has a payoff that depends on more than one underlying. A modest example would be an option that gives you at expiration the value of the higher performing out of two stocks. Another example would be a contract that pays the average of the values of 20 stocks at expiration provided that value is above a specified strike level. These contracts can be valued straightforwardly by Monte Carlo simulation as long as there is no early exercise feature. You would not use finite-difference methods because of the high dimensionality. If the contract is European, non-path dependent with all of the underlyings following lognormal random walks with constant parameters, then there is a closed-form formula for the value of the contract, and this can be calculated by numerical integration (quadrature). Basket options are popular in foreign exchange for those with exposure to multiple exchange rates. They can also be used as options on your own index. Although pricing these contracts can be theoretically straightforward they depend crucially on the correlation between the underlyings. These correlations can be very difficult to estimate since they can be quite unstable.
is one where the holder has the right to exercise on certain dates or periods rather than only at expiration (European exercise) or at any time (American exercise). Bermudan options cannot be worth less than their European equivalent and cannot be worth more than their American equivalent.
has a payoff that is discontinuous. For example a binary call pays off a specified amount if the underlying ends above the strike at expiration and is otherwise worthless. A one-touch pays off the specified amount as soon as the strike is reached; it can be thought of as an American version of the European binary. These contracts are also called digitals.
is a forward contract, usually FX, where the holder can terminate the contract at certain times if he so wishes.
is a periodic option in which the strike gets reset to the worst of the underlying and the previous strike. Similar to a cliquet option, but cheaper.
is an option to buy the underlying asset for a specified price, the strike or exercise price, at (European) or before (American) a specified data, the expiry or expiration. The underlying can be any security. They are bought to benefit from upward moves in the underlying, or if volatility is believed to be higher than implied. In the latter case the buyer would delta hedge the option to eliminate exposure to direction. Calls are written for the opposite reasons, of course. Also a holder of the underlying stock might write a call to gain some premium in a market where the stock is not moving much. This is called covered call writing. Simultaneous buying of the stock and writing a call is a buy-write strategy. For calls on lognormal underlyings in constant or time-dependent volatility worlds there are closed-form expressions for prices. With more complicated underlyings or volatility models these contracts can be priced by Monte Carlo or finite difference, the latter being more suitable if there is early exercise.
Other contracts may have call features or an embedded call. For example, a bond may have a call provision allowing the issuer to buy it back under certain conditions at specified times. If the issuer has this extra right then it may decrease the value of the contract, so it might be less than an equivalent contract without the call feature. Sometimes the addition of a call feature does not affect the value of a contract, this would happen when it is theoretically never optimal to exercise the call option. The simplest example of this is an
American versus a European call on a stock without any dividends. These both have the same theoretical value since it is never optimal to exercise early.