For Asian options, the payoff is determined by the average underlying price over some pre-set period of time.
There are two types of Asian options: Average Price Option (fixed strike), where the strike price is predetermined and the averaging price of the underlying asset is used for payoff calculation; and Average Strike Option (floating strike), where the averaging price of the underlying asset over the duration becomes the strike price.
One advantage of Asian options is that these reduce the risk of market manipulation of the underlying instrument at maturity.
This can be an advantage for corporations that are subject to the Financial Accounting Standards Board revised Statement No.
123, which required that corporations expense employee stock options.
[2] 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 a 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."
[3][4][5][6] There are numerous permutations of Asian option; the most basic are listed below: The Average
) we have the average given by There also exist Asian options with geometric average; in the continuous case, this is given by A discussion of the problem of pricing Asian options with Monte Carlo methods is given in a paper by Kemna and Vorst.
[7] In the path integral approach to option pricing,[8] the problem for geometric average can be solved via the Effective Classical potential [9] of Feynman and Kleinert.
[10] Rogers and Shi solve the pricing problem with a PDE approach.
[11] A Variance Gamma model can be efficiently implemented when pricing Asian-style options.
Then, using the Bondesson series representation to generate the variance gamma process can increase the computational performance of the Asian option pricer.
[12] Within jump diffusions and stochastic volatility models, the pricing problem for geometric Asian options can still be solved.
[13] For the arithmetic Asian option in Lévy models, one can rely on numerical methods[13] or on analytic bounds.
[14] We are able to derive a closed-form solution for the geometric Asian option; when used in conjunction with control variates in Monte Carlo simulations, the formula is useful for deriving fair values for the arithmetic Asian option.
, we find that the integrals are equivalent - this will be useful later on in the derivation.
Using martingale pricing, the value of the European Asian call with geometric averaging
However, it is easy to verify with Itô isometry that the integral is normally distributed as:
Now it is possible the calculate the value of the European Asian call with geometric averaging!
Going through the same process as is done with the Black-Scholes model, we are able to find that:
In fact, going through the same arguments for the European Asian put with geometric averaging
This implies that there exists a version of put-call parity for European Asian options with geometric averaging:
There are some variations that are sold in the over-the-counter market.
For example, BNP Paribas introduced a variation, termed conditional Asian option, where the average underlying price is based on observations of prices over a pre-specified threshold.
A conditional Asian put option has the payoff
Such an option offers a cheaper alternative than the classic Asian put option, as the limitation on the range of observations reduces the volatility of average price.
It is typically sold at the money and last for up to five years.
The pricing of conditional Asian option is discussed by Feng and Volkmer.