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Review of Derivatives Research

Erschienen in:

06.02.2017

A unified approach for the pricing of options relating to averages

verfasst von: Hideharu Funahashi, Masaaki Kijima

Erschienen in: Review of Derivatives Research | Ausgabe 3/2017

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Abstract

In this paper, we consider generalized Asian options and propose a unified approximation method for the pricing of such options when the underlying process is a diffusion. Through numerical examples, we show that our approximation method is accurate enough to be used in practice for the pricing of any type of Asian options that has been treated separately in the literature. Comparisons are made with the existing methods in the literature to support the usefulness of our method.

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See, e.g., Fusai et al. (2008) for the usage of Asian options in the actual markets.

For an extensive literature review, we refer to Cai and Kou (2012).

A symmetry result between the floating-strike case and the fixed-strike case is first obtained in Henderson and Wojakowski (2002). This symmetry has been extended to various setting as a duality by Vecer and Xu (2004) and further by Eberlein et al. (2008).

An extension to the stochastic volatility case is straightforward by using the method given in Funahashi (2014) and omitted.

If a dividend rate d is considered, we simply replace r by \(r-d\) and the same arguments can apply.

The Hermite polynomials are defined by \(h_{n}(x) = (-1)^{n} \mathrm{e}^{x^2/2} \frac{\mathrm{d}^{n}}{\mathrm{d}x^{n}} \mathrm{e}^{-x^2/2}\), \(n=1,2, \dots \), with \(h_0(x)=1\). For example, we have \(h_{1}(x)=x\), \(h_{2}(x)=x^{2} - 1\), \(h_{3}(x)=x^{3} - 3x\), etc.

Throughout this paper, the benchmark values are computed by MC simulations with 300,000 trials and 20,000 time steps for all the cases.

The PDFs of the averages are more concentrated and becomes more symmetric around the mean compared with the stock price \(S_T\).

As shown in Cai and Kou (2012), the double Laplace inversion method agrees with the eigenfunction expansion method to ten decimal points.

In practice, daily-sampled Asian options are most frequently traded. For one-year maturity options, the number of sampling is given by \(N=250\).

The case \(\eta =1\) corresponds to the GBM case, i.e. the Black–Scholes setting.

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Titel: A unified approach for the pricing of options relating to averages
verfasst von: Hideharu Funahashi
Masaaki Kijima
Publikationsdatum: 06.02.2017
Verlag: Springer US
Erschienen in: Review of Derivatives Research / Ausgabe 3/2017
Print ISSN: 1380-6645
Elektronische ISSN: 1573-7144
DOI: https://doi.org/10.1007/s11147-017-9128-4