In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These options—as well as others where the payoff is calculated similarly—are referred to as "vanilla options". Options where the payoff is calculated differently are categorized as "exotic options". Exotic options can pose challenging problems in valuation and hedging. The key difference between American and European options relates to when the options can be exercised: A European option may be exercised only at the expiration date of the option, i.e. at a single pre-defined point in time. An American option on the other hand may be exercised at any time before the expiration date. For both, the payoff—when it occurs—is given by for a call option for a put option where is the strike price and is the spot price of the underlying asset. Option contracts traded on futures exchanges are mainly American-style, whereas those traded over-the-counter are mainly European. Most stock and equity options are American options, while indexes are generally represented by European options. Commodity options can be either style. Traditional monthly American options expire the third Saturday of every month (or the third Friday if the first of the month begins on a Saturday). They are closed for trading the Friday prior. European options traditionally expire the Friday prior to the third Saturday of every month. Therefore, they are closed for trading the Thursday prior to the third Saturday of every month. Assuming an arbitrage-free market, a partial differential equation known as the Black-Scholes equation can be derived to describe the prices of derivative securities as a function of few parameters. Under simplifying assumptions of the widely adopted Black model, the Black-Scholes equation for European options has a closed-form solution known as the Black-Scholes formula.