A preconditioned iterative method for coupled fractional partial differential equation in European option pricing

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-01-03 DOI:10.1515/math-2023-0169

Shuang Wu, Lot-Kei Chou, Xu Chen, Siu-Long Lei

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引用次数: 0

Abstract

Recently, regime-switching option pricing based on fractional diffusion models has been used, which explains many significant empirical facts about financial markets better. There are many methods to solve the problem, but to the best of our knowledge, effective preconditioners for the second-order schemes have not been proposed. Thus, in this article, an implicit numerical scheme is developed for a regime-switching European option pricing problem under a multi-state tempered fractional model. The scheme is proven to be unconditionally stable and converges quadratically in space and linearly in time. Besides, the resulting linear system is solved using an iterative method, and a preconditioner is proposed to accelerate the rate of convergence. The preconditioner is constructed through circulant approximations to the Toeplitz blocks due to the coefficient matrix, which is is a block matrix with Toeplitz blocks. The spectral analysis of the preconditioned matrix is given, which demonstrates that the spectrum of the preconditioned matrix is clustered around 1. Numerical examples show the efficiency of the proposed method, and an empirical study is also provided.

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欧式期权定价中耦合分式偏微分方程的预条件迭代法

最近，基于分数扩散模型的制度转换期权定价得到了应用，它能更好地解释金融市场的许多重要经验事实。解决该问题的方法有很多，但就我们所知，尚未提出有效的二阶方案预处理。因此，本文针对多状态节制分式模型下的制度切换欧式期权定价问题，开发了一种隐式数值方案。该方案被证明是无条件稳定的，并且在空间上呈二次收敛，在时间上呈线性收敛。此外，还使用迭代法求解了所得到的线性系统，并提出了一个加速收敛速度的前置条件器。由于系数矩阵是一个带有 Toeplitz 块的块矩阵，因此通过对 Toeplitz 块的环状近似来构建预处理器。给出了预处理矩阵的频谱分析，结果表明预处理矩阵的频谱集中在 1 附近。数值示例显示了所提方法的效率，并提供了实证研究。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.