Computational Economics

, Volume 53, Issue 1, pp 259–287 | Cite as

Radial Basis Functions with Partition of Unity Method for American Options with Stochastic Volatility

  • Reza MollapouraslEmail author
  • Ali Fereshtian
  • Michèle Vanmaele


In this article, we price American options under Heston’s stochastic volatility model using a radial basis function (RBF) with partition of unity method (PUM) applied to a linear complementary formulation of the free boundary partial differential equation problem. RBF-PUMs are local meshfree methods that are accurate and flexible with respect to the problem geometry and that produce algebraic problems with sparse matrices which have a moderate condition number. Next, a Crank–Nicolson time discretisation is combined with the operator splitting method to get a fully discrete problem. To better control the computational cost and the accuracy, adaptivity is used in the spatial discretisation. Numerical experiments illustrate the accuracy and efficiency of the proposed algorithm.


Radial basis function Partition of unity Operator splitting American option pricing Stochastic volatility Heston’s model 



The first author would like to thank the Department of Applied Mathematics, Computer Science and Statistics of Ghent University and the FWO Scientific Research Network Stochastic Modelling with Applications in Financial Markets for some financial support during his scientific research stay at that department. The authors thank professor in ’t Hout for some fruitful discussion.


  1. Apel, T., Winkler, G., & Wystup, U. (2001). Valuation of options in Heston’s stochastic volatility model using finite element methods. In J. Hakala & U. Wystup (Eds.), Foreign exchange risk (pp. 283–303). London: Risk Books.Google Scholar
  2. Babuška, I., & Melenk, J. M. (1997). The partition of unity method. International Journal for Numerical Methods in Engineering, 40(4), 727–758.Google Scholar
  3. Ballestra, L. V., & Pacelli, G. (2011). Computing the survival probability density function in jump-diffusion models: A new approach based on radial basis functions. Engineering Analysis with Boundary Elements, 35(9), 1075–1084.Google Scholar
  4. Ballestra, L. V., & Pacelli, G. (2012). A radial basis function approach to compute the first-passage probability density function in two-dimensional jump-diffusion models for financial and other applications. Engineering Analysis with Boundary Elements, 36(11), 1546–1554.Google Scholar
  5. Ballestra, L. V., & Pacelli, G. (2013). Pricing European and American options with two stochastic factors: A highly efficient radial basis function approach. Journal of Economic Dynamics and Control, 37(6), 1142–1167.Google Scholar
  6. Ballestra, L. V., & Sgarra, C. (2010). The evaluation of American options in a stochastic volatility model with jumps: An efficient finite element approach. Computers & Mathematics with Applications, 60(6), 1571–1590.Google Scholar
  7. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.Google Scholar
  8. Borici, A., & Lüthi, H.-J. (2002). Pricing American put options by fast solutions of the linear complementarity problem. In E. J. Kontoghiorghes, B. Rustem, & S. Siokos (Eds.), Computational methods in decision-making, economics and finance (pp. 325–338). US: Springer.Google Scholar
  9. Buhmann, M. D. (2000). Radial basis functions. Acta Numerica, 9, 1–38. 01.Google Scholar
  10. Buhmann, M. D. (2003). Radial basis functions: theory and implementations. New York: Cambridge University Press.Google Scholar
  11. Cavoretto, R., & De Rossi, A. (2012). Spherical interpolation using the partition of unity method: An efficient and flexible algorithm. Applied Mathematics Letters, 25(10), 1251–1256.Google Scholar
  12. Cavoretto, R., & De Rossi, A. (2014). A meshless interpolation algorithm using a cell-based searching procedure. Computers & Mathematics with Applications, 67(5), 1024–1038.Google Scholar
  13. Chen, Y., Gottlieb, S., Heryudono, A., & Narayan, A. (2016). A reduced radial basis function method for partial differential equations on irregular domains. Journal of Scientific Computing, 66(1), 67–90.Google Scholar
  14. Clarke, N., & Parrott, K. (1999). Multigrid for American option pricing with stochastic volatility. Applied Mathematical Finance, 6(3), 177–195.Google Scholar
  15. Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica, 53(2), 385–408.Google Scholar
  16. Dehghan, M., & Shokri, A. (2007). A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions. Computers & Mathematics with Applications, 54(1), 136–146.Google Scholar
  17. Dehghan, M., & Shokri, A. (2009). Numerical solution of the nonlinear Klein-Gordon equation using radial basis functions. Journal of Computational and Applied Mathematics, 230(2), 400–410.Google Scholar
  18. Duan, Y. (2008). A note on the meshless method using radial basis functions. Computers & Mathematics with Applications, 55(1), 66–75.Google Scholar
  19. Fang, F., & Oosterlee, C. W. (2011). A Fourier-based valuation method for Bermudan and barrier options under Heston’s model. SIAM Journal on Financial Mathematics, 2(1), 439–463.Google Scholar
  20. Fasshauer, G. E. (2007). Meshfree approximation methods with Matlab. Singapore: World Scientific Publishing Co.Google Scholar
  21. Fasshauer, G. E., Khaliq, A. Q. M., & Voss, D. A. (2004). Using meshfree approximation for multiasset American options. Journal of the Chinese Institute of Engineers, 27(4), 563–571.Google Scholar
  22. Haentjens, T., & in ’t Hout, K. J. (2015). ADI schemes for pricing American options under the Heston model. Applied Mathematical Finance, 22(3), 207–237.Google Scholar
  23. Hardy, R. L. (1971). Multiquadric equations of topography and other irregular surfaces. Journal of Geophysical Research, 76(8), 1905–1915.Google Scholar
  24. Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6(2), 327–343.Google Scholar
  25. Hu, H.-Y., Li, Z.-C., & Cheng, A. H.-D. (2005). Radial basis collocation methods for elliptic boundary value problems. Computers & Mathematics with Applications, 50(1), 289–320.Google Scholar
  26. Hull, J., & White, A. (1987). The pricing of options on assets with stochastic volatilities. The Journal of Finance, 42(2), 281–300.Google Scholar
  27. Ikonen, S., & Toivanen, J. (2004). Operator splitting methods for American option pricing. Applied Mathematics Letters, 17(7), 809–814.Google Scholar
  28. Ikonen, S., & Toivanen, J. (2008). Efficient numerical methods for pricing American options under stochastic volatility. Numerical Methods for Partial Differential Equations, 24(1), 104–126.Google Scholar
  29. Ikonen, S., & Toivanen, J. (2009). Operator splitting methods for pricing American options under stochastic volatility. Numerische Mathematik, 113(2), 299–324.Google Scholar
  30. in ’t Hout, K. J., & Foulon, S. (2010). ADI finite difference schemes for option pricing in the Heston model with correlation. International Journal of Numerical Analysis and Modeling, 7(2), 303–320.Google Scholar
  31. Kansa, E. J. (1990a). Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics-I: surface approximations and partial derivative estimates. Computers & Mathematics with Applications, 19(8), 127–145.Google Scholar
  32. Kansa, E. J. (1990b). Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics-II: solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers & Mathematics with Applications, 19(8), 147–161.Google Scholar
  33. Kunoth, A., Schneider, C., & Wiechers, K. (2012). Multiscale methods for the valuation of American options with stochastic volatility. International Journal of Computer Mathematics, 89(9), 1145–1163.Google Scholar
  34. Kwok, Y. K. (1998). Mathematical models of financial derivatives. Singapore: Springer.Google Scholar
  35. Larsson, E., & Fornberg, B. (2003). A numerical study of some radial basis function based solution methods for elliptic PDEs. Computers & Mathematics with Applications, 46(5), 891–902.Google Scholar
  36. Larsson, E., Gomes, S. M., Heryudono, A., & Safdari-Vaighani, A. (2013). Radial basis function methods in computational finance. In Proceedings of the 13th international conference on computational and mathematical methods in science and engineering, CMMSE 2013.Google Scholar
  37. Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141–183.Google Scholar
  38. Oosterlee, C. W. (2003). On multigrid for linear complementary problems with application to American-style options. Electronic Transactions on Numerical Analysis, 15, 165–185.Google Scholar
  39. Oosterlee, C. W., Leentvaar, C. C. W., & Huang, X. (2005). Accurate American option pricing by grid stretching and high order finite differences. Technical report, DIAM, Delft University of Technology.Google Scholar
  40. O’Sullivan, C., & O’Sullivan, S. (2013). Pricing European and American options in the Heston model with accelerated explicit finite differencing methods. International Journal of Theoretical and Applied Finance, 16(03), 1350015.Google Scholar
  41. Powell, M. J. D. (1992). The theory of radial basis functions approximation in 1990. In W. A. Light (Ed.), Advances in numerical analysis. Vol. II. Wavelets, subdivision algorithms and radial basis functions (pp. 105–210). London: Oxford Univ. Press.Google Scholar
  42. Safdari-Vaighani, A., Heryudono, A., & Larsson, E. (2015). A radial basis function partition of unity collocation method for convection-diffusion equations arising in financial applications. Journal of Scientific Computing, 64(2), 341–367.Google Scholar
  43. Seydel, R. (2009). Tools for Computational Finance (4th ed.). Berlin, Heidelberg: Springer.Google Scholar
  44. Shcherbakov, V., & Larsson, E. (2016). Radial basis function partition of unity methods for pricing vanilla basket options. Computers & Mathematics with Applications, 71(1), 185–200.Google Scholar
  45. Shepard, D. (1968). A two-dimensional interpolation function for irregularly-spaced data. In Proceedings of the 1968 23rd ACM national conference, ACM ’68 (pp. 517 – 524). ACM: New York, NY, USA.Google Scholar
  46. Tatari, M., & Dehghan, M. (2010). A method for solving partial differential equations via radial basis functions: Application to the heat equation. Engineering Analysis with Boundary Elements, 34(3), 206–212.Google Scholar
  47. Tavella, D., & Randall, C. (2000). Pricing financial instruments: The finite difference method. New York: Wiley.Google Scholar
  48. Wendland, H. (1995). Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics, 4(1), 389–396.Google Scholar
  49. Wendland, H. (2002). Fast evaluation of radial basis functions: Methods based on partition of unity. In C. K. Chui, L. L. Schumaker, & J. Stöckler (Eds.), Approximation theory X: Wavelets, splines, and applications (pp. 473–483). Nashville: Vanderbilt University Press.Google Scholar
  50. Wendland, H. (2005). Scattered Data Approximation. Number 17 in Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, New York.Google Scholar
  51. Zhu, S.-P., & Chen, W.-T. (2010). A new analytical approximation for European puts with stochastic volatility. Applied Mathematics Letters, 23(6), 687–692.Google Scholar
  52. Zhu, S.-P., & Chen, W.-T. (2011). A predictor-corrector scheme based on the ADI method for pricing American puts with stochastic volatility. Computers & Mathematics with Applications, 62(1), 1–26.Google Scholar
  53. Zvan, R., Forsyth, P. A., & Vetzal, K. R. (1998). Penalty methods for American options with stochastic volatility. Journal of Computational and Applied Mathematics, 91(2), 199–218.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Reza Mollapourasl
    • 1
    • 3
    Email author
  • Ali Fereshtian
    • 1
  • Michèle Vanmaele
    • 2
  1. 1.School of MathematicsShahid Rajaee Teacher Training UniversityLavizan, TehranIran
  2. 2.Department of Applied Mathematics, Computer Science and StatisticsGhent UniversityGhentBelgium
  3. 3.Department of MathematicsOregon State UniversityCorvallisUSA

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