High-Order Compact Methods for the American Option Pricing Problem

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



In this thesis, we seek for an effective numerical solution for the American option pricing problem. The Black-Scholes equation is transformed to the diffusion equation on an unbounded domain. An artificial boundary is introduced to limit this domain. The diffusion operator is discretized by using the different schemes for the interior grid with different boundary conditions. The considered high order compact schemes Method 1 proposed by MacCartin, Method 2 proposed by Tangman and R3-Methods belong to weighted scheme of the approximation for the pure heat equation. Moreover, they coincide and this method is called the optimal weighted scheme. This scheme is unconditionally stable and it has the fourth order approximation in space and the second in time. Also, we compare Crank-Nicolson and five-point two-level non-compact stencils with compact method. They show worth numerical result in comparison with high order compact method. The Crank-Nicolson scheme with different boundary conditions is inferior to the non-compact Heun's method. But the Heun's scheme is conditionally stable, which is a big disadvantage for this scheme. On the right boundary of the truncated computational domain we use the Dirichlet boundary condition. The boundary conditions on the left boundary have influence for the numerical solution of the heat equation. The numerical results show that the combination of high order compact schemes with the Han and Wu boundary condition is more accurate to the exact solution which obtained by using the Binomial method with large number of the steps.


  1. E. Cryer, The solution of a quadratic programming problem using systematic overrelaxation, SIAM J. Control 9 (1971), 385-392.
  2. B. Düring, Black-Scholes Type Equations: Mathematical Analysis, Parameter Identification and Numerical Solution, Dissertation, Universität Mainz, 2005.
  3. M. Ehrhardt, Discrete Artificial Boundary Conditions, Dissertation, Technische Universität Berlin, 2001.
  4. P.A. Forsyth and K.R. Vetzal, Quadratic Convergence for Valuing American Options using a Penalty Method, SIAM J. Sci. Comput. 23 (2002), 2095-2122.
  5. H. Han and X. Wu, A fast numerical method for the Black-Scholes equation of American options, SIAM J. Numer. Anal. 41 (2003), 2081-2095.
  6. J.-Z. Huang, M.G. Subrahmanyam and G.G. Yu, Pricing and Hedging American Options: A Recursive Integration Method, Review of Financial Studies 9 (1996), 277-300.
  7. T. Kim and Y. Kwon, Finite-Difference Bisection Algorithms for free boundaries of American Call and Put Options, Preprint, Pohang University of Science and Technology, South Korea, 2002.
  8. B.J. McCartin and S.M. Labadie, Accurate and efficient pricing of vanilla stock options via the Crandall- Douglas scheme, Applied Mathematics and Computation 143 (2003), 39-60.
  9. B. Mayfield, Non-local boundary conditions for the Schrödinger equation, Ph.D. Thesis, University of Rhode Island, Providence, RI, 1989.
  10. B. F. Nielsen, O. Skavhaug and A. Tveito, Penalty and front-fixing methods for the numerical solution of American option problems, J. Comp. Finance 5 (2002), 69-97.
  11. A. Rigal, High Order Difference Schemes for Unsteady One-Dimensional Diffusion-Convection Problems, J. Comput. Phys. 114 (1994), 59-76.
  12. R. Smith, Optimal and near-optimal advection-diffusion finite- difference schemes. II Unsteadiness and non-uniform grid, Proc. The Royal Society, 456 (1999), 489-502.
  13. D.Y. Tangman, A. Gopaul and M. Bhuruth, Numerical pricing of options using high-order compact finite difference schemes, J. Comput. Appl. Math. 218 (2008), 270-280.
  14. D.Y. Tangman, A. Gopaul and M. Bhuruth, A Fast high-order finite difference algorithm for pricing American options, J. Comput. Appl. Math. 222 (2008), 17-29.
  15. P. Wilmott, S. Howison and J. Dewynne, The mathematics of financial derivatives, University of Cambridge, USA, 1995.
  16. J. Zhao, M. Davison and R.M. Corless, Compact finite difference method for American option pricing, J. Comput. Appl. Math. 206 (2007), 306-321.

mailbox ehrhardt@math.uni-wuppertal.de