Before discussing how options can be valued, we need to be able to distinguish the different types of options that are traded in financial markets. In general, options (when bought) provide the right to buy or sell and asset at a certain price fixed now at (or before) some date in the future. The value of option contracts is thus always as least as high as, and often higher than that of forward contracts. This is because forward contracts are an obligation to buy or sell whereas options provide a right.
Call and put options
Call options provide the buyer the right to buy a certain asset at a price fixed before (or on) expiration of the option contract. Call options can thus be used as an alternative to buying shares when having bullish expectations. Put options, on the other hand, provide the buyer a right to sell a certain asset at a price fixed before (or on) expiration of the option contract. Put options can thus be used to buy protection for shares you own, or to speculate on a downward market. The seller of either call or put options is thus obliged to sell (in case of call options) or buy (in case of put options) the underlying securities when asked to. In exchange for assuming this risk, the option seller receives a premium from the option buyer (i.e. the option price).
European, American and Bermudan options
The option style is important, both for valuation purposes as well as for risk management. The option style refers to when option buyers can exercise their options. European style options are the most simple. These options can only be exercised at maturity. American-style options on the other hand can be exercised at or before maturity. As a result, American-style options are (most of the time but no always) worth than European option. Bermudan options lie somewhere in between European- and American-style options. Bermudan options can be exercised at maturity or at certain fixed dates, and therefore not at every point in time between now and expiry.
Asian options provide a payoff at maturity that is a function of the average price it realized during a certain period, over a particular threshold. As a result of its backward-looking nature, simulation methods are needed to value Asian options.
Binary options provide a fixed payoff. Either the buyer (long binary) obtains nothing if the price at maturity is not higher than the strike price, or the buyer receives a fixed amount if the price ends up above the strike price. The exact amount by which the closing price exceeds the strike price is not important here. Whether it is 1 cent, 1 dollar, or 10 dollar does not affect the cashflow received. Because of its odd payoff structure, binary options are strongly related to pure speculation or gambling.
Lookback options are a partical kind of exotic options that provide at maturity a payoff in function of the most favorable realized price over a certain period. Due to their floating strike price, these kind of options always end up in the money and are thus more expensive.
There exist a huge amount of different option types in the market. With respect to option exercise, options can either be European, American or Bemudan style. All of these can be either cal or put options, which can be bought or sold. Among the exotic option family, binary options and Asian options are the most commonly known.