TY - JOUR

T1 - Pricing Asian option by the FFT with higher-order error convergence rate under Lévy processes

AU - Chiu, Chun Yuan

AU - Dai, Tian-Shyr

AU - Lyuu, Yuh Dauh

PY - 2015/2/1

Y1 - 2015/2/1

N2 - 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.

AB - 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.

KW - Asian option

KW - Fast Fourier Transform

KW - Newton-Cotes integration formula

KW - Pricing

UR - http://www.scopus.com/inward/record.url?scp=84920167529&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2014.12.002

DO - 10.1016/j.amc.2014.12.002

M3 - Article

AN - SCOPUS:84920167529

VL - 252

SP - 418

EP - 437

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

ER -