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Using Chebyshev Polynomials to Approximate Partial Differential Equations

Author

Listed:
  • Guglielmo Maria Caporale
  • Mario Cerrato

Abstract

This paper suggests a simple method based on a Chebyshev approximation at Chebyshev nodes to approximate partial differential equations. It consists in determining the value function by using a set of nodes and basis functions. We provide two examples: pricing a European option and determining the best policy for shutting down a machine. The suggested method is flexible, easy to programme and efficient. It is also applicable in other fields, providing efficient solutions to complex systems of partial differential equations.

Suggested Citation

  • Guglielmo Maria Caporale & Mario Cerrato, 2008. "Using Chebyshev Polynomials to Approximate Partial Differential Equations," CESifo Working Paper Series 2308, CESifo.
    • Handle: RePEc:ces:ceswps:_2308
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    References listed on IDEAS

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    1. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    2. Avinash K. Dixit & Robert S. Pindyck, 1994. "Investment under Uncertainty," Economics Books, Princeton University Press, edition 1, number 5474.
    3. Abadir, Karim M. & Rockinger, Michael, 2003. "Density Functionals, With An Option-Pricing Application," Econometric Theory, Cambridge University Press, vol. 19(5), pages 778-811, October.
    4. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, April.
    5. Dangl, Thomas & Wirl, Franz, 2004. "Investment under uncertainty: calculating the value function when the Bellman equation cannot be solved analytically," Journal of Economic Dynamics and Control, Elsevier, vol. 28(7), pages 1437-1460, April.
    6. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    7. Avinash Dixit, 1992. "Investment and Hysteresis," Journal of Economic Perspectives, American Economic Association, vol. 6(1), pages 107-132, Winter.
    8. Elias Tzavalis & Shijun Wang, 2003. "Pricing American Options under Stochastic Volatility: A New Method Using Chebyshev Polynomials to Approximate the Early Exercise Boundary," Working Papers 488, Queen Mary University of London, School of Economics and Finance.
    9. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
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    Cited by:

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    2. Om Prakash Yadav & Shashwati Ray, 2021. "A novel method of preprocessing and modeling ECG signals with Lagrange–Chebyshev interpolating polynomials," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 12(3), pages 377-390, June.
    3. Alejandro Mosiño, 2012. "Using Chebyshev Polynomials to Approximate Partial Differential Equations: A Reply," Computational Economics, Springer;Society for Computational Economics, vol. 39(1), pages 13-27, January.
    4. Mosiño, Alejandro, 2012. "Producing energy in a stochastic environment: Switching from non-renewable to renewable resources," Resource and Energy Economics, Elsevier, vol. 34(4), pages 413-430.
    5. Polychronis Manousopoulos & Michalis Michalopoulos, 2015. "Term structure of interest rates estimation using rational Chebyshev functions," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 38(2), pages 119-146, October.

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    More about this item

    Keywords

    European options; Chebyshev polynomial approximation; Chebyshev nodes;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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