A high-order compact method for nonlinear Black-Scholes option pricing equations with transaction costs

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



Over the last several years there were numerous discussions about the Black-Scholes model and its appliance to the nonlinear case. In the linear model there exists a lot of restrictions such as frictionless, liquid and complete market. But all these assumptions are not fulfilled in reality. One of the most important issues that has a great influence on the option pricing strategy are transaction costs. Several authors proposed relaxing hedging conditions dealing with transaction costs, e.g. Leland, Boyle & Vorst, Barles & Soner or the Risk Adjusted Pricing Methodology (RAPM) proposed by Kratka in [17] and improved by Jandacka and Sevcovic in [14]. Although the Leland model has been criticized by Kabanov, it has played a significant role in financial mathematics.

In this work we will consider compact schemes for European and American options and work closer with the transaction cost model of Barles and Soner. Instead of solving the singular differential equation of Barles and Soner we propose to use some fixpoint properties of the Barles/Soner volatlity function psi described recently in [6]. As compact schemes cannot be directly applied to American type options and multi-dimensional problems, we will try to employ them using a fixed domain transformation explained in [2]. We will also consider the method of Liao and Khaliq [19] for solving nonlinear Black-Scholes equation with transaction costs.


  1. J. Ankudinova, The numerical solution of nonlinear Black-Scholes equations, Diploma Thesis, Technische Universität Berlin, Berlin, Germany, 2008.
  2. J. Ankudinova and M. Ehrhardt, Fixed domain transformations and split-step finite difference schemes for Nonlinear Black-Scholes equations for American Options, Chapter 8 in M. Ehrhardt (ed.) Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing. Nova Science Publishers, Inc., Hauppauge, NY 11788, pp. 243-273.
  3. G. Barles and H.M. Soner, Option pricing with transaction costs and a nonlinear Black-Scholes equation, Finance Stochast. 2 (1998), 369-397.
  4. F. Black and M.S. Scholes, The pricing of options and corporate liabilities, J.Political Economy 81 (1973), 637-654.
  5. P. Boyle and T. Vorst, Option replication in discrete time with transaction costs, Journal of Finance 47 (1973), 271-293.
  6. R. Company, E. Navarro, J.R. Pintos and E. Ponsoda, Numerical solution of linear and nonlinear Black-Scholes option pricing equations, Comput. Math. Appl. 56 (2008), 813-821.
  7. E. Cryer, The solution of a quadratic programming problem using systematic overrelaxation, SIAM J. Control 9 (1971), 385-392.
  8. B. Düring, Black-Scholes Type Equations: Mathematical Analysis, Parameter Identification and Numerical Solution, Dissertation, Universität Mainz, 2005.
  9. B. Düring, M. Fournié and A. Jüngel, High order compact finite difference schemes for a nonlinear Black-Scholes equation, Intern. J. Theor. Appl. Finance 6 (2003), 767-789.
  10. M. Ehrhardt and R.E. Mickens A fast, stable and accurate numerical method for the Black-Scholes equation of American options, Int. J. Theoret. Appl. Finance 11 (2008), 471-501.
  11. 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.
  12. 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.
  13. 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.
  14. M. Jandacka and D. Sevcovic, On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile, J. Appl. Math. 3 (2005), 235-258.
  15. Y. Kabanov and M. Safarian, On Leland's strategy of option pricing with transaction costs, Finance Stoch. 1 (1997).
  16. 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.
  17. M. Kratka, No mystery behind the smile, Risk 9 (1998), 67-71.
  18. M. Leland, Option pricing and replication with Transaction Costs, Journal of Finance 5 (1985), 1283-1301.
  19. W. Liao and A.Q.M. Khaliq, High order compact scheme for solving nonlinear Black-Scholes equation with transaction cost, International Journal of Computer Mathematics (2009), to appear.
  20. 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.
  21. C. W. Oosterlee, C.C.W. Leentvaar, A. Almendral, Pricing options with discrete dividends by high order finite differences and grid stretching, ECCOMAS 2004, P. Neittaanmaekki et al. (eds.), a CD Rom, Jyvaskyla, Finland.
  22. A. Rigal, High Order Difference Schemes for Unsteady One-Dimensional Diffusion-Convection Problems, J. Comput. Phys. 114 (1994), 59-76.
  23. D. Sevcovic, An iterative algorithm for evaluating approximations to the optimal exercise boundary for a nonlinear Black-Scholes equation, Canad. Appl. Math. Quarterly 15 (2007), 77-79.

mailbox ehrhardt@math.uni-wuppertal.de