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A Spectral Element Method for Option Pricing Under Regime-Switching with Jumps

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Abstract

In this paper, we propose the spectral element method to price European, digital, butterfly, American, discrete and continuous barrier options in a Markovian jump-diffusion regime-switching economy. The spectral element method discretisation is considered for the approximation of the spatial derivatives in a system of partial integro-differential equations and is chosen because it possesses spectral accuracy such that highly accurate option prices can be obtained using a small number of grid discretisation nodes. Essentially, the spectral element method consists of splitting the computational domain into as many elements as needed and approximating the basis functions by high-order orthogonal polynomials within each element. In order to sustain the high-order convergence in time, we also use an exponential time integration scheme to solve the semi-discrete system. Our numerical examples support our error analysis and indicate that the spectral element method converges exponentially for the values and the hedging parameters of the regime-dependent options. Therefore, the proposed scheme provides a viable alternative to the finite difference or finite element methods which usually converge only algebraically.

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Notes

  1. A higher number of regime-switching states implies more model parameters such that calibration algorithms might not converge to the correct solution when the parsimony of the model is destroyed.

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Authors and Affiliations

  1. Department of Mathematics, University of Mauritius, Réduit, Mauritius

    Geraldine Tour, Nawdha Thakoor & Désiré Yannick Tangman

  2. School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

    Jingtang Ma

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  1. Geraldine Tour
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  2. Nawdha Thakoor
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  3. Jingtang Ma
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The research of Geraldine Tour was supported by a postgraduate research scholarship from the Tertiary Education Commission. The work of Jingtang Ma was supported by National Natural Science Foundation of China (Grant No. 11671323), Program for New Century Excellent Talents in University of China (Grant No. NCET-12-0922), and the Fundamental Research Funds for the Central Universities of China (JBK1805001). The work was done when J. Ma visited University of Mauritius in August 2019 and D. Y. Tangman visited SWUFE in December 2019.

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Tour, G., Thakoor, N., Ma, J. et al. A Spectral Element Method for Option Pricing Under Regime-Switching with Jumps. J Sci Comput 83, 61 (2020). https://doi.org/10.1007/s10915-020-01252-7

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