# 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. There are two types of Asian Fixed Strike option, the Asian Fixed Strike call and the Asian Fixed Strike put. In general they do not differ in definition, only in how the pay-off is calculated.

One advantage of Asian options is that these reduce the risk of market manipulation of the underlying instrument at maturity (Kemna & 1990 1077).[1] 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.[2]

## 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.[3][4][5][6]

## 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
${\displaystyle C(T)={\text{max}}\left(A(0,T)-K,0\right),}$
where A denotes the average price for the period [0, T], and K is the strike price. The equivalent put option is given by
${\displaystyle P(T)={\text{max}}\left(K-A(0,T),0\right).}$
• The floating strike (or floating rate) Asian call option has the payout
${\displaystyle C(T)={\text{max}}\left(S(T)-kA(0,T),0\right),}$
where S(T) is the price at maturity and k is a weighting, usually 1 so often omitted from descriptions. The equivalent put option payoff is given by
${\displaystyle P(T)={\text{max}}\left(kA(0,T)-S(T),0\right).}$

## Types of averaging

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

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

For the case of discrete monitoring (with monitoring at the times ${\displaystyle 0=t_{0},t_{1},t_{2},\dots ,t_{n}=T}$ and ${\displaystyle t_{i}=i\cdot {\frac {T}{n}}}$) we have the average given by

${\displaystyle A(0,T)={\frac {1}{n}}\sum _{i=0}^{n-1}S(t_{i}).}$

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

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

## 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.[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]

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.[12]

Within Lévy models the pricing problem for geometrically 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]

## References

1. ^ Kemna et al. 1990, p 1077
2. ^ FASB (2004). Share-based payment (Report). Financial Accounting Standards Board.
3. ^ William Falloon; David Turner, eds. (1999). "The evolution of a market". Managing Energy Price Risk. London: Risk Books.
4. ^ Wilmott, Paul (2006). "25". Paul Wilmott on Quantitative Finance. John Wiley & Sons. p. 427.
5. ^ Palmer, Brian (July 14, 2010), Why Do We Call Financial Instruments "Exotic"? Because some of them are from Japan., Slate
6. ^ Glyn A. Holton (2013). "Asian Option (Average Option)". Risk Encyclopedia. Archived from the original on 2013-12-06. Retrieved 2013-08-10. 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." "[I]n 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.
7. ^ Kemna, A.G.Z.; Vorst, A.C.F.; Rotterdam, E.U.; Instituut, Econometrisch (1990), A Pricing Method for Options Based on Average Asset Values
8. ^ Kleinert, H. (2009), Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, archived from the original on 2009-04-24, retrieved 2010-01-10
9. ^ Feynman R.P., Kleinert H. (1986), "Effective classical partition functions" (PDF), Physical Review A, 34 (6): 5080–5084, Bibcode:1986PhRvA..34.5080F, doi:10.1103/PhysRevA.34.5080, PMID 9897894
10. ^ Devreese J.P.A.; Lemmens D.; Tempere J. (2010), "Path integral approach to Asianoptions in the Black-Scholes model", Physica A, 389 (4): 780–788, arXiv:0906.4456, Bibcode:2010PhyA..389..780D, doi:10.1016/j.physa.2009.10.020
11. ^ Rogers, L.C.G.; Shi, Z. (1995), "The value of an Asian option" (PDF), Journal of Applied Probability, 32 (4): 1077–1088, doi:10.2307/3215221, JSTOR 3215221, archived from the original (PDF) on 2009-03-20, retrieved 2008-11-28
12. ^ Mattias Sander. Bondesson's Representation of the Variance Gamma Model and Monte Carlo Option Pricing. Lunds Tekniska Högskola 2008
13. ^ a b Fusai, Gianluca.; Meucci, Attilio (2008), "Pricing discretely monitored Asian options under Lévy processes", J. Bank. Finan., 32 (10): 2076–2088, doi:10.1016/j.jbankfin.2007.12.027
14. ^ Lemmens, Damiaan; Liang, Ling Zhi; Tempere, Jacques; De Schepper, Ann (2010), "Pricing bounds for discrete arithmetic Asian options under Lévy models", Physica A: Statistical Mechanics and its Applications, 389 (22): 5193–5207, Bibcode:2010PhyA..389.5193L, doi:10.1016/j.physa.2010.07.026