Jan Vecer
New Pricing of Asian Options

    Asian options are securities with payoff which depends on the average of the underlying stock price over certain time interval. Since no general analytical solution for the price of the Asian option is known, a variety of techniques have been developed to analyze arithmetic average Asian options. There is enormous literature devoted to study of this option. A number of approxima-tions that produce closed form expressions have appeared, most recently in Thompson (1999), who provides tight analytical bounds for the Asian option price. Geman and Yor (1993) computed the Laplace transform of the Asian option price, but numerical inversion remains problematic for low volatility and/or short maturity cases as shown by Fu, Madan and Wang (1998). Monte Carlo simulation works well, but it can be computationally expensive without the enhancement of variance reduction techniques and one must account for the inherent discretization bias resulting from the approximation of continuous time processes through discrete sampling as shown by Broadie,
Glasserman and Kou (1999).

    In general, the price of an Asian option can be found by solving a partial differential equation (PDE) in two space dimensions (see Ingersoll (1987)), which is prone to oscillatory solutions. Ingersoll also observed that the two-dimensional PDE for a oating strike Asian option can be reduced to a one-dimensional PDE. Rogers and Shi (1995) have formulated a one-dimensional PDE that can model both oating and fixed strike Asian options. However this one-dimensional PDE is difficult to solve numerically since the diffusion term is very small for values of interest on the finite difference grid. The Dirac delta function also appears as a coefficient of the PDE in the case of the oating strike option. There are several articles devoted to enhance the numerical performance of this type of PDE. Andreasen (1998) applied Rogers and Shi�fs reduction to discretely sampled Asian option. 

    Very recently there are several independent efforts to unify pricing techniques for different types of options and relate these techniques to pricing Asian option. Lipton (1999) noticed similarity of pricing equations for the passport, lookback and the Asian option, again using Rogers and Shi�fs reduction. Shreve and Vecer (2000) developed techniques for pricing options on a traded account, which include all options which could be replicated by a self-financing trading in the underlying asset. These option include European, passport, vacation, as well as Asian options. Numerical techniques for pricing contracts of this type are described in Vecer (2000). Hoogland and Neumann (2001) developed alternative framework for pricing various types of options using scale invariance methods. They noticed in their very recent work (2000) close links een pricing Asian options and options on the stock paying cash dividend and derived more general semianalytic solutions for Asian option prices.

    This paper provides even simpler and unifying approach for pricing Asian options, for both discrete and continuous arithmetic average. The resulting one-dimensional PDE for the price of the Asian option can be easily implemented to give extremely fast and accurate results. Moreover, this approach easily incorporates cases of continuous or discrete dividends.

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