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Second Order Accurate IMEX Methods for Option Pricing Under Merton and Kou Jump-Diffusion Models

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Abstract

In this paper three implicit-explicit (IMEX) time semi-discrete methods, namely IMEX-BDF1, IMEX-BDF2 and CN-LF, are developed for solving parabolic partial integro-differential equations which arise in option pricing theory when the underlying asset follows a jump diffusion process. It is shown that IMEX-BDF2 and CN-LF are stable and second order accurate, whereas IMEX-BDF1 is stable but only first order accurate. After time semi-discretization, the resulting linear differential equations are solved by using a cubic B-spline collocation method. The methods so developed have computational complexity of \(O(MNlog_{2}(M))\) for Merton model and of \(O(MN)\) for Kou model, where \(N\) denotes the number of time steps and \(M\) the number of collocation points. Some numerical examples, for pricing European options under Merton and Kou jump-diffusion models with constant as well as variable volatility, are presented to demonstrate the stability, convergence and computational complexity of the methods.

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Acknowledgments

The authors gratefully acknowledge the comments/suggestions of the referee which have greatly improved the paper. The work of authors [Lok Pati Tripathi, Alpesh Kumar] is supported by Council of Scientific and Industrial Research(CSIR), India.

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  1. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India

    Mohan K. Kadalbajoo, Lok Pati Tripathi & Alpesh Kumar

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  1. Mohan K. Kadalbajoo
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  2. Lok Pati Tripathi
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  3. Alpesh Kumar
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Correspondence to Lok Pati Tripathi.

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Kadalbajoo, M.K., Tripathi, L.P. & Kumar, A. Second Order Accurate IMEX Methods for Option Pricing Under Merton and Kou Jump-Diffusion Models. J Sci Comput 65, 979–1024 (2015). https://doi.org/10.1007/s10915-015-0001-z

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  • DOI: https://doi.org/10.1007/s10915-015-0001-z

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