The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

EFFICIENT AND ACCURATE FINITE DIFFERENCE METHOD FOR THE FOUR UNDERLYING ASSET ELS

  • Received : 2021.04.28
  • Accepted : 2021.08.23
  • Published : 2021.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this study, we consider an efficient and accurate finite difference method for the four underlying asset equity-linked securities (ELS). The numerical method is based on the operator splitting method with non-uniform grids for the underlying assets. Even though the numerical scheme is implicit, we solve the system of discrete equations in explicit manner using the Thomas algorithm for the tri-diagonal matrix resulting from the system of discrete equations. Therefore, we can use a relatively large time step and the computation of the ELS option pricing is fast. We perform characteristic computational test. The numerical test confirm the usefulness of the proposed method for pricing the four underlying asset equity-linked securities.

Keywords

References

  1. W. Chen & S. Wang: A 2nd-order ADI finite difference method for a 2D fractional BlackScholes equation governing European two asset option pricing. Math. Comput. Simul. 171 (2020), 279-293. https://doi.org/10.1016/j.matcom.2019.10.016
  2. R. Cont & E. Voltchkova: A finite difference scheme for option pricing in jump diffusion and exponential Lvy models. SIAM J. Numer. Anal. 43 (2005), no. 4, 1596-1626. https://doi.org/10.1137/S0036142903436186
  3. D.J. Duffy: Finite difference methods in financial engineering: a partial differential equation approach. John Wiley and Sons, New York, 2013.
  4. B. Dring, M. Fourni & Heuer, C: High-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids. J. Comput. Appl. Math. 271 (2014), 247-266. https://doi.org/10.1016/j.cam.2014.04.016
  5. P. Glasserman: Monte Carlo methods in financial engineering. Springer Science and Business Media 53, 2013.
  6. S. Ikonen & J. Toivanen: Operator splitting methods for American option pricing. Appl. Math. Lett. 17 (2004), no. 7, 809-814. https://doi.org/10.1016/j.aml.2004.06.010
  7. H. Jang, S. Kim, J. Han, S. Lee, J. Ban, H. Han, C. Lee, D. Jeong & J. Kim: Fast Monte Carlo simulation for pricing equity-linked securities. Comput. Econ. 56 (2019), 865-882. https://doi.org/10.1007/s10614-019-09947-2
  8. H. Jang et al.: Fast ANDROID implimentation of Monte Carlo simulation for pricing equity-linked securities. J. Korean Soc. Ind. Appl. Math. 24 (2020), no. 1, 79-84. https://doi.org/10.12941/JKSIAM.2020.24.079
  9. D. Jeong & 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
  10. D.R. Jeong, I.S. Wee & J.S. Kim: An operator splitting method for pricing the ELS option. J. Korean Soc. Ind. Appl. Math. 14 (2010), no. 3, 175-187. https://doi.org/10.12941/JKSIAM.2010.14.3.175
  11. Y. Kim, H.O. Bae & H.K. Koo: Option pricing and Greeks via a moving least square meshfree method. Quant. Finance 14 (2014), no. 10, 1753-1764. https://doi.org/10.1080/14697688.2013.845686
  12. J. Kim, T. Kim, J. Jo, Y. Choi, S. Lee, H. Hwang, M. Yoo, & D. Jeong: A practical finite difference method for the three-dimensional Black-Scholes equation. Eur. J. Oper. Res. 252 (2016), no. 1, 183-190. https://doi.org/10.1016/j.ejor.2015.12.012
  13. Y. Kwon & Y. Lee: A second-order finite difference method for option pricing under jump-diffusion models. SIAM J. Numer. Anal. 49 (2011), no. 6, 2598-2617. https://doi.org/10.1137/090777529
  14. C. Lee, J. Lyu, E. Park, W. Lee, S. Kim, D. Jeong & J. Kim: Super-Fast computation for the three-asset equity-linked securities using the finite difference method. Math. 8 (2020), no. 3, 307:1-13.
  15. L. Li & G. Zhang: Error analysis of finite difference and Markov chain approximations for option pricing. Math. Finan. 28 (2018), no. 3, 877-919. https://doi.org/10.1111/mafi.12161
  16. J. Persson & L. von Persson: Pricing European multi-asset options using a space-time adaptive FD-method. Comput. Vis. Sci. 10 (2007), no. 4, 173-183. https://doi.org/10.1007/s00791-007-0072-y
  17. N. Rambeerich, D.Y. Tangman, M.R. Lollchund & M. Bhuruth: High-order computational methods for option valuation under multifactor models. Eur. J. Oper. Res. 224 (2013), no. 1, 219-226. https://doi.org/10.1016/j.ejor.2012.07.023
  18. C. Reisinger & G. Wittum: On multigrid for anisotropic equations and variational inequalities pricing multi-dimensional European and American options. Comput. Vis. Sci. 7 (2004), no. 3-4, 189-197. https://doi.org/10.1007/s00791-004-0149-9
  19. F. Soleymani: Efficient semi-discretization techniques for pricing European and American basket options. Comput. Econ. 53 (2019), no. 4, 1487-1508. https://doi.org/10.1007/s10614-018-9819-4
  20. F. Soleymani & A. Akgl: Improved numerical solution of multi-asset option pricing problem: A localized RBF-FD approach. Chaos, Solitons & Fractals. 119 (2019), 298-309. https://doi.org/10.1016/j.chaos.2019.01.003
  21. J. Wang, J. Ban, J. Han, S. Lee & D. Jeong: Mobile platform for pricing of equity-linked securities. J. Korean Soc. Ind. Appl. Math. 21 (2017), no. 3, 181-202. https://doi.org/10.12941/jksiam.2017.21.181
  22. J. Wang, Y. Yan, W. Chen, W. Shao & W. Tang: Equity-linked securities option pricing by fractional Brownian motion. Chaos, Solitons & Fractals 144 (2021), 110716. https://doi.org/10.1016/j.chaos.2021.110716
  23. M. Yoo, D. Jeong, S. Seo & J. Kim: A comparison study of explicit and implicit numerical methods for the equity-linked securities. Honam Math. J. 37 (2015), no. 4, 441-455. https://doi.org/10.5831/HMJ.2015.37.4.441