American option

is one where the holder has the right to exercise at any time before expiration and receive the payoff. Many contracts have such early exercise American features. Mathematically, early exercise is the same as conversion of a convertible bond. These contracts are priced assuming that the holder exercises so as to give the contract its highest value. Therefore a comparison must be made between the value of the option assuming you don't exercise and what you would get if you immediately exercised. This makes finite differences a much more natural numerical method for pricing such contracts than Monte Carlo.

Asian option

is an option whose payoff depends on the average value of the underlying during some period of the option's life. The average can be defined in many ways, as an arithmetic or geometric mean, for example, and can use a large set of data points in a continuously sampled Asian or only a smaller set, in the discretely sampled Asian. In an Asian tail the averaging only occurs over a short period before option expiration. There are closed-form formula for some of the simpler Asian options based on geometric averages, and approximations for others. Otherwise they can be priced using Monte Carlo methods, or sometimes by finite differences. Because the average of an asset price path is less volatile than the asset path itself these options can be cheaper than their equivalent vanillas, but this will obviously depend on the nature of the payoff. These contracts are very common in the commodity markets because users of commodities tend to be exposed to prices over a long period of time, and hence their exposure is to the average price.

Asset swap

is the exchange of one asset for interest payments for a specified period.

Balloon option

is an option where the quantity of option bought will increase if certain conditions are met, such as barriers being triggered.

Barrier option

has a payoff that depends on whether or not a specified level of the underlying is reached before expiration. In an 'out' option if the level is reached (triggered) then the option immediately becomes worthless. In an 'in' option the contract is worthless unless the level is triggered before expiration. An 'up' option is one where the trigger level is above the initial price of the underlying and a 'down' option is one where the trigger level is below the initial price of the underlying. Thus one talks about contracts such as the 'up-and-in call' which will have the same payoff as a call option but only if the barrier is hit from below. In these contracts one must specify the barrier level, whether it is in or out, and the payoff at expiration. A double barrier option has both an upper and a lower barrier. These contracts are bought by those with very specific views on the direction of the underlying, and its probability of triggering the barrier. These contracts are weakly path dependent. There are formulae for many types of barrier option, assuming that volatility is constant. For more complicated barrier contracts or when volatility is not constant these contracts must be valued using numerical methods. Both Monte Carlo and finite differences can be used but the latter is often preferable.

Basis swap

is an exchange of floating interest payments of one tenor for floating interest payments of another tenor, a six-month rate for a two-year rate for example. Since the two payments will generally move together if the yield curve experiences parallel shifts, the basis swap gives exposure to non-parallel movements in the yield curve such as flattening or steepening. More generally basis swap refers to any exchange in which the two floating rates are closely related, and therefore highly correlated.