Efficient numerical pricing of American options based on multiple shooting method: a PDE approach

References

• Black F, Scholes M. The pricing of options and corporate liabilities. J Polit Econ. 1973;81:637–659.
   Web of Science ®Google Scholar
• Jiang L. Mathematical modeling and methods of option pricing. Singapore: Tongji University, World Scientific; 2004.
   Google Scholar
• Wilmott P, Dewynne J, Howison S. Option pricing mathematical models and computation. Oxford: Oxford Financial Press; 1993.
   Google Scholar
• Ballestra LV, Cecere L. Pricing American options under the constant elasticity of variance model: an extension of the method by Barone–Adesi and Whaley. Finance Res Lett. 2015;14:45–55.
   Web of Science ®Google Scholar
• Cheng J, Zhu SP, Liao SJ. An explicit series approximation to the optimal exercise boundary of American put options. Commun Nonlinear Sci Numer Simul. 2010;15:1148–1158.
   Web of Science ®Google Scholar
• Chockalingam A, Muthuraman K. An approximate moving boundary method for American option pricing. Eur J Oper Res. 2015;240:431–438.
   Web of Science ®Google Scholar
• Company R, Egorova VN, Jodar L. Constructing positive reliable numerical solution for American call options: a new front-fixing approach. J Comput Appl Math. 2016;291:422–431.
   Web of Science ®Google Scholar
• Company R, Egorova VN, Jodar L, et al. Finite difference methods for pricing American put option with rationality parameter: numerical analysis and computing. J Comput Appl Math. 2016;304:1–17.
   Web of Science ®Google Scholar
• Pooley DM, Vetzal KR, Forsyth PA. Convergence remedies for non-smooth payoffs in option pricing. J Comput Finance. 2003;6:25–40.
   Google Scholar
• Rad JA, Parand K, Pallestra LV. Pricing European and American options by radial basis points interpolation. Appl Math Comput. 2015;251:363–377.
   Web of Science ®Google Scholar
• Rathish Kumar BV, Mehra M. A wavelet Taylor Galerkin method for parabolic and hyperbolic partial differential equations. Int J Comput Methods. 2005;2(1):75–97.
   Web of Science ®Google Scholar
• Zhang K, Song H, Li J. Front-fixing FEMs for the pricing of American options based on a PML technique. Appl Anal. 2014;94:472–479.
   Web of Science ®Google Scholar
• Zhang R, Zhang Q, Song H. An efficient finite element method for pricing American multi-asset put options. Commun Nonlinear Sci Numer Simul. 2015;29:25–36.
   Web of Science ®Google Scholar
• Nielsen BF, Skavhaug O, Tveito A. Penalty and front-fixing methods for the numerical solution of American option problems. J Comput Finance. 2002;5(4):69–97.
   Google Scholar
• Nielsen BF, Skavhaug O, Tveito A. Penalty methods for the numerical solution of American multi-asset option problems. J Comput Appl Math. 2008;222:3–16.
   Web of Science ®Google Scholar
• Ragusa MA. On weak solutions of ultraparabolic equations. Nonlinear Anal Theory Methods Appl. 2001;47(1):503–511.
   Web of Science ®Google Scholar
• Sun Z, Liu Z, Yang X. On power penalty methods for linear complementarity problems arising from American option pricing. J Glob Optim. 2015;63(1):165–180.
   Web of Science ®Google Scholar
• Yang P, Xu ZL. Numerical valuation of European and American options under Merton's model. Appl Anal. 2021. DOI:10.1080/00036811.2021.2016717
   Web of Science ®Google Scholar
• Zhang K, Wang S. Pricing American bond options using a penalty method. Automatica. 2012;48:472–479.
   Web of Science ®Google Scholar
• Zhang K, Wang S, Yang XQ, et al. A power penalty approach to numerical solutions of two-asset American options. Numer Math Theory Methods Appl. 2009;2:202–223.
   Google Scholar
• Zhou HJ, Yiu KFC, Li LK. Evaluating American put options on zero-coupon bonds by a penalty method. J Comput Appl Math. 2011;235:3921–3931.
   Web of Science ®Google Scholar
• Khaliq AQM, Voss DA, Kazmi K. Adaptive θ-methods for pricing American options. J Comput Appl Math. 2008;222:210–227.
   Web of Science ®Google Scholar
• Persson J, Sydow LV. Pricing American options using a space–time adaptive finite difference method. Math Comput Simul. 2010;80:1922–1935.
   Web of Science ®Google Scholar
• Vaquero JM, Khaliq AQM, Kleefeld B. Stabilized explicit Runge–Kutta methods for multi-asset American options. Comput Math Appl. 2014;67:1293–1308.
   Web of Science ®Google Scholar
• Stoer J, Bulirsch R. Introduction to numerical analysis. New York: Springer; 2002.
   Google Scholar
• Riley JD, Morrison DD, Zancanaro JF. Multiple shooting method for two-point boundary value problems. Commun ACM. 1962;1(5):613–614.
   Google Scholar
• Keller HB. Numerical solution of two point boundary value problems. Philadelphia (PA): SIAM: Society for Industrial and Applied Mathematics; 1976.
   Google Scholar
• Abdi-Mazraeh S, Khani A, Irandoust-Pakchin S. Multiple shooting method for solving Black–Scholes equation. Comput Econ. 2020;56:723–746.
   Web of Science ®Google Scholar
• Duffy DJ. Finite difference methods in financial engineering a partial differential equation approach. Chichester: John Wiley; 2006.
   Google Scholar
• Gander W, Gander MJ, Kwok F. Scientific computing an introduction using maple and MATLAB. New Delhi: Springer International Publishing; 2014.
   Google Scholar
• Ballestra LV, Cecere L. A numerical method to estimate the parameters of the CEV model implied by American option prices: evidence from NYSE. Chaos Solitons Fractals. 2016;88:100–106.
   Web of Science ®Google Scholar
• Chen W, Xu X, Zhu S. A predictor-corrector approach for pricing American options under the finite moment log-stable model. Appl Numer Math. 2015;97:15–29.
   Web of Science ®Google Scholar
• Zvan R, Forsyth PA, Vetzal KR. Penalty methods for American options with stochastic volatility. J Comput Appl Math. 1998;91:199–218.
   Web of Science ®Google Scholar
• Canuto C, Hussaini MY, Quarteroni A, et al. Spectral method in fluid dynamics. Berlin: Springer–Verlag; 1988.
   Google Scholar
• Chebyshev PL. Sur les questions de minima qui se rattachent a la représentation approximative des fonctions. J Comput Appl Math. 1998;91:199–291.
   Google Scholar
• Khater AH, Shamardan AB, Callebaut DK, et al. Chebyshev spectral collocation methods for nonlinear isothermal magnetostatic atmospheres. J Comput Appl Math. 2000;115:309–329.
   Web of Science ®Google Scholar
• Mahson JC, Handscomb DC. Chebyshev polynomials. New York: Chapman and Hall, CRC Press Company; 2003.
   Google Scholar
• Javidi M, Golbabai A. A new domain decomposition algorithm for generalized Burger–Huxley equation based on Chebyshev polynomials and preconditioning. Chaos Solit Fractals. 2009;39:849–857.
   Web of Science ®Google Scholar
• Phillips GM. Interpolation and approximation by polynomials. Berlin, Heidelberg: Springer–Verlag; 2003.
   Google Scholar
• Baltensperger R, Berrut JP. The errors in calculating the pseudospectral differentiation matrices for Chebyshev–Gauss–Lobatto point. Comput Math Appl. 1999;37:41–48.
   Web of Science ®Google Scholar
• Javidi M. Spectral collocation method for the solution of the generalized Burger–Fisher equation. Appl Math Comput. 2006;174:345–352.
   Web of Science ®Google Scholar
• Javidi M. A modified Chebyshev pseudo-spectral DD algorithm for the GBH equation. Comput Math Appl. 2011;62:3366–3377.
   Web of Science ®Google Scholar
• Black K. Polynomial collocation using a domain decomposition solution to parabolic PDE's via the penalty method and explicit/implicit time marching. J Sci Comput. 1992;7:313–338.
   Google Scholar
• Welfert BD. A remark on pseudospectral differentiation matrices. Tempe: Department of Mathematics, Arizona State University; 1992.
   Google Scholar
• Carbonell C, Iturria-Medina Y, Jimenez JC. Multiple shooting-local linearization method for the identification of dynamical systems. Commun Nonlinear Sci Numer Simul. 2016;37:292–304.
   Web of Science ®Google Scholar
• Dueias E, England R, Lopez-Estrada J. Multiple shooting with dichotomically stable formulae for linear boundary-value problems. Comput Math Appl. 1999;38:143–159.
   Web of Science ®Google Scholar
• Zhang W. Improved implementation of multiple shooting for BVPs [MSc thesis]. Computer Science Department, University of Toronto; 2012.
   Google Scholar
• Golbabai A, Ballestra LV, Ahmadian D. Superconvergence of the finite element solutions of the Black–Scholes equation. Finance Res Lett. 2013;10:17–26.
   Web of Science ®Google Scholar
• Chicone C. Ordinary differential equations with applications. Heidelberg: Springer; 2005.
   Google Scholar
• Lambert J. Computational methods in ordinary differential equations. New York: John Wiley; 1973.
   Google Scholar
• Abdi-Mazraeh S, Khani A. An efficient computational algorithm for pricing European, barrier and American options. Comput Appl Math. 2018;37:4856–4876.
   Google Scholar
• Huang SJY. Implementation of general linear methods for stiff ordinary differential equations [PhD thesis]. Auckland University; 2005.
   Google Scholar
• Huang JZ, Subrahmanyam MG, Yu GG. Pricing and hedging American options: a recursive integration method. Rev Financ Stud. 1996;9:277–300.
   Web of Science ®Google Scholar
• Abbas MI, Ragusa MA. On the hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain function. Symmetry. 2021;13:264. DOI:10.3390/sym13020264
   Google Scholar
• Abbas MI, Ragusa MA. Solvability of Langevin equations with two Hadamard fractional derivatives via Mittag–Leffler functions. Appl Anal. 2020;99:1–19.
   Google Scholar
• Akdemir AO, Butt SI, Nadeem M, et al. New general variants of Chebyshev type inequalities via generalized fractional integral operators. Mathematics. 2021;9:122. DOI:10.3390/math9020122
   Web of Science ®Google Scholar
• Baleanu D, Jajarmi A, Mohammadi H, et al. A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative. Chaos Solitons Fractals. 2020;134:Article ID 109705.
   PubMed Web of Science ®Google Scholar
• Matar MM, Abbas MI, Alzabut J, et al. Investigation of the p-Laplacian nonperiodic nonlinear boundary value problem via generalized caputo fractional derivatives. Adv Differ Equ. 2021;2021:68.
   Web of Science ®Google Scholar
• Mohammadi H, Kumar S, Rezapour S, et al. A theoretical study of the Caputo–Fabrizio fractional modeling for hearing loss due to mumps virus with optimal control. Chaos Solit Fractals. 2021;144:Article ID 110668.
   Web of Science ®Google Scholar
• Rezapour S, Mohammadi H, Jajarmi A. A new mathematical model for Zika virus transmission. Adv Differ Equ. 2020;2020:589.
   Web of Science ®Google Scholar