Superconvergence estimates of finite element methods for American options

Qun Lin; Tang Liu; Shu Hua Zhang

Applications of Mathematics (2009)

  • Volume: 54, Issue: 3, page 181-202
  • ISSN: 0862-7940

Abstract

top
In this paper we are concerned with finite element approximations to the evaluation of American options. First, following W. Allegretto etc., SIAM J. Numer. Anal. 39 (2001), 834–857, we introduce a novel practical approach to the discussed problem, which involves the exact reformulation of the original problem and the implementation of the numerical solution over a very small region so that this algorithm is very rapid and highly accurate. Secondly by means of a superapproximation and interpolation postprocessing analysis technique, we present sharp L 2 -, L ∞ -norm error estimates and an H 1 -norm superconvergence estimate for this finite element method. As a by-product, the global superconvergence result can be used to generate an efficient a posteriori error estimator.

How to cite

top

Lin, Qun, Liu, Tang, and Zhang, Shu Hua. "Superconvergence estimates of finite element methods for American options." Applications of Mathematics 54.3 (2009): 181-202. <http://eudml.org/doc/37815>.

@article{Lin2009,
abstract = {In this paper we are concerned with finite element approximations to the evaluation of American options. First, following W. Allegretto etc., SIAM J. Numer. Anal. 39 (2001), 834–857, we introduce a novel practical approach to the discussed problem, which involves the exact reformulation of the original problem and the implementation of the numerical solution over a very small region so that this algorithm is very rapid and highly accurate. Secondly by means of a superapproximation and interpolation postprocessing analysis technique, we present sharp $L^2$-, $L^\{\infty \}$-norm error estimates and an $H^1$-norm superconvergence estimate for this finite element method. As a by-product, the global superconvergence result can be used to generate an efficient a posteriori error estimator.},
author = {Lin, Qun, Liu, Tang, Zhang, Shu Hua},
journal = {Applications of Mathematics},
keywords = {American options; variational inequality; finite element methods; optimal and superconvergent estimates; interpolation postprocessing; a posteriori error estimators; American options; variational inequality; finite element methods; optimal and superconvergent estimates; interpolation postprocessing; a posteriori error estimators},
language = {eng},
number = {3},
pages = {181-202},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Superconvergence estimates of finite element methods for American options},
url = {http://eudml.org/doc/37815},
volume = {54},
year = {2009},
}

TY - JOUR
AU - Lin, Qun
AU - Liu, Tang
AU - Zhang, Shu Hua
TI - Superconvergence estimates of finite element methods for American options
JO - Applications of Mathematics
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 3
SP - 181
EP - 202
AB - In this paper we are concerned with finite element approximations to the evaluation of American options. First, following W. Allegretto etc., SIAM J. Numer. Anal. 39 (2001), 834–857, we introduce a novel practical approach to the discussed problem, which involves the exact reformulation of the original problem and the implementation of the numerical solution over a very small region so that this algorithm is very rapid and highly accurate. Secondly by means of a superapproximation and interpolation postprocessing analysis technique, we present sharp $L^2$-, $L^{\infty }$-norm error estimates and an $H^1$-norm superconvergence estimate for this finite element method. As a by-product, the global superconvergence result can be used to generate an efficient a posteriori error estimator.
LA - eng
KW - American options; variational inequality; finite element methods; optimal and superconvergent estimates; interpolation postprocessing; a posteriori error estimators; American options; variational inequality; finite element methods; optimal and superconvergent estimates; interpolation postprocessing; a posteriori error estimators
UR - http://eudml.org/doc/37815
ER -

References

top
  1. Allegretto, W., Barone-Adesi, G., Dinenis, E., Lin, Y., Sorwar, G., A new approach to check the free boundary of single factor interest rate put option, Finance 20 (1999), 153-168. (1999) 
  2. Allegretto, W., Barone-Adesi, G., Elliott, R. J., 10.1080/13518479500000009, European J. Finance 1 (1995), 69-78. (1995) DOI10.1080/13518479500000009
  3. Allegretto, W., Lin, Y., Yang, H., 10.1137/S0036142900370137, SIAM J. Numer. Anal. 39 (2001), 834-857. (2001) Zbl0996.91064MR1860447DOI10.1137/S0036142900370137
  4. Allegretto, W., Lin, Y., Yang, H., A fast and highly accurate numerical method for the evaluation of American options, Dyn. Contin. Discrete Impuls. Syst., Ser. B Appl. Algorithms 8 (2001), 127-138. (2001) Zbl1108.91034MR1824289
  5. Badea, L., Wang, J., A new formulation for the valuation of American options. I. Solution uniqueness. II. Solution existence, Anal. Sci. Comput. (Eun-Jae Park, Jongwoo Lee, eds.) 5 (2000), 3-16, 17-33. (2000) 
  6. Barone-Adesi, G., Whaley, R. E., 10.1111/j.1540-6261.1987.tb02569.x, J. Finance 42 (1987), 301-320. (1987) DOI10.1111/j.1540-6261.1987.tb02569.x
  7. Black, F., Scholes, M., 10.1086/260062, J. Polit. Econ. 81 (1973), 637-659. (1973) Zbl1092.91524DOI10.1086/260062
  8. Boyle, P., Broadie, M., Glasserman, P., 10.1016/S0165-1889(97)00028-6, J. Econ. Dyn. Control 21 (1997), 1267-1321. (1997) Zbl0901.90007MR1470283DOI10.1016/S0165-1889(97)00028-6
  9. Brennan, M. J., Schwartz, E. S., 10.2307/2326779, J. Finance 32 (1997), 449-462. (1997) DOI10.2307/2326779
  10. Broadie, M., Detemple, J., 10.1093/rfs/9.4.1211, Rev. Financial Studies 9 (1996), 1211-1250. (1996) DOI10.1093/rfs/9.4.1211
  11. Crank, J., Free and Moving Boundary Problems, Clarendon Press Oxford (1984). (1984) Zbl0547.35001MR0776227
  12. Duffy, D. J., Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, John Wiley & Sons Hoboken (2006). (2006) MR2286409
  13. Eaves, B. C., 10.1007/BF01584073, Math. Program. 1 (1971), 68-75. (1971) Zbl0227.90044MR0287901DOI10.1007/BF01584073
  14. Elliott, C. M., Ockendon, J. R., Weak and Variational Methods for Moving Boundary Problems, Pitman Boston-London-Melbourne (1982). (1982) Zbl0476.35080MR0650455
  15. Fetter, A., 10.1007/BF01408576, Numer. Math. 50 (1987), 557-565. (1987) MR0880335DOI10.1007/BF01408576
  16. Feistauer, M., 10.1080/01630568908816293, Numer. Funct. Anal. Optimization 10 (1989), 91-110. (1989) Zbl0668.65008MR0978805DOI10.1080/01630568908816293
  17. Han, W., Chen, X., An Introduction to Variational Inequalities: Elementary Theory, Numerical Analysis and Applications, Higher Education Press Beijing (2007). (2007) MR2791918
  18. Huang, J., Subrahmanyam, M. C., Yu, G. G., 10.1093/rfs/9.1.277, Rev. Financial Studies 9 (1996), 277-300. (1996) DOI10.1093/rfs/9.1.277
  19. Hull, J., Option, Futures and Other Derivative Securities, 2nd edition, Prentice Hall New Jersey (1993). (1993) 
  20. Jaillet, P., Lamberton, D., Lapeyre, B., 10.1007/BF00047211, Acta Appl. Math. 21 (1990), 263-289. (1990) Zbl0714.90004MR1096582DOI10.1007/BF00047211
  21. Jiang, L., Dai, M., 10.1137/S0036142902414220, SIAM J. Numer. Anal. 42 (2004), 1094-1109. (2004) Zbl1159.91392MR2113677DOI10.1137/S0036142902414220
  22. Jiang, L., Dai, M., Convergence of the explicit difference scheme and binomial tree method for American options, J. Comput. Math. 22 (2004), 371-380. (2004) MR2056293
  23. Jiang, L., Mathematical Modeling and Methods of Options Pricing, Higher Education Press Beijing (2003). (2003) MR1318688
  24. Johnson, H. E., 10.2307/2330809, J. Financial and Quantitative Anal. 18 (1983), 141-148. (1983) DOI10.2307/2330809
  25. Křížek, M., Neittaanmäki, P., Bibliography on superconvergence, In: Proc. Conf. Finite Element Methods: Superconvergence, Post-processing and A Posteriori Estimates, Lecture Notes in Pure and Appl. Math. 196 M. Křížek et al. Marcel Dekker New York (1998), 315-348. (1998) MR1602730
  26. Kwok, Y. K., Mathematical Models of Financial Derivatives, Springer Singapore (1998). (1998) Zbl0931.91018MR1645143
  27. Lin, Q., Yan, N., The Construction and Analysis of High Efficiency Finite Element Methods, Hebei University Publishers Baoding (1996), Chinese. (1996) 
  28. Lin, Q., Zhang, S., 10.1023/A:1022264125558, Appl. Math. 42 (1997), 1-21. (1997) Zbl0902.65090MR1426677DOI10.1023/A:1022264125558
  29. Liu, M., Wang, J., Pricing American options by domain decomposition methods, In: Iterative Methods in Scientific Computation J. Wang, H. Allen, H. Chen, L. Mathew IMACS Publication (1998). (1998) 
  30. Liu, T., Zhang, P., Numerical methods for option pricing problems, J. Syst. Sci. & Math. Sci. 12 (2003), 12-20. (2003) MR2034582
  31. Marcozzi, M. D., 10.1137/S1064827599364647, SIAM J. Sci. Comput. 22 (2001), 1865-1884. (2001) Zbl0980.60047MR1813301DOI10.1137/S1064827599364647
  32. MacMillan, L. W., Analytic approximation for the American put option, Adv. in Futures and Options Res. 1 (1986), 1149-1159. (1986) 
  33. McKean, H. P., Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics, Industrial Management Rev. 6 (1965), 32-39. (1965) 
  34. Merton, R. C., 10.2307/3003143, Bell J. Econom. and Management Sci. 4 (1973), 141-183. (1973) MR0496534DOI10.2307/3003143
  35. Sanchez, A. M., Arcangéli, R., 10.1007/BF01389473, Numer. Math. 45 (1984), 301-321 French. (1984) Zbl0587.41018MR0766187DOI10.1007/BF01389473
  36. Topper, J., Financial Engineering with Finite Elements, John Wiley & Sons Hoboken (2005). (2005) 
  37. Underwood, R., Wang, J., 10.1016/S1468-1218(01)00028-1, Nonlinear. Anal., Real World Appl. 3 (2002), 259-274. (2002) Zbl1011.91049MR1893977DOI10.1016/S1468-1218(01)00028-1
  38. Vuik, C., 10.1007/BF01386423, Numer. Math. 57 (1990), 453-471. (1990) MR1063805DOI10.1007/BF01386423
  39. Wilmott, P., Dewynne, J., Howison, S., Option Pricing: Mathematical Models and Computation, Financial Press Oxford (1995). (1995) Zbl0844.90011MR1357666
  40. Zhang, T., The numerical methods for American options pricing, Acta Math. Appl. Sin. 25 (2002), 113-122. (2002) MR1926728

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.