Taiwanese Journal of Mathematics

A Method-of-lines Approach for Solving American Option Problems

Min-Sun Horng, Tzyy-Leng Horng, and Chih-Yuan Tien

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The early exercise property of American option changes the original Black-Scholes equation to an inequality that cannot be solved via traditional finite difference method. Therefore, finding the early exercise boundary prior to spatial discretization is a must in each time step. This overhead slows down the computation and the accuracy of solution relies on if the early exercise boundary can be accurately located. A simple numerical method based on finite difference and method of lines is proposed here to overcome this difficulty in American option valuation. Our method averts the otherwise necessary procedure of locating the optimal exercise boundary before applying finite difference discretization. The method is efficient and flexible to all kinds of pay-off. Computations of American put, American call with dividend, American strangle, two-factor American basket put option, and two-factor convertible bond with embedded call and put options are demonstrated to show the efficiency of the current method.

Article information

Taiwanese J. Math., Volume 23, Number 5 (2019), 1253-1270.

Received: 2 April 2018
Revised: 2 October 2018
Accepted: 23 October 2018
First available in Project Euclid: 8 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65M06: Finite difference methods 65M20: Method of lines 91G60: Numerical methods (including Monte Carlo methods) 62P05: Applications to actuarial sciences and financial mathematics 97M30: Financial and insurance mathematics

American option method of lines finite difference method American strangle option two-factor American basket put option callable and putable convertible bond


Horng, Min-Sun; Horng, Tzyy-Leng; Tien, Chih-Yuan. A Method-of-lines Approach for Solving American Option Problems. Taiwanese J. Math. 23 (2019), no. 5, 1253--1270. doi:10.11650/tjm/181010. https://projecteuclid.org/euclid.twjm/1541667765

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  • K. Amin and A. Khanna, Convergence of American option values from discrete- to continuous-time financial models, Math. Finance 4 (1994), no. 4, 289–304.
  • F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ. 81 (1973), no. 3, 637–654.
  • P. Bogacki and L. F. Shampine, A 3(2) pair of Runge-Kutta formulas, Appl. Math. Lett. 2 (1989), no. 4, 321–325.
  • M. J. Brennan and E. S. Schwartz, The valuation of American put options, J. Finance 32 (1977), no. 2, 449–462.
  • ––––, Finite difference methods and jump processes arising in the pricing of contingent claims: a synthesis, J. Financ. Quant. Anal. 13 (1978), no. 3, 461–474.
  • M. Broadie and J. Detemple, American option valuation: new bounds, approximations, and a comparison of existing methods, Rev. Financ. Stud. 9 (1996), no. 4, 1211–1250.
  • P. Carr and D. Faguet, Fast Accurate Valuation of American Options, Cornell University, 1994.
  • C. Chiarella and A. Ziogas, Evaluation of American strangles, J. Econom. Dynam. Control 29 (2005), no. 1-2, 31–62.
  • J. C. Cox, S. A. Ross and M. Rubinstein, Option pricing: A simplified approach, J. Financ. Econ. 7 (1979), no. 3, 229–263.
  • J. de Frutos, A finite element method for two factor convertible bonds, in: Numerical Methods in Finance, 109–128, Springer, 2005.
  • P. W. Duck, D. P. Newton, M. Widdicks and Y. Leung, Enhancing the accuracy of pricing American and Bermudan options, J. Derivative 12 (2005), no. 4, 34–44.
  • G. E. Fasshauer, A. Q. M. Khaliq and D. A. Voss, Using mesh free approximation for multi-asset American option problems, J. Chinese Inst. Engrs. 27 (2004), 563–571.
  • M. C. Fu, Optimization via simulation: A review, Ann. Oper. Res. 53 (1994), no. 1, 199–247.
  • ––––, A tutorial review of techniques for simulation optimization, in: Proceedings of the 1994 Winter Simulation Conference, 149–156, Orlando, Florida, 1994.
  • R. Geske and H. E. Johnson, The American put options valued analytically, J. Finance 39 (1984), no. 5, 1511–1524.
  • R. Geske and K. Shastri, Valuation by approximation: A comparison of alternative option valuation techniques, J. Financ. Quant. Anal. 20 (1985), no. 1, 45–71.
  • D. Goldenberg and R. Schmidt, Estimating the early exercise boundary and pricing American options, working paper, Rensselaer Polytechnic Institute, 1995.
  • P. Jaillet, D. Lamberton and B. Lapeyre, Variational inequalities and the pricing of American options, Acta Appl. Math. 21 (1990), no. 3, 263–289.
  • H. E. Johnson, An analytic approximation for the American put price, J. Financ. Quant. Anal. 18 (1983), no. 1, 141–148.
  • N. Ju and R. Zhong, An approximate formula for pricing American options, J. Derivatives 7 (1999), no. 2, 31–40.
  • O. A. Liskovets, Method of Lines, J. Differ. Equ. 1 (1965), 1308–1317.
  • L. W. MacMillan, An analytical approximation for the American put prices, Advances in Futures and Options Research 1 (1986), 119–139.
  • R. C. Merton, Theory of rational option pricing, Bell J. Econ. Manag. Sci. 4 (1973), no. 1, 141–183.
  • G. H. Meyer and J. van der Hoek, The evaluation of American options with the method of lines, Adv. Fut. Opt. Res. 9 (1997), 265–286.
  • B. F. Nielsen, O. Skavhaug and A. Tveito, Penalty methods for the numerical solution of American multi-asset option problems, J. Comput. Appl. Math. 222 (2008), no. 1, 3–16.
  • E. S. Schwartz, The valuation of warrants: Implementing a new approach, J. Financ. Econ. 4 (1977), no. 1, 79–93.