.