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An explicit series approximation to the optimal exercise boundary of American put options. (English) Zbl 1221.91053
Summary: We derive an explicit series approximation solution for the optimal exercise boundary of an American put option by means of a new analytical method for strongly nonlinear problems, namely the homotopy analysis method (HAM). The Black--Scholes equation subject to the moving boundary conditions for an American put option is transferred into an infinite number of linear sub-problems in a fixed domain through the deformation equations. Different from perturbation/asymptotic approximations, the HAM approximation can be applicable for options with much longer expiry. Accuracy tests are made in comparison with numerical solutions. It is found that the current approximation is as accurate as many numerical methods. Considering its explicit form of expression, it can bring great convenience to the market practitioners.

MSC:
 91G60 Numerical methods in mathematical finance 65M99 Numerical methods for IVP of PDE 91G20 Derivative securities
Full Text:
References:
 [1] Cox, J.; Ross, S.; Rubinstein, M.: Option pricing: a simplified approach, J financ econ 7, 229-263 (1979) · Zbl 1131.91333 · doi:10.1016/0304-405X(79)90015-1 [2] Broadie, M.; Detemple, J.: American option valuation: new bounds, approximations, and a comparison of existing methods, Rev financ stud 9, No. 4, 1211-1250 (1996) [3] Grant, D.; Vora, G.; Weeks, D.: Simulation and the early-exercise option problem, J financ eng 5, 211-227 (1996) [4] Longstaff, F.; Schwartz, E. S.: A radial basis function method for solving options pricing model, Rev financ stud 14, 113-147 (2001) [5] Jaillet, P.; Lamberton, D.; Lapeyre, B.: Variational inequalities and the pricing of American options, Acta applicandae math 21, 263-289 (1990) · Zbl 0714.90004 · doi:10.1007/BF00047211 [6] Dempster M. Fast numerical valuation of American, exotic and complex options. Colchester, England: Department of Mathematics research report, University of Essex; 1994. · Zbl 1009.91017 [7] Brennan, M.; Schwartz, E.: The valuation of American put options, J financ 32, 449-462 (1977) [8] Wu, L.; Kwok, Y. K.: A front-fixing finite difference method for the valuation of American options, J financ eng 6, 83-97 (1997) [9] Allegretto, W.; Lin, Y.; Yang, H.: Simulation and the early-exercise option problem, Discr contin dyn syst ser B, appl algor 8, 127-136 (2001) [10] Hon, Y. C.; Mao, X. Z.: A radial basis function method for solving options pricing model, J financ eng 8, 31-49 (1997) [11] Chen XF, Chadam J, Stamicar R. The optimal exercise boundary for American put options: analytic and numerical approximations. Working paper. University of Pittsburgh; 2000. Available from: http://www.math.pitt.edu/-xfc/Option/CCSFinal.ps. [12] Broadie, M.; Detemple, J.: Recent advances in numerical methods for pricing derivative securities, Numerical methods in finance (1997) · Zbl 0898.90029 [13] Kim, I. J.: The analytic valuation of American options, Rev financ stud 3, 547-572 (1990) [14] Carr, P.; Jarrow, R.; Myneni, R.: Alternative characterizations of American put options, Math financ 2, 87-106 (1992) · Zbl 0900.90004 · doi:10.1111/j.1467-9965.1992.tb00040.x [15] Van Moerbeke, P.: An optimal stopping problem with linear reward, Acta math 132, 111-151 (1974) · Zbl 0297.60027 · doi:10.1007/BF02392110 [16] Blanchet, A.: On the regularity of the free boundary in the parabolic obstacle problem. Application to American options, Nonlinear anal 65, 1362-1378 (2006) · Zbl 1109.35121 · doi:10.1016/j.na.2005.10.009 [17] Barles, G.; Burdeau, J.; Romano, M.; Samsoen, N.: Critical stock price near expiration, Math financ 5, No. 2, 77-95 (1995) · Zbl 0866.90029 · doi:10.1111/j.1467-9965.1995.tb00103.x [18] Kuske, R. A.; Keller, J. B.: Optional exercise boundary for an American put option, Appl math financ 5, 107-116 (1998) · Zbl 1009.91025 · doi:10.1080/135048698334673 [19] Allobaidi, G.; Mallier, R.: On the optimal exercise boundary for an American put option, J appl math 1, No. 1, 39-45 (2001) · Zbl 0976.91029 · doi:10.1155/S1110757X01000018 [20] Evans, J. D.; Kuske, R.; Keller, J. B.: American options on asserts with dividends near expiry, Math financ 12, No. 3, 219-237 (2002) · Zbl 1031.91047 · doi:10.1111/1467-9965.02008 [21] Zhang JE, Li TC. Pricing and hedging American options analytically: a perturbation method. Working paper. The University of Hong Kong; 2006. [22] Knessl, C.: A note on a moving boundary problem arising in the American put option, Stud appl math 107, 157-183 (2001) · Zbl 1152.91522 · doi:10.1111/1467-9590.00183 [23] Chen XF, Chadam J. A mathematical analysis for the optimal exericise boundary American put option. Working paper. University of Pittsburgh; 2005. Available from: http://www.pitt.edu/-chadam/papers/2CC9-30-05.pdf. [24] Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University; 1992. [25] Liao, S. J.: Beyond perturbation: introduction to the homotopy analysis method, (2003) [26] Liao, S. J.; Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations, Stud appl math 119, 297-355 (2007) [27] Liao, S. J.: Notes on the homotopy analysis method some definitions and theorems, Commun nonlinear sci numer simul 14, No. 4, 983-997 (2009) · Zbl 1221.65126 · doi:10.1016/j.cnsns.2008.04.013 [28] Liao, S. J.; Cheung, K. F.: Homotopy analysis of nonlinear progressive waves in deep water, J eng math 45, No. 2, 105-116 (2003) · Zbl 1112.76316 · doi:10.1023/A:1022189509293 [29] Liao, S. J.: A new branch of solutions of boundary-layer flows over a permeable stretching plate, Int J non-linear mech 42, 819-830 (2007) · Zbl 1200.76046 · doi:10.1016/j.ijnonlinmec.2007.03.007 [30] Hayat, T.; Abbas, Z.; Sajid, M.; Asghar, S.: The influence of thermal radiation on MHD flow of a second grade fluid, Int J heat mass transfer 50, 931-941 (2007) · Zbl 1124.80325 · doi:10.1016/j.ijheatmasstransfer.2006.08.014 [31] Abbasbandy, S.: Solitary wave equations to the Kuramoto -- Sivashinsky equation by means of the homotopy analysis method, Nonlinear dyn 52, 35-40 (2007) · Zbl 1173.35646 · doi:10.1007/s11071-007-9255-9 [32] Mustafa Inc. On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method. Phys Lett A 2007;365:412 -- 15. · Zbl 1203.65275 · doi:10.1016/j.physleta.2007.01.069 [33] Cheng, J.; Liao, S. J.; Mohapatra, R. N.; Vajravelu, K.: Series solutions of nano boundary layer flows by means of the homotopy analysis method, J math anal appl 343, No. 1, 233-245 (2008) · Zbl 1135.76016 · doi:10.1016/j.jmaa.2008.01.050 [34] Cheng J, Cang J, Liao SJ. On the interaction of deep water waves and exponential shear currents. ZAMP. Online. · Zbl 1173.76007 · doi:10.1007/s00033-008-7050-1 [35] Zhu, S. P.: An exact and explicit solution for the valuation of American put options, Quant financ 6, 229-242 (2006) · Zbl 1136.91468 · doi:10.1080/14697680600699811 [36] Landau, H. G.: Heat conduction in melting solid, Quart appl math 8, 81-94 (1950) · Zbl 0036.13902 [37] Carr P, Faguet D. Fast accurate valuation of American options. Working paper. Cornell University; 1994. [38] Bunch, D. S.; Johnson, H.: The American put option and its critical stock price, J financ 5, 2333-2356 (2000) [39] Van Dyke, M.: Extension of goldstein’s series for the Oseen drag of a sphere, J fluid mech 44, 365-372 (1970) · Zbl 0217.25203 · doi:10.1017/S0022112070001878 [40] Huang, J. Z.; Marti, G. S.; Yu, G. G.: Pricing and hedging American options: a recursive integration method, Rev financ stud 9, 277-300 (1996) [41] Geske, R.; Johnson, H. E.: The American put option valued analytically, J financ 5, 1511-1523 (1984)