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

Pricing Asian options is a long standing hard problem since 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 option payoff can be expressed as a recursive function of sums of independent random variables. As a result, the density function of the option payoff can be efficiently approximated by the Fast Fourier Transform (FFT). The advantage of this approach is that we can evaluate Asian options under the more general Lévy process than the lognormal diffusion process. This paper shows that the pricing error of this approach can be decomposed into the truncation error, the integration error, and the interpolation error. We also prove that the pricing results generated by previous algorithms that follow the FFT approach converge quadratically. To improve the error convergence rate, our algorithm reduces the integration error by the higher-order Newton-Cotes formulas and new integration rules derived from the Lagrange interpolating polynomial. The interpolation error are also reduced by the higher-order Newton divideddifference interpolation formula. As a result, our algorithm can be sped up by the FFT to achieve the same time complexity as previous algorithms, but with a faster convergence rate. Numerical results are given to verify the efficiency and the fast convergence of our algorithm.