The style or family of a financial option is a general term denoting the class into which the option falls, usually defined by the manner in which the option may be exercised. The two great families are european and american.
A european option may be exercised by only at the maturity of the option, i.e. at a single point in time
An american option on the other hand may be exercised at any time before expiry. Oddly, american options are very rarely exercised. That is because any option has a time value and is therefore worth more unexercised. Buyers who wish to realise the full value of their option will therefore prefer sell it on a recognised market, either an over the counter (OTC) or listed market.
European options are typically valued using the Black-Scholes or Black 76 formulas. American options are more difficult to value, and a choice of models are available (for example Whaley, binomial options model, Monte Carlo and others).
A bermudan option is an option where the buyer has the right to exercise at a set (always discretely spaced) number of times. For example a typical bermudan swaption might confer the opportunity to enter into an interest rate swap. The option holder might decide to enter into the swap at the first exercise date (and so enter into, say, a ten-year swap) or defer and have the opportunity to enter in six months time (and so enter a nine-year and six-month swap). Most exotic interest rate options are of bermudan style.
A quanto option is an option on some underlying in one currency but paid in another currency. For example a standard call option on IBM, which is denominated in dollars pays $MAX(S-K,0) (where S is the stock price at maturity and K is the strike). A quanto stock option might pay £MAX(S-K,0). The pricing of such options naturally needs to take into account the correlation between the exchange rate of the two currencies involved and the underlying stock price.
An asian option is an option where the payoff is not determined by the underlying price at maturity but by the average underlying price over some pre-set period of time. For example an asian call option might pay MAX(DAILY_AVERAGE_OVER_LAST_THREE_MONTHS(S) - K , 0).
Some other options styles that have occasionally attracted the interest of the market or theoreticians.
A lookback option is an option where the option owner has the right to buy (respectively sell) the underlying instrument at its lowest (respectively highest) price over some preceding period.
A russian option is a lookback option which runs for perpetuity. That is, there is no end to the period into which the owner can look back.
A game option or israeli option is an option where the writer has the opportunity to cancel the option he has offered, but must pay the payoff at that point plus a penalty fee.
The payoff of a parisian option is dependent of the amount of time the option has spent above or below a strike price.
Barrier options involve a mechanism where if a price is crossed by the underlying, the option either can be exercised or can no longer be exercised. They can be four different types:
Up and in options can be exercised only if the underlying goes above a certain price.
Up and out options cannot be exercised if the underlying goes above a certain price.
Down and in options can be exercised only if the underlying goes below a certain price.
Down and out options cannot be exercised if the underlying goes below a certain price.
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