Recent Advances in Numerical Solution of HJB Equations Arising in Option Pricing

  • Song WangEmail author
  • Wen Li
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)


This paper provides a brief survey on some of the recent numerical techniques and schemes for solving Hamilton-Jacobi-Bellman equations arising in pricing various options. These include optimization methods in both infinite and finite dimensions and discretization schemes for nonlinear parabolic PDEs.


Price Option Reservation Price Linear Complementarity Problem Penalty Method American Option 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Achdou, Y., Pironneau, O.: Computational Methods for Option Pricing. SIAM series in Applied Math, Philadelphia (2005)zbMATHCrossRefGoogle Scholar
  2. 2.
    Allegretto, W., Lin, Y., Yang, H.: Finite element error estimates for a nonlocal problem in American option valuation. SIAM J. Numer. Anal. 39, 834–857 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Angermann, L., Wang, S.: Convergence of a fitted finite volume method for European and American Option Valuation. Numerische Mathematik 106, 1–40 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Barles, G., Soner, H.M.: Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance Stochast. 2(4), 369–397 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Barles, G., Souganidis, P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Analysis 4, 271–283 (1991)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Bensoussan, A., Lions, J.L.: Applications of Variational Inequalities in Stochastic Control. North-Holland, Amsterdam (1978)Google Scholar
  7. 7.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)zbMATHCrossRefGoogle Scholar
  8. 8.
    Boyle, P.P., Vorst, T.: Option replication in discrete time with transaction costs. J. Finan. XLVI(1), 271–293 (1992)CrossRefGoogle Scholar
  9. 9.
    Cartea, A., del-Castillo-Negrete, D.: Fractional diffusion models of option prices in markets with jumps. Physica A 374(2), 749–763 (2007)CrossRefGoogle Scholar
  10. 10.
    Chen, W., Wang, S.: A penalty method for a fractional order parabolic variational inequality governing American put option valuation. Comp. Math. Appl. 67(1), 77–90 (2014)CrossRefGoogle Scholar
  11. 11.
    Chen, W., Wang, S.: A finite difference method for pricing European and American options under a geometric Levy process. J. Ind. Manag. Optim. 11, 241–264 (2015)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Chernogorova, T., Valkov, R.: Finite volume difference scheme for a degenerate parabolic equation in the zero-coupon bond pricing. Math. Comp. Model. 54, 2659–2671 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Company, R., Navarro, E., Pintos, J.R., Ponsoda, E.: Numerical solution of linear and nonlinear Black-Scholes option pricing equation. Comp. Math. Appl. 56, 813–821 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Crandall, M.G., Lions, P.L.: Viscosity Solution of Hamilton-Jacobi-Equations. Trans. Am. Math. Soc. 277, 1–42 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Damgaard, A.: Computation of Reservation Prices of Options with Proportional Transaction Costs. J. of Econ. Dyn. Control 30, 415–444 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Davis, M.H.A., Panas, V.G., Zariphopoulou, T.: European Option Pricing with Transaction Costs. SIAM J. Control Optim. 31, 470–493 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Davis, M.H.A., Zariphopoulou, T.: In mathemtical finance. In: Davis, M.H.A., et al. (eds.) American Options and Transaction Fees, pp. 47–62. Springer, New York (1995)Google Scholar
  18. 18.
    Hanson, F.B., Yan, G.: American put option pricing for stochastic-volatility, jump-diffusion models. In: Proceedings of 2007 American Control Conference, pp. 384–389 (2007)Google Scholar
  19. 19.
    Hoggard, T., Whalley, A.W., Wilmott, P.: Hedging option portfolios in the presence of transaction costs. Adv. Futures Options Research 7, 21–35 (1994)Google Scholar
  20. 20.
    Huang, C.C., Wang, S.: A Power Penalty approach to a nonlinear complementarity problem. Oper. Res. Lett. 38, 72–76 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Huang, C.C., Wang, S.: A penalty method for a mixed nonlinear complementarity problem. Nonlinear Anal. TMA 75, 588–597 (2012)zbMATHCrossRefGoogle Scholar
  22. 22.
    Huang, C.-S., Hung, C.-H., Wang, S.: A fitted finite volume method for the valuation of options on assets with stochastic volatilities. Comput. 77, 297–320 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Jandačka, M., Ševčovič, D.: On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile. J. Appl. Math. 3, 235–258 (2005)CrossRefGoogle Scholar
  24. 24.
    Leland, H.E.: Option pricing and replication with transaction costs. J. Finan. 40, 1283–1301 (1985)CrossRefGoogle Scholar
  25. 25.
    Koleva, M.N., Vulkov, L.G.: A positive flux limited difference scheme for the uncertain correlation 2D Black-Scholes problem, J. Comp. Appl. Math. (in press)Google Scholar
  26. 26.
    Lesmana, D.C., Wang, S.: An upwind finite difference method for a nonlinear Black-Scholes equation governing European option valuation. Appl. Math. Comp. 219, 8818–8828 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lesmana, D.C., Wang, S.: Penalty approach to a nonlinear obstacle problem governing American put option valuation under transaction costs. Appl. Math. Comp. 251, 318–330 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Li, W., Wang, S.: Penalty approach to the HJB equation arising in European stock option pricing with proportional transaction Costs. J. Optim. Theory Appl. 143, 279–293 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Li, W., Wang, S.: Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme. J. Ind. Manag. Optim. 9, 365–398 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Li, W., Wang, S.: A numerical method for pricing European options with proportional transaction costs. J. Glob. Optim. 60, 59–78 (2014)zbMATHCrossRefGoogle Scholar
  31. 31.
    Lyons, T.J.: Uncertain volatility and the risk free synthesis of derivatives. Appl. Math. Finance 2, 117–133 (1995)CrossRefGoogle Scholar
  32. 32.
    Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manage. Sci. 4, 141–183 (1973)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Miller, J.J.H., Wang, S.: A new non-conforming Petrov-Galerkin method with triangular elements for a singularly perturbed advection-diffusion problem. IMA J. Numer. Anal. 14, 257–276 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Miller, J.J.H., Wang, S.: An exponentially fitted finite element volume method for the numerical solution of 2D unsteady incompressible flow problems. J. Comput. Phys. 115, 56–64 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Nielsen, B.F., Skavhaug, O., Tveito, A.: Penalty and front-fixing methods for the numerical solution of American option problems. J. Comp. Fin. 5, 69–97 (2001)Google Scholar
  36. 36.
    Valkov, R.: Fitted finite volume method for a generalized Black Scholes equation transformed on finite interval. Numer. Algorithms 65, 195–220 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Vazquez, C.: An upwind numerical approach for an American and European option pricing model. Appl. Math. Comp. 97, 273–286 (1998)zbMATHCrossRefGoogle Scholar
  38. 38.
    Wang, G., Wang, S.: Convergence of a finite element approximation to a degenerate parabolic variational inequality with non-smooth data Arising from American option valuation. Optim. Method Softw. 25, 699–723 (2010)zbMATHCrossRefGoogle Scholar
  39. 39.
    Wang, S.: A novel fitted finite volume method for the Black-Scholes equation governing option pricing. IMA J. Nuner. Anal. 24, 669–720 (2004)Google Scholar
  40. 40.
    Wang, S.: A power penalty method for a finite-dimensional obstacle problem with derivative constraints. Optim. Lett. 8, 1799–1811 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Wang, S.: A penalty approach to a discretized double obstacle problem with derivative constraints. J. Glob. Optim. (2014). doi: 10.1007/s10898-014-0262-3 Google Scholar
  42. 42.
    Wang, S., Huang, C.-S.: A power penalty method for a nonlinear parabolic complementarity problem. Nonlinear Anal. TMA 69, 1125–1137 (2008)zbMATHCrossRefGoogle Scholar
  43. 43.
    Wang, S., Yang, X.Q.: A power penalty method for linear complementarity problems. Oper. Res. Lett. 36, 211–214 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Wang, S., Yang, X.Q.: A power penalty method for a bounded nonlinear complementarity problem. Optimization (2014). doi: 10.1080/02331934.2014.967236
  45. 45.
    Wang, S., Yang, X.Q., Teo, K.L.: Power penalty method for a linear complementarity problem arising from American option valuation. J. Optim. Theory Appl. 129(2), 227–254 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Wang, S., Zhang, S., Fang, Z.: A superconvergent fitted finite volume method for black scholes equations governing European and American option valuation. Numer. Meth. Part. D.E. 31, 1190 (2015)CrossRefGoogle Scholar
  47. 47.
    Wilmott, P., Dewynne, J., Howison, S.: Option Pricing Math. Models Comput. Oxford Financial Press, Oxford (1993)Google Scholar
  48. 48.
    Zhang, K., Wang, S.: A computational scheme for uncertain volatility model in option pricing. Appl. Numer. Math. 59, 1754–1767 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Zhang, K., Wang, S.: Convergence property of an interior penalty approach to pricing American option. J. Ind. Manag. Optim. 7, 435–447 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Zhang, K., Wang, S.: Pricing American bond options using a penalty method. Automatica 48, 472–479 (2012)zbMATHCrossRefGoogle Scholar
  51. 51.
    Zhang, K., Wang, S., Yang, X.Q., Teo, K.L.: A power penalty approach to numerical solutions of two-asset American options. Numer. Math. Theory Methods Appl. 2, 127–140 (2009)MathSciNetGoogle Scholar
  52. 52.
    Zhou, Y.Y., Wang, S., Yang, X.Q.: A penalty approximation method for a semilinear parabolic double obstacle problem. J. Glob. Optim. 60, 531–550 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    Zvan, R., Forsyth, P.A., Vetzal, K.R.: Penalty methods for American options with stochastic volatility. Comput. Appl. Math. 91, 199–218 (1998)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsCurtin UniversityPerthAustralia

Personalised recommendations