Advertisement

Applications of Mathematics

, Volume 62, Issue 6, pp 607–632 | Cite as

DG method for the numerical pricing of two-asset European-style Asian options with fixed strike

  • Jiří HozmanEmail author
  • Tomáš Tichý
Article
  • 85 Downloads

Abstract

The evaluation of option premium is a very delicate issue arising from the assumptions made under a financial market model, and pricing of a wide range of options is generally feasible only when numerical methods are involved. This paper is based on our recent research on numerical pricing of path-dependent multi-asset options and extends these results also to the case of Asian options with fixed strike. First, we recall the three-dimensional backward parabolic PDE describing the evolution of European-style Asian option contracts on two assets, whose payoff depends on the difference of the strike price and the average value of the basket of two underlying assets during the life of the option. Further, a suitable transformation of variables respecting this complex form of a payoff function reduces the problem to a two-dimensional equation belonging to the class of convection-diffusion problems and the discontinuous Galerkin (DG) method is applied to it in order to utilize its solving potentials. The whole procedure is accompanied with theoretical results and differences to the floating strike case are discussed. Finally, reference numerical experiments on real market data illustrate comprehensive empirical findings on Asian options.

Keywords

option pricing discontinuous Galerkin method Asian option basket option fixed strike 

MSC 2010

65M60 35Q91 91G60 91G80 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Y. Achdou, O. Pironneau: Computational Methods for Option Pricing. Frontiers in Applied Mathematics 30, Society for Industrial and Applied Mathematics, Philadelphia, 2005.Google Scholar
  2. [2]
    B. Alziary, J.-P. Décamps, P.-F. Koehl: A P. D. E. approach to Asian options: analytical and numerical evidence. J. Bank. Financ. 21 (1997), 613–640.CrossRefGoogle Scholar
  3. [3]
    F. Black, M. Scholes: The pricing of options and corporate liabilities. J. Polit. Econ. 81 (1973), 637–654.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    P. Boyle, M. Broadie, P. Glasserman: Monte Carlo methods for security pricing. J. Econ. Dyn. Control 21 (1997), 1267–1321.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    P. G. Ciarlet: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications 4, North-Holland Publishing Company, Amsterdam, 1978.Google Scholar
  6. [6]
    R. Cont, P. Tankov: Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series, Chapman and Hall/CRC, Boca Raton, 2004.Google Scholar
  7. [7]
    J. C. Cox, S. A. Ross, M. Rubinstein: Option pricing: a simplified approach. J. Financ. Econ. 7 (1979), 229–263.CrossRefzbMATHGoogle Scholar
  8. [8]
    V. Dolejší, M. Feistauer: Discontinuous Galerkin Method. Analysis and Applications to Compressible Flow. Springer Series in Computational Mathematics 48, Springer, Cham, 2015.Google Scholar
  9. [9]
    F. Dubois, T. Lelièvre: Efficient pricing of Asian options by the PDE approach. J. Comput. Finance 8 (2005), 55–63.CrossRefGoogle Scholar
  10. [10]
    E. Eberlein, A. Papapantoleon: Equivalence of floating and fixed strike Asian and lookback options. Stochastic Processes Appl. 115 (2005), 31–40.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    J. K. Hale: Ordinary Differential Equations. Pure and Applied Mathematics 21, Wiley-Interscience a division of John Wiley & Sons, New York, 1969.Google Scholar
  12. [12]
    J. M. Harrison, D. M. Kreps: Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory 20 (1979), 381–408.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    E. G. Haug: The Complete Guide to Option Pricing Formulas, McGraw-Hill, New York, 2006.Google Scholar
  14. [14]
    F. Hecht: New development in freefem++. J. Numer. Math. 20 (2012), 251–265.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    V. Henderson, R. Wojakowski: On the equivalence of floating- and fixed-strike Asian options. J. Appl. Probab. 39 (2002), 391–394.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    J. Hozman: Analysis of the discontinuous Galerkin method applied to the European option pricing problem. AIP Conference Proceedings 1570 (2013), 227–234.CrossRefGoogle Scholar
  17. [17]
    J. Hozman, T. Tichý: Black-Scholes option pricing model: Comparison of h-convergence of the DG method with respect to boundary condition treatment. ECON - Journal of Economics, Management and Business 24 (2014), 141–152.Google Scholar
  18. [18]
    J. Hozman, T. Tichý: On the impact of various formulations of the boundary condition within numerical option valuation by DG method. Filomat 30 (2016), 4253–4263.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    J. Hozman, T. Tichý: DG method for numerical pricing of multi-asset Asian options— The case of options with floating strike. Appl. Math., Praha 62 (2017), 171–195.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    J. Hozman, T. Tichý, D. Cvejnová: A discontinuous Galerkin method for two-dimensional PDE models of Asian options. AIP Conference Proceedings 1738 (2016), Article no. 080011.Google Scholar
  21. [21]
    J. E. Ingersoll, Jr.: Theory of Financial Decision Making, Rowman & Littlefield Publishers, New Jersey, 1987.Google Scholar
  22. [22]
    J. L. Lions, E. Magenes: Non-Homogeneous Boundary Value Problems and Applications. Vol. I. Die Grundlehren der mathematischenWissenschaften 181, Springer, Berlin, 1972.Google Scholar
  23. [23]
    R. C. Merton: Theory of rational option pricing. Bell J. Econ. Manage. Sci. 4 (1973), 141–183.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    W. H. Reed, T. R. Hill: Triangular mesh methods for the neutron transport equation. Conf. Report, National Topical Meeting on Mathematical Models and Computational Techniques for Analysis of Nuclear Systems, Ann Arbor 1973. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, New Mexico, 1973.Google Scholar
  25. [25]
    K. Rektorys: Variational Methods in Engineering Problems and in Problems of Mathematical Physics, Nakladatelsví Technické Literatury, Praha, 1974. (In Czech.)zbMATHGoogle Scholar
  26. [26]
    B. Rivière: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. Theory and Implementation. Frontiers in Applied Mathematics 35, Society for Industrial and Applied Mathematics, Philadelphia, 2008.Google Scholar
  27. [27]
    L. C. G. Rogers, Z. Shi: The value of an Asian option. J. Appl. Probab. 32 (1995), 1077–1088.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    J. Večeř: A new PDE approach for pricing arithmetic average Asian options. J. Comput. Finance 4 (2001), 105–113.CrossRefGoogle Scholar
  29. [29]
    J. Večeř: Unified pricing of Asian options. Risk 15 (2002), 113–116.Google Scholar
  30. [30]
    P. Wilmott, J. Dewynne, J. Howison: Option Pricing: Mathematical Models and Computation, Financial Press, Oxford, 1993.zbMATHGoogle Scholar
  31. [31]
    E. Zeidler: Nonlinear Functional Analysis and Its Applications, II/A: Linear Monotone Operators. Springer, New York, 1990.CrossRefzbMATHGoogle Scholar
  32. [32]
    R. Zvan, P. A. Forsyth, K. Vetzal: Robust numerical methods for PDE models of Asian options. J. Comput. Finance 1 (1998), 39–78.CrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Didactics of MathematicsTechnical University of Liberec, Faculty of Science, Humanities and EducationLiberecCzech Republic
  2. 2.Department of FinanceVŠB—Technical University of Ostrava, Faculty of EconomicsOstravaCzech Republic

Personalised recommendations