A transform-based method for pricing Asian options under general two-dimensional models

Abstract

We propose a unified transform-based method, which we call the extended double spiral (EDS) method, for pricing arithmetic Asian options under general two-dimensional (2D) models that nest regime-switching Lévy models, stochastic volatility (SV) models with Lévy jumps, and time-changed Lévy models. We first construct a new single backward induction in the state space that relaxes the restriction of the independent increments of the log-asset price. Second, we build an exact and explicit double backward induction in the Fourier space based on this single backward induction, a combination of the 1D Fourier transform method and a key function characterizing the 2D model, and the double spiral method. Third, we develop a unified EDS algorithm to recursively implement this double backward induction via the fast Fourier transform (FFT), various quadrature rules, asymmetric truncation boundaries, and so on. Extensive numerical results across a broad class of 2D models, monitoring frequencies, option moneyness, and model parameters demonstrate that our method is remarkably accurate, efficient, robust, simple to implement, and widely applicable.

Keywords:

• Arithmetic Asian options
• Two-dimensional models
• Extended double spiral method
• Fast Fourier transform

JEL Classifications:

• C00
• C63
• G13

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Here and subsequently, the state space and the Fourier space refer to the component Y.

2 As a remark, in the continuous case, we need the transformation from ${\nu }^{\prime }$ to $\ln n{u}^{\prime }$ to calculate (17(17) $\begin{array}{rl}{\mathcal{Q}}_{h,M,\tilde{M}}^{\left(i\right)}g(\text{β},\nu )& :=\frac{1}{2\pi }{\int }_E\left(\sum _{m=-\tilde{M}}^M{\overline{\mathrm{\Gamma }}}^{\left(i\right)}\right(\text{β}-mh\left)g^{\left({\alpha }_i\right)}\right(mh,{\nu }^{\prime }\left)h\right)\\ & \times {\mathrm{\Psi }}_{\mathrm{\Delta }}(-\text{β}+\mathrm{i}{\alpha }_{3-i};\nu ,{\nu }^{\prime })\mathrm{d}{\nu }^{\prime }\end{array}$(17) ) when the left tail of the density function of the process v in the function Ψ grows rapidly.

3 The derivations originate from a manuscript by Cai and Zeng (2023).

Additional information

Funding

Pingping Zeng would like to acknowledge the support from the National Natural Science Foundation of China (Grant Nos. 11701266 and 12171228), and the Philosophy and Social Science Planning Project of Guangdong Province, China (Grant No. GD20XGL31).