Article

Full entry | PDF   (0.3 MB)
Keywords:
American options; variational inequality; finite element methods; optimal and superconvergent estimates; interpolation postprocessing; a posteriori error estimators
Summary:
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. {\it 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.
References:
[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.
[2] Allegretto, W., Barone-Adesi, G., Elliott, R. J.: Numerical evaluation of the critical price and American options. European J. Finance 1 (1995), 69-78. DOI 10.1080/13518479500000009
[3] Allegretto, W., Lin, Y., Yang, H.: Finite element error estimates for a nonlocal problem in American option valuation. SIAM J. Numer. Anal. 39 (2001), 834-857. DOI 10.1137/S0036142900370137 | MR 1860447 | Zbl 0996.91064
[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. MR 1824289 | Zbl 1108.91034
[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.
[6] Barone-Adesi, G., Whaley, R. E.: Efficient analytic approximation of American option values. J. Finance 42 (1987), 301-320. DOI 10.1111/j.1540-6261.1987.tb02569.x
[7] Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81 (1973), 637-659. DOI 10.1086/260062 | Zbl 1092.91524
[8] Boyle, P., Broadie, M., Glasserman, P.: Monte Carlo methods for security pricing. J. Econ. Dyn. Control 21 (1997), 1267-1321. DOI 10.1016/S0165-1889(97)00028-6 | MR 1470283 | Zbl 0901.90007
[9] Brennan, M. J., Schwartz, E. S.: The valuation of American put options. J. Finance 32 (1997), 449-462. DOI 10.2307/2326779
[10] Broadie, M., Detemple, J.: American option valuation: New bounds, approximations, and a comparison of existing methods. Rev. Financial Studies 9 (1996), 1211-1250. DOI 10.1093/rfs/9.4.1211
[11] Crank, J.: Free and Moving Boundary Problems. Clarendon Press Oxford (1984). MR 0776227 | Zbl 0547.35001
[12] Duffy, D. J.: Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach. John Wiley & Sons Hoboken (2006). MR 2286409
[13] Eaves, B. C.: On the basic theorem of complementarity. Math. Program. 1 (1971), 68-75. DOI 10.1007/BF01584073 | MR 0287901 | Zbl 0227.90044
[14] Elliott, C. M., Ockendon, J. R.: Weak and Variational Methods for Moving Boundary Problems. Pitman Boston-London-Melbourne (1982). MR 0650455 | Zbl 0476.35080
[15] Fetter, A.: $L^{\infty}$-error estimate for an approximation of a parabolic variational inequality. Numer. Math. 50 (1987), 557-565. DOI 10.1007/BF01408576 | MR 0880335
[16] Feistauer, M.: On the finite element approximation of functions with noninteger derivatives. Numer. Funct. Anal. Optimization 10 (1989), 91-110. DOI 10.1080/01630568908816293 | MR 0978805 | Zbl 0668.65008
[17] Han, W., Chen, X.: An Introduction to Variational Inequalities: Elementary Theory, Numerical Analysis and Applications. Higher Education Press Beijing (2007). MR 2791918
[18] Huang, J., Subrahmanyam, M. C., Yu, G. G.: Pricing and hedging American options: A recursive integration method. Rev. Financial Studies 9 (1996), 277-300. DOI 10.1093/rfs/9.1.277
[19] Hull, J.: Option, Futures and Other Derivative Securities, 2nd edition. Prentice Hall New Jersey (1993).
[20] Jaillet, P., Lamberton, D., Lapeyre, B.: Variational inequalities and the pricing of American options. Acta Appl. Math. 21 (1990), 263-289. DOI 10.1007/BF00047211 | MR 1096582 | Zbl 0714.90004
[21] Jiang, L., Dai, M.: Convergence of binomial tree methods for European/American path-dependent options. SIAM J. Numer. Anal. 42 (2004), 1094-1109. DOI 10.1137/S0036142902414220 | MR 2113677 | Zbl 1159.91392
[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. MR 2056293
[23] Jiang, L.: Mathematical Modeling and Methods of Options Pricing. Higher Education Press Beijing (2003). MR 1318688
[24] Johnson, H. E.: An analytic approximation for the American put price. J. Financial and Quantitative Anal. 18 (1983), 141-148. DOI 10.2307/2330809
[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. MR 1602730
[26] Kwok, Y. K.: Mathematical Models of Financial Derivatives. Springer Singapore (1998). MR 1645143 | Zbl 0931.91018
[27] Lin, Q., Yan, N.: The Construction and Analysis of High Efficiency Finite Element Methods. Hebei University Publishers Baoding (1996), Chinese.
[28] Lin, Q., Zhang, S.: An immediate analysis for global superconvergence for integrodifferential equations. Appl. Math. 42 (1997), 1-21. DOI 10.1023/A:1022264125558 | MR 1426677 | Zbl 0902.65090
[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).
[30] Liu, T., Zhang, P.: Numerical methods for option pricing problems. J. Syst. Sci. & Math. Sci. 12 (2003), 12-20. MR 2034582
[31] Marcozzi, M. D.: On the approximation of optimal stopping problems with application to financial mathematics. SIAM J. Sci. Comput. 22 (2001), 1865-1884. DOI 10.1137/S1064827599364647 | MR 1813301 | Zbl 0980.60047
[32] MacMillan, L. W.: Analytic approximation for the American put option. Adv. in Futures and Options Res. 1 (1986), 1149-1159.
[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.
[34] Merton, R. C.: Theory of rational option pricing. Bell J. Econom. and Management Sci. 4 (1973), 141-183. DOI 10.2307/3003143 | MR 0496534
[35] Sanchez, A. M., Arcangéli, R.: Estimations des erreurs de meilleure approximation polynomiale et d'interpolation de Lagrange dans les espaces de Sobolev d'ordre non entier. Numer. Math. 45 (1984), 301-321 French. DOI 10.1007/BF01389473 | MR 0766187 | Zbl 0587.41018
[36] Topper, J.: Financial Engineering with Finite Elements. John Wiley & Sons Hoboken (2005).
[37] Underwood, R., Wang, J.: An integral representation and computation for the solution of American options. Nonlinear. Anal., Real World Appl. 3 (2002), 259-274. DOI 10.1016/S1468-1218(01)00028-1 | MR 1893977 | Zbl 1011.91049
[38] Vuik, C.: An $L^2$-error estimate for an approximation of the solution of a parabolic variational inequality. Numer. Math. 57 (1990), 453-471. DOI 10.1007/BF01386423 | MR 1063805
[39] Wilmott, P., Dewynne, J., Howison, S.: Option Pricing: Mathematical Models and Computation. Financial Press Oxford (1995). MR 1357666 | Zbl 0844.90011
[40] Zhang, T.: The numerical methods for American options pricing. Acta Math. Appl. Sin. 25 (2002), 113-122. MR 1926728

Partner of