Pricing Asian option by the FFT with higher-order error convergence rate under Lévy processes

Pricing Asian options is a long-standing hard problem; there is no analytical formula for the probability density of its payoff even when the process of the underlying asset follows the simple lognormal diffusion process. It is known that the density function of a discretely-sampled Asian option’s p...

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Veröffentlicht in:Applied mathematics and computation Jg. 252; S. 418 - 437
Hauptverfasser: Chiu, Chun-Yuan, Dai, Tian-Shyr, Lyuu, Yuh-Dauh
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier Inc 01.02.2015
Schlagworte:
Algorithms
Asian
Asian option
Convergence
Density
Errors
Fast Fourier Transform
Interpolation
Mathematical analysis
Mathematical models
Newton–Cotes integration formula
Pricing
Fast Fourier Transform
Newton–Cotes integration formula
Asian option
Pricing
ISSN:0096-3003, 1873-5649
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Zusammenfassung:Pricing Asian options is a long-standing hard problem; there is no analytical formula for the probability density of its payoff even when the process of the underlying asset follows the simple lognormal diffusion process. It is known that the density function of a discretely-sampled Asian option’s payoff can be efficiently approximated by the Fast Fourier Transform (FFT). As a result, we can accurately price the option under more general Lévy processes. This paper shows that the pricing error of this approach, called the FFT approach, can be decomposed into the truncation error, the integration error, and the interpolation error. We prove that previous algorithms that follow the FFT approach converge quadratically. To improve the error convergence rate, our proposed algorithms reduce the integration error by the higher-order Newton–Cotes formulas and new integration rules derived from the Lagrange interpolating polynomial. The interpolation error is reduced by the higher-order Newton divided-difference interpolation formula. Consequently, our algorithms can be sped up by the FFT to achieve the same time complexity as previous algorithms, but with a faster error convergence rate. Numerical results are given to verify the efficiency and the fast convergence of our algorithms.
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ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2014.12.002