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A Kernel-Based Algorithm for Numerical Solution of Nonlinear PDEs in Finance

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Large-Scale Scientific Computing (LSSC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7116))

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Abstract

We present an algorithm for approximate solutions to certain nonlinear model equations from financial mathematics, using kernels techniques (fundamental solution, Green’s function) for the linear Black-Scholes operator as a basis of the computation. Numerical experiments for comparison the accuracy of the algorithms with other known numerical schemes are discussed. Finally, observations are given.

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References

  1. Abe, R., Ishimura, N.: Existence of solutions for the nonlinear partial differential equation arising in the optimal investment problem. Proc. Japan Acad., Ser. A 84, 11–14 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agliardi, R., Popivanov, P., Slavova, A.: Nonhypoellipticity and comparison principle for partial differential equations of Black-Scholes type. Nonl. Anal.: Real Word Appl. 12(3), 1429–1436 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ankudinova, J., Ehrhardt, M.: On tne numerical solution of nonlinear Black-Scholes equations. Int. J. of Comp. and Math. with Appl. 56, 779–812 (2008)

    MathSciNet  Google Scholar 

  4. Avellaneda, M., Parás, A.: Managing the volatility risk of portfolios of derivative securities: the Lagrangian uncertain volatility model. Appl. Math. Fin. 3(1), 21–52 (1996)

    Article  MATH  Google Scholar 

  5. Barles, G., Soner, H.M.: Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance and Stochastics 2(4), 369–397 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bordag, L., Chmakova, A.: Explicit solutions for a nonlinear model of financial derivatives. Int. J. Theor. Appl. Finance 10, 1–21 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, X., Yamamoto, T.: On the convergence of some quasi-Newton methods for nonlinear equations with nondifferentiable operators. Computing 49, 87–94 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Company, R., Navarro, E., Pintos, J., Ponsoda, E.: Numerical solution of linear and nonlinear Black-Scholes option pricing equations. Int. J. of Comp. and Math. with Appl. 56, 813–821 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Düring, B., Fournié, M., Jüngel, A.: Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation. ESAIM: M2AN 38(2), 359–369 (2004)

    Article  MATH  Google Scholar 

  10. Epperson, J.: A kernel-based method for parabolic equations with nonlinear convection terms. J. Comp. and Appl. Math. 36, 275–288 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Frey, R., Patie, P.: Risk management for derivatives in illiquid markets: a simulation-study. In: Sandmann, K., Schon̈bucher, P. (eds.) Adv. in Fin. and Stoch., pp. 137–159. Springer, Berlin (2002)

    Google Scholar 

  12. Heider, P.: Numerical Methods for Non-Linear Black-Scholes Equations. Appl. Math. Fin. 17(1), 59–81 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ishimura, N., Koleva, M.N., Vulkov, L.G.: Numerical Solution of a Nonlinear Evolution Equation for the Risk Preference. In: Dimov, I., Dimova, S., Kolkovska, N. (eds.) NMA 2010. LNCS, vol. 6046, pp. 445–452. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  14. Jandačka, M., Ševčovič, D.: On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile. J. of Appl. Math. (3), 235–258 (2005)

    Google Scholar 

  15. Leland, H.E.: Option pricing and replication with transactions costs. J. of Finance 40(5), 1283–1301 (1985)

    Google Scholar 

  16. Ševčovič, D., Stehlíková, B., Mikula, K.: Analytical and Numerical Methods for Pricing Financial Derivatives. Nova Sci. Publ., NY (2010)

    Google Scholar 

  17. Seydel, R.: Tools for Computational Finance. Springer, Berlin (2003)

    Google Scholar 

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Author information

Authors and Affiliations

  1. University of Rousse, 8 Studentska Str., Rousse, 7017, Bulgaria

    Miglena N. Koleva & Lubin G. Vulkov

Authors
  1. Miglena N. Koleva
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  2. Lubin G. Vulkov
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Editor information

Editors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev, Block 25A, 1113, Sofia, Bulgaria

    Ivan Lirkov  & Svetozar Margenov  & 

  2. Department of Informatics and Mathematical Modelling, Technical University of Denmark, Richard Petersens Plads, Building 321,, 2800, Kongens Lyngby, Denmark

    Jerzy Waśniewski

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Koleva, M.N., Vulkov, L.G. (2012). A Kernel-Based Algorithm for Numerical Solution of Nonlinear PDEs in Finance. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2011. Lecture Notes in Computer Science, vol 7116. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29843-1_64

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  • DOI: https://doi.org/10.1007/978-3-642-29843-1_64

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-29842-4

  • Online ISBN: 978-3-642-29843-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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