Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
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PATH AVERAGED OPTION VALUE CRITERIA FOR SELECTING BETTER OPTIONS

  • KIM, JUNSEOK (DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY) ;
  • YOO, MINHYUN (DEPARTMENT OF FINANCIAL ENGINEERING, KOREA UNIVERSITY) ;
  • SON, HYEJU (DEPARTMENT OF ECONOMICS, SOOKMYUNG WOMEN'S UNIVERSITY) ;
  • LEE, SEUNGGYU (DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY) ;
  • KIM, MYEONG-HYEON (BUSINESS SCHOOL, KOREA UNIVERSITY) ;
  • CHOI, YONGHO (DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY) ;
  • JEONG, DARAE (DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY) ;
  • KIM, YOUNG ROCK (MAJOR IN MATHEMATICS EDUCATION, HANKUK UNIVERSITY OF FOREIGN STUDIES)
  • Received : 2016.05.27
  • Accepted : 2016.06.11
  • Published : 2016.06.25
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Abstract

In this paper, we propose an optimal choice scheme to determine the best option among comparable options whose current expectations are all the same under the condition that an investor has a confidence in the future value realization of underlying assets. For this purpose, we use a path-averaged option as our base instrument in which we calculate the time discounted value along the path and divide it by the number of time steps for a given expected path. First, we consider three European call options such as vanilla, cash-or-nothing, and asset-or-nothing as our comparable set of choice schemes. Next, we perform the experiments using historical data to prove the usefulness of our proposed scheme. The test suggests that the path-averaged option value is a good guideline to choose an optimal option.

Keywords

References

  1. F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-659. https://doi.org/10.1086/260062
  2. C. Wang, S. Zhou, and J. Yang, The pricing of vulnerable options in a fractional Brownian motion environment, Discrete Dyn. Nat. Soc., 2015 (2015), 579213.
  3. D.J. Duffy, Finite Difference Methods in Financial Engineering, John Wiley & Sons, New York, NY, USA, (2006).
  4. H. Han and X. Wu, A fast numerical method for the Black-Scholes equation of American options, SIAM J. Numer. Anal., 41(6) (2003), 2081-2095. https://doi.org/10.1137/S0036142901390238
  5. D. Jeong, T. Ha, M. Kim, J. Shin, I.H. Yoon, and J. Kim, An adaptive finite difference method using far-field boundary conditions for the Black-Scholes equation, B. Korean Math. Soc., 51(4) (2014), 1087-1100. https://doi.org/10.4134/BKMS.2014.51.4.1087
  6. D. Jeong and J. Kim, A comparison study of ADI and operator splitting methods on option pricing models, J. Comput. Appl. Math., 247 (2013), 162-171. https://doi.org/10.1016/j.cam.2013.01.008
  7. D. Jeong, J. Kim, and I.S. Wee, An accurate and efficient numerical method for Black-Scholes equations, Commun. Korean Math. Soc., 24(4) (2009), 617-628. https://doi.org/10.4134/CKMS.2009.24.4.617
  8. D. Jeong, I.S.Wee, and J. Kim, An operator splitting method for pricing the ELS option, J. KSIAM, 14 (2010), 175-187.
  9. R. Seydel, Tools for Computational Finance, Springer, Berlin, Germany, (2003).
  10. D. Tavella and C. Randall, Pricing Financial Instruments: The Finite Difference Method, John Wiley & Sons, New York, NY, USA, (2000).
  11. J. Topper, Financial Engineering with Finite Elements, John Wiley & Sons, Chichester, UK, (2005).
  12. P. Wilmott, J. Dewynne, and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, UK, (1993).
  13. D. Jeong, S. Seo, H. Hwang, D. Lee, Y. Choi, and J. Kim, Accuracy, robustness, and efficiency of the linear boundary condition for the Black-Scholes equations, Discrete Dyn. Nat. Soc., 2015 (2015) 359028.
  14. Y. Achdou and N. Tchou, Variational analysis for the Black and Scholes equation with stochastic volatility, ESAIM-Math. Model. Num., 36)(3) (2002), 373-395. https://doi.org/10.1051/m2an:2002018
  15. A. Ern, S. Villeneuve, and A. Zanette, Adaptive finite element methods for local volatility European option pricing, Int. J. Theor. Appl. Financ., 7(6) (2004), 659-684. https://doi.org/10.1142/S0219024904002669
  16. N. Rambeerich, D.Y. Tangman, M.R. Lollchund, and M. Bhuruth, High-order computational methods for option valuation under multifactor models, Eur. J. Oper. Res., 224(1) (2013), 219-226. https://doi.org/10.1016/j.ejor.2012.07.023
  17. P.A. Forsyth and K.R. Vetzal, Quadratic convergence for valuing American options using a penalty method, SIAM J. Sci. Comput., 23(6) (2002), 2095-2122. https://doi.org/10.1137/S1064827500382324
  18. S. Wang, A novel fitted finite volume method for the Black-Scholes equation governing option pricing, IMA J. Numer. Anal., 24(4) (2004), 699-720. https://doi.org/10.1093/imanum/24.4.699
  19. S. Wang, S. Zhang, and Z. Fang, A super convergent fitted finite volume method for Black-Scholes equations governing European and American option valuation, Numer. Meth. Part. D. E., 31(4) (2015), 1190-1208. https://doi.org/10.1002/num.21941
  20. P. Carr and D.B. Madan, Option valuation using the fast Fourier transform, J. Comput. Financ., 2(4) (1999), 61-73. https://doi.org/10.21314/JCF.1999.043
  21. A. Cerny, Introduction to fast Fourier transform in finance, J. Deriv., 12(1) (2004), 73-88. https://doi.org/10.3905/jod.2004.434538
  22. M.A.H. Dempster and S.S.G. Hong, Spread option valuation and the fast Fourier transform, Springer Finance, Springer Finance, Springer, Berlin, (2002), 203-220.
  23. C.C. Hsu, S.K. Lin, and T.F. Chen, Pricing and hedging European energy derivatives: a case study of WTI oil options, Asia-Pac. J. Financ. St., 43(3) (2014), 317-355. https://doi.org/10.1111/ajfs.12050
  24. T. Sakuma and Y. Yamada, Application of homotopy analysis method to option pricing under Levy processes, Asia-Pac. Financ. Mark., 21(1) (2014), 1-14. https://doi.org/10.1007/s10690-013-9175-2
  25. E.G. Haug, The Complete Guide to Option Pricing Formulas, New York, McGraw-Hill, (1998).
  26. M. Rubinstein, One for another, Risk, 4(7) (1991), 30-32.
  27. Korea Exchange, Historical KOSPI 200 index option price, http://www.krx.co.kr/m3/m32/m321/JHPKOR0300201.jsp, (2015).