Bergische Universität Wuppertal
Fachbereich Mathematik und Naturwissenschaften
Angewandte Mathematik - Numerische Analysis


An Exponentially Fitted Finite Volume Method for the Numerical Pricing of Options under Jump Diffusion Processes

Masterarbeit Wirtschaftsmathematik

Master's Thesis in Financial Mathematics, Halmstad University, Sweden



In this thesis we investigate a new strategy of Zhang and Wang to construct exponentially fitted finite volume schemes. We apply this approach in the context of options under jump diffusion processes and certain nonlinear Black-Scholes equations, e.g. modeling transactions costs. Especially we discuss the numerical treatment of the case of small volatility, i.e. the convection dominated case. The results are compared with the point-distributed finite volume scheme of Zvan, Vetzal and Forsyth.


Option pricing, Finite volume method, Exponential Fitting, nonlinear partial integro differential equation, Il'in scheme, Scharfetter-Gummer scheme, El-Mistikawy-Werle scheme, partial exponential fitting, complete exponential fitting


  1. A. Anderson and J. Andresen, Jump diffusion process: Volatility smile fitting and numerical methods for option pricing, Rev. Derivatives Res. 4 (2000), 231-262.
  2. L. Angermann and S. Wang, Convergence of a fitted finite volume method for European and American option valuation, Numer. Math. 106 (2007), 1-40.
  3. D.J. Duffy, Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, John Wiley & Sons, Chichester, 2006.
  4. A.M. Il'in, A difference scheme for a differential equation with small parameter affecting the highest derivative, Mat. Zametki 6 (1969), 237-248 (in Russian).
  5. C.-Y. Hsu, Adaptive finite volume methods for pricing European-style Asian options, Master's thesis, Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan, 2005.
  6. C.-S. Huang, C.-H. Hung and S. Wang, A fitted finite volume method for the valuation of options on assets with stochastic volatilities, Computing 77 (2006), 297-320.
  7. J.J.H. Miller and S. Wang, A new non-conforming Petrov-Galerkin method with triangular elements for a singularly perturbed advection-diffusion problem, IMA J. Numer. Anal. 14 (1994), 257-276.
  8. J.J.H. Miller and S. Wang, An exponentially fitted finite element volume method for the numerical solution of 2D unsteady incompressible flow problems, J. Comput. Phys. 115 (1994), 56-64.
  9. H.-G. Roos, Ten ways to generate the Il'in and related schemes, J. Comput. Appl. Math. 53 (1994), 43-59.
  10. G.I. Shiskin, Grid approximation of singularly perturbed elliptic and parabolic equations, Second Doctoral Thesis, Keldysh Institute, Moscow, 1990.
  11. S. Wang, A novel fitted finite volume method for the Black-Scholes equation governing option pricing, IMA J. Numer. Anal. 24 (2004), 699-720.
  12. K. Zhang and S. Wang, Pricing options under jump diffusion processes with fitted finite volume method, Appl. Math. Comput. 201 (2008), 398-413.
  13. K. Zhang and S. Wang, A computational scheme for uncertain volatility model in option pricing, Appl. Numer. Math. 59 (2009), 1754-1767.
  14. R. Zvan, K.R. Vetzal and P.A. Forsyth, PDE methods for pricing barrier options, J. Economic Dynamics & Control 24 (2000), 1563-1590.
  15. R. Zvan, P.A. Forsyth, K.R. Vetzal, A finite volume approach for contingent claims valuation, IMA J. Numer. Anal. 21 (2001), 703-731.

University of Wuppertal
Faculty of Mathematics and Natural Sciences
Department of Mathematics
Applied Mathematics & Numerical Analysis Group

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