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Implicit-explicit Runge–Kutta methods for pricing financial derivatives in state-dependent regime-switching jump-diffusion models

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Abstract

In this paper, we have devised a novel class of implicit-explicit Runge–Kutta methods for the valuation of financial derivatives under state-dependent regime-switching jump-diffusion models. The developed methods utilize an implicit technique to solve the problem without performing inversion of the coefficient matrix at each state of the economy. The pricing of European options under the regime-switching jump-diffusion process is formulated by coupled partial integro-differential equations, whereas the pricing framework for American options is based on addressing coupled linear complementary problems. The operator splitting technique is employed along with implicit-explicit methods to solve the linear complementarity problems. Consistency and convergence results of the developed methods are theoretically established using the discrete \(\ell ^{2}\)-norm. The accuracy and efficiency of developed methods are validated by solving the European and American options at different states of the economy under regime-switching jump-diffusion models. Finally, the computed results are compared with the recently proposed methods mentioned in the literature.

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Acknowledgements

Vikas Maurya gratefully acknowledges the financial support from the Council of Scientific and Industrial Research (09/1088(0006) /2019-EMR-I), Government of India, New Delhi.

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Authors and Affiliations

  1. Mathematics and Computing Laboratory, Department of Mathematical Sciences, Rajiv Gandhi Institute of Petroleum Technology, Jais, Amethi, UP, 229304, India

    Vikas Maurya, Ankit Singh & Manoj K. Rajpoot

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  1. Vikas Maurya
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  2. Ankit Singh
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  3. Manoj K. Rajpoot
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VM Conceptualization, computations, and manuscript writing. AS Conceptualization and computations. MKR Supervision, methodology, and original draft preparation. All authors reviewed the manuscript

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Correspondence to Manoj K. Rajpoot.

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Appendix

Appendix

1.1 The free parameter values used in the numerical simulation for pricing the American put option under three-state RSMJD model

State-1: \(\alpha _{1}=\alpha _{2}=10.0\), \(\alpha _{3}=5.0\), \( \beta _{1}=\beta _{2}=\beta _{3}=\gamma _{1}=\gamma _{2}=\gamma _{3} =1.0\).

State-2: \(\alpha _{1}=5.0\), \(\alpha _{2}=\alpha _{3}=2.0\), \( \beta _{1}=\beta _{2}=\beta _{3}=\gamma _{1}=\gamma _{2}=\gamma _{3} =1.0\).

State-3: \(\alpha _{1}=5.0\), \( \alpha _{2}=\alpha _{3}=\beta _{1}=\beta _{2}=\beta _{3}=\gamma _{1}=\gamma _{2}=\gamma _{3} =1.0\).

1.2 The free parameter values used in the numerical simulation for pricing the European and American put options under the five-state RSKJD model

European put option, State-3: \(\alpha _{3}=-3.25\), \(\alpha _{1}=\alpha _{2}=\alpha _{4}=\alpha _{5}=\beta _{1}=\beta _{2}=\beta _{3}=\beta _{4}=\beta _{5} =\gamma _{1}=\gamma _{2}=\gamma _{3} =\gamma _{4} =\gamma _{5} =1.0\).

American put option, State-5: \(\alpha _{1}=\alpha _{3}=\alpha _{4} = 5.0\), \(\alpha _{2}=\alpha _{5}=\beta _{1}=\beta _{2}=\beta _{3}=\beta _{4}=\beta _{5} =\gamma _{1}=\gamma _{2}=\gamma _{3} =\gamma _{4} =\gamma _{5} =1.0\).

American trader’s put option: \(\alpha _{1}=\alpha _{2}=\alpha _{3}=\alpha _{4}=\alpha _{5}=\beta _{1}=\beta _{2}=\beta _{3}=\beta _{4}=\beta _{5} =\gamma _{1}=\gamma _{2}=\gamma _{3}=1\), \(\gamma _{4}=10, \gamma _{5} =15.0\).

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Maurya, V., Singh, A. & Rajpoot, M.K. Implicit-explicit Runge–Kutta methods for pricing financial derivatives in state-dependent regime-switching jump-diffusion models. J. Appl. Math. Comput. 70, 1601–1632 (2024). https://doi.org/10.1007/s12190-024-02020-8

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