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
SN - 0096-3003
VL - 252
SP - 418
EP - 437
JO - Applied Mathematics and Computation
JF - Applied Mathematics and Computation
ER -