Asian Option (Average Option)

Explained:

An Asian option (also called an average option) is an option whose payoff is linked to the average value of the underlier on a specific set of dates during the life of the option. There are two basic forms:

An average rate option (or average price option) is a cash-settled option whose payoff is based on the difference between a the average value of the underlier during the life of the option and a fixed strike.

An average strike option is a cash settled or physically settled option. It is structured like a vanilla option except that its strike is set equal to the average value of the underlier over the life of the option.

Both forms can be structured as puts or calls. Exercise is generally European, but it is possible to specify early exercise provisions based upon an average-to-date. Averages can be calculated arithmetically:

[1]

or geometrically:

[2]

They can also be weighted with some weights w_i:

	[3]

[4]

David Spaughton tells the story of how Asian options got their name. He and Mark Standish worked for Bankers Trust in 1987. They were in Tokyo on business when they developed the first commercially used pricing formula for options linked to the average price of crude oil. Because they were in Asia, they called the options "Asian options." (see Falloon and Turner (1999))

End-users of commodities or energies tend to be exposed to average prices over time, so Asian options may appeal to them. Asian options are also popular with corporations, such as exporters, who have ongoing currency exposures. Asian options are also attractive because they tend to be less expensive—sell at lower premiums—than comparable vanilla puts or calls. This is because the volatility in the average value of an underlier tends to be lower than the volatility of the value of the underlier. Also, in situations where the underlier is thinly traded or there is the potential for its price to be manipulated, an Asian option offers some protection. It is more difficult to manipulate the average value of an underlier over an extended period of time than it is to manipulate it just at the expiration of an option.

Exact analytic formulas for average rate options don't exist. This is primarily due to the fact that the arithmetic average of a set of lognormal random variables has a distribution that is largely intractable. Analytic approximations have been proposed by: Turnbull and Wakeman (1991), Levy (1992) and Curran (1992). If the underlier is assumed lognormal, then its geometric average is lognormal. In a classic paper, Kemna and Vorst (1990) use this fact to derive an analytic solution for the price of a geometric average rate option. They use that solution as a control variate for a Monte Carlo solution for the price of an arithmetic average rate option. See also Wilmott, Dewynne and Howison (2000) and Seydel (2002) for numerical solutions based on exact differential equations. Tavella and Randell (2000) focus specifically on finite differences. Klassen (2001) discusses the use of binomial methods.

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