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Mathematical Analysis and its Applications pp 439–450Cite as

Efficient Meshfree Method for Pricing European and American Put Options on a Non-dividend Paying Asset

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 143))

Abstract

We develop efficient meshfree method based on radial basis functions (RBFs) to solve European and American option pricing problems arising in computational finance. The application of RBFs leads to system of differential equations which are then solved by a time integration \(\theta \)-method. The main difficulty in pricing the American options lies in the fact that these options are allowed to be exercised at any time before their expiry. Such an early exercise right purchased by the holder of the option results into a free boundary problem. Following the approach of Nielsen et al. [B.F. Nielsen, O. Skavhaug and A. Tveito, Penalty methods for the numerical solution of American multi-asset option problems. J. Comput. Appl. Math. 222, 3–16 (2008)], we use a small penalty term to remove the free boundary. The method is analyzed for stability. Numerical results describing the payoff functions and option values are also present. We also compute the two important Greeks, delta and gamma, of these options.

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References

  1. Black, F., Scholes, M.: Pricing of options and corporate liabilities. J. Pol. Eco. 81(3), 637–654 (1973)

    Article  Google Scholar 

  2. Broadie, M., Detemple, J.: American option valuation: new bounds, approximations, and a comparison of existing methods. Rev. Fin. Stud. 9, 1211–1250 (1996)

    Article  Google Scholar 

  3. Chawla, M.M., AL-Zanaidi, M.A., Evans, D.J.: Generralized trapezoidal formulas for valuing American options. Int. J. Comp. Math. 81(3), 375–381 (2004)

    Google Scholar 

  4. Fasshauer, G.E., Khaliq, A.Q.M., Voss, D.A.: Using meshfree approximation for multi-asset American option. J. Chin. Inst. Eng. 27(4), 563–571 (2004)

    Article  Google Scholar 

  5. Hull, J.C.: Options, Futures, and Other Derivatives. Prentice Hall, Upper Saddle River (2009)

    Google Scholar 

  6. Ju, N.: Pricing an American option by approximating its early exercise boundary as a multipiece exponential function. Rev. Fin. Stud. 11, 627–646 (1998)

    Article  Google Scholar 

  7. Kallast, S., Kivinukk, A.: Pricing and hedging American options using approximations by Kim integral equations. Eur. Fin. Rev. 7(361–383), 361–383 (2003)

    Article  MATH  Google Scholar 

  8. Kansa, E.J.: Multiquadrics—a scatered data approximation scheme with applications to computional Fluid-Dynamics-I. Comput. Math. Appl. 19, 127–145 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kansa, E.J.: Multiquadrics—a scatered data approximation scheme with applications to computional Fluid-Dynamics-II. Comput. Math. Appl. 19, 147–161 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Khaliq, A.Q.M., Voss, D.A., Kazmi, S.H.K.: A linearly implicit predictor-corrector scheme for pricing American options using a penalty method approach. J. Bank. Fin. 30, 489–502 (2006)

    Article  Google Scholar 

  11. Khaliq, A.Q.M., Voss, D.A., Kazmi, S.H.K.: Adaptive \(\theta \)-methods for pricing American options. J. Comput. Appl. Math. 222, 210–227 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Merton, R.C.: Theory of rational option pricing. Bell J. Eco. Man. Sci. 4, 141–183 (1973)

    Article  MathSciNet  Google Scholar 

  13. Merton, R.C.: Option pricing when the underlying stocks are discontinuous. J. Fin. Eco. 5, 125–144 (1976)

    Article  Google Scholar 

  14. Tatari, M., Dehghan, M.: A method for solving partial differential equations via radial basis functions: application to the heat equation. Eng. Anal. Bound. Elem. 34, 206–212 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wilmott, P., Howison, S., Dewynne, J.: The Mathematical Financial Derivatives: A Student Introduction. Cambridge University Press, Oxford (1995)

    Book  Google Scholar 

  16. Wua, Z., Hon, Y.C.: Convergence error estimate in solving free boundary diffusion problem by radial basis functions method. Eng. Anal. Bound. Elem. 27, 73–79 (2003)

    Article  Google Scholar 

  17. Zhao, J., Davison, M., Corless, R.M.: Compact finite difference method for American option pricing. J. Comput. Appl. Math. 206, 306–321 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

We thank the anonyms referees for their valuable comments and suggestions. The research of KCP was supported by the South African National Research Foundation. AOMS acknowledges the financial support of AL-Neelain University, Sudan.

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of the Western Cape, Bellville, 7535, South Africa

    Kailash C. Patidar & Abdelmgid O. M. Sidahmed

Authors
  1. Kailash C. Patidar
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  2. Abdelmgid O. M. Sidahmed
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Correspondence to Kailash C. Patidar .

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Editors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    P. N. Agrawal

  2. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    R. N. Mohapatra

  3. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    Uaday Singh

  4. Dept. of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada

    H. M. Srivastava

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Patidar, K.C., Sidahmed, A.O.M. (2015). Efficient Meshfree Method for Pricing European and American Put Options on a Non-dividend Paying Asset. In: Agrawal, P., Mohapatra, R., Singh, U., Srivastava, H. (eds) Mathematical Analysis and its Applications. Springer Proceedings in Mathematics & Statistics, vol 143. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2485-3_36

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