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COMPARATIVE STUDY OF NUMERICAL ALGORITHMS FOR THE ARITHMETIC ASIAN OPTION

  • WANG, JIAN (DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY) ;
  • BAN, JUNGYUP (DEPARTMENT OF FINANCIAL ENGINEERING, KOREA UNIVERSITY) ;
  • LEE, SEONGJIN (DEPARTMENT OF FINANCIAL ENGINEERING, KOREA UNIVERSITY) ;
  • YOO, CHANGWOO (DEPARTMENT OF FINANCIAL ENGINEERING, KOREA UNIVERSITY)
  • Received : 2018.02.28
  • Accepted : 2018.03.13
  • Published : 2018.03.25

Abstract

This paper presents the numerical valuation of the arithmetic Asian option by using the operator-splitting method (OSM). Since there is no closed-form solution for the arithmetic Asian option, finding a good numerical algorithm to value the arithmetic Asian option is important. In this paper, we focus on a two-dimensional PDE. The OSM is famous for dealing with plural-dimensional PDE using finite difference discretization. We provide a detailed numerical algorithm and compare results with MCS method to show the performance of the method.

References

  1. Duffy, Daniel J. Finite Difference methods in financial engineering: a Partial Differential Equation approach. John Wiley & Sons, 2013.
  2. GLASSERMAN, Paul. Monte Carlo methods in financial engineering. Springer Science & Business Media, 2013.
  3. SEYDEL, Rudiger; SEYDEL, Rudiger. Tools for computational finance. Berlin: Springer, 2006.
  4. TURNBULL, Stuart M.; WAKEMAN, Lee Macdonald. A quick algorithm for pricing European average options. Journal of financial and quantitative analysis, 1991, 26.3: 377-389. https://doi.org/10.2307/2331213
  5. Vorst, Ton. VORST, Ton. Prices and hedge ratios of average exchange rate options. International Review of Financial Analysis, 1992, 1.3: 179-193. https://doi.org/10.1016/1057-5219(92)90003-M
  6. LEVY, Edmond. Pricing European average rate currency options. Journal of International Money and Finance, 1992, 11.5: 474-491. https://doi.org/10.1016/0261-5606(92)90013-N
  7. LEVY, Edmond; TURNBULL, Stuart. Average intelligence. Risk, 1992, 5.2: 53-57.
  8. CEN, Zhongdi; XU, Aimin; LE, Anbo. A hybrid finite difference scheme for pricing Asian options. Applied Mathematics and Computation, 2015, 252: 229-239. https://doi.org/10.1016/j.amc.2014.12.007
  9. INGERSOLL, Jonathan E. Theory of financial decision making. Rowman & Littlefield, 1987.
  10. BLACK, Fischer; SCHOLES, Myron. The pricing of options and corporate liabilities. Journal of political economy, 1973.
  11. WILMOTT, Paul; DEWYNNE, Jeff; HOWISON, Sam. Option pricing: mathematical models and computation. Oxford financial press, 1993.
  12. ROGERS, L. Chris G.; SHI, Zo. The value of an Asian option. Journal of Applied Probability, 1995, 32.4: 1077-1088. https://doi.org/10.2307/3215221
  13. FOSCHI, Paolo; PAGLIARANI, Stefano; PASCUCCI, Andrea. Approximations for Asian options in local volatility models. Journal of Computational and Applied Mathematics, 2013, 237.1: 442-459. https://doi.org/10.1016/j.cam.2012.06.015
  14. S. Shreve, Stochastic calculus for finance II: Continuous-time models, Springer Science and Business Media, Pennsylvania (2004)
  15. KANGRO, Raul; NICOLAIDES, Roy. Far Field Boundary Conditions for Black-Scholes Equations. SIAM Journal on Numerical Analysis, 2000, 38.4: 1357-1368. https://doi.org/10.1137/S0036142999355921
  16. OOSTERLEE, Cornelis W. On multigrid for linear complementarity problems with application to Americanstyle options. Electronic Transactions on Numerical Analysis, 2003, 15.1: 165-185.
  17. PERSSON, Jonas; VON PERSSON, Lina. Pricing European multi-asset options using a space-time adaptive FD-method. Computing and Visualization in Science, 2007, 10.4: 173-183. https://doi.org/10.1007/s00791-007-0072-y
  18. TAVELLA, Domingo; RANDALL, Curt. Pricing financial instruments: The finite difference method. John Wiley & Sons, 2000.
  19. ZVAN, Robert; VETZAL, Kenneth R.; FORSYTH, Peter A. PDE methods for pricing barrier options. Journal of Economic Dynamics and Control, 2000, 24.11-12: 1563-1590. https://doi.org/10.1016/S0165-1889(00)00002-6
  20. IKONEN, Samuli; TOIVANEN, Jari. Operator splitting methods for American option pricing. Applied mathematics letters, 2004, 17.7: 809-814. https://doi.org/10.1016/j.aml.2004.06.010
  21. Using MATLAB, The MathWorks Inc., Natick, MA, 1998, http://www.mathworks.com/.
  22. XU, Yongjia; LAI, Yongzeng; YAO, Haixiang. Efficient simulation of Greeks of multiasset European and Asian style options by Malliavin calculus and quasi-Monte Carlo methods. Applied mathematics and computation, 2014, 236: 493-511. https://doi.org/10.1016/j.amc.2014.03.057
  23. UEKI, Taro. Brownian bridge and statistical error estimation in Monte Carlo reactor calculation. Journal of Nuclear Science and Technology, 2013, 50.8: 762-773. https://doi.org/10.1080/00223131.2013.808003
  24. JACOBSON, Sheldon H. Variance and bias reduction techniques for the harmonic gradient estimator1. Applied Mathematics and Computation, 1993, 55.2-3: 153-186. https://doi.org/10.1016/0096-3003(93)90019-B
  25. DANG, Duy-Minh; JACKSON, Kenneth R.; MOHAMMADI, Mohammadreza. Dimension and variance reduction for Monte Carlo methods for high-dimensional models in finance. Applied Mathematical Finance, 2015, 22.6: 522-552. https://doi.org/10.1080/1350486X.2015.1110492