# Asian option

An Asian option (or average value option) is a special type of option contract. For Asian options the payoff is determined by the average underlying price over some pre-set period of time. This is different from the case of the usual European option and American option, where the payoff of the option contract depends on the price of the underlying instrument at exercise; Asian options are thus one of the basic forms of exotic options.

One advantage of Asian options is that these reduce the risk of market manipulation of the underlying instrument at maturity (Kemma & 1990 1077). Another advantage of Asian options involves the relative cost of Asian options compared to European or American options. Because of the averaging feature, Asian options reduce the volatility inherent in the option; therefore, Asian options are typically cheaper than European or American options. This can be an advantage for corporations that are subject to the Financial Accounting Standards Board (2004 FASB) revised Statement No. 123, which required that corporations expense employee stock options.

## Etymology

In the 1980s Mark Standish was with the London-based Bankers Trust working on fixed income derivatives and proprietary arbitrage trading. David Spaughton worked as systems analyst in the financial markets with Bankers Trust since 1984 when the Bank of England first gave licences for banks to do foreign exchange options in the London market. In 1987 Standish and Spaughton were in Tokyo on business when "they developed the first commercially used pricing formula for options linked to the average price of crude oil." They called this exotic option, the Asian option, because they were in Asia.

## Permutations of Asian option

There are numerous permutations of Asian option; the most basic are listed below:

Fixed strike (also known as an average rate) Asian call payout

$P(T) = \text{max}\left( A(0,T) - K, 0 \right),$

where A denotes the average, and K the strike. The equivalent put option is given by

$P(T) = \text{max}\left( K - A(0,T), 0 \right).$

The floating strike (or floating rate) Asian call option has the payout

$P(T) = \text{max}\left( S(T) - k A(0,T), 0 \right),$

where k is a weighting, usually 1 so often omitted from descriptions. The equivalent put option payoff is given by

$P(T) = \text{max}\left( k A(0,T) - S(T), 0 \right).$

## Types of averaging

The Average $A$ may be obtained in many ways. Conventionally, this means an arithmetic average. In the continuous case, this is obtained by

$A(0,T) = \frac{1}{T} \int_{0}^{T} S(t) dt.$

For the case of discrete monitoring (with monitoring at the times $t_1, t_2, \dots, t_n$) we have the average given by

$A(0,T) = \frac{1}{N} \sum_{i=1}^{N} S(t_i).$

There also exist Asian options with geometric average; in the continuous case, this is given by

$A(0,T) = \exp \left( \frac{1}{T} \int_{0}^{T} \ln( S(t)) dt \right).$

## Pricing of Asian options

A discussion of the problem of pricing Asian options with Monte Carlo methods is given in a paper by Kemna and Vorst.

In the path integral approach to option pricing , the problem for geometric average can be solved via the Effective Classical potential  of Feynman and Kleinert.

Rogers and Shi solve the pricing problem with a PDE approach .

Variance Gamma model can be efficiently implemented when pricing Asian style options. Then using the Bondesson series representation for generating the variance gamma process shows to increase performance when pricing this type of option.

Within Lévy models the pricing problem for geometrically Asian options can still be solved. For the arithmetic Asian option in Lévy models one can rely on numerical methods or on analytic bounds .