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Primal-Dual Active Set Method for American Lookback Put Option Pricing

Part of: Partial differential equations Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  07 September 2017

Haiming Song ,
Xiaoshen Wang ,
Kai Zhang  and
Qi Zhang
Haiming Song
Affiliation:
Department of Mathematics, Jilin University Changchun, Jilin, 130012, China
Xiaoshen Wang
Affiliation:
Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, Arkansas, 72204, USA
Kai Zhang*
Affiliation:
Department of Mathematics, Jilin University Changchun, Jilin, 130012, China
Qi Zhang
Affiliation:
School of Science, Shenyang University of Technology, Shenyang, Liaoning, 110870, China
*
*Corresponding author. Email address:kzhang@jlu.edu.cn (K. Zhang)
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Abstract

The pricing model for American lookback options can be characterised as a two-dimensional free boundary problem. The main challenge in this problem is the free boundary, which is also the main concern for financial investors. We use a standard technique to reduce the pricing model to a one-dimensional linear complementarity problem on a bounded domain and obtain a corresponding variational inequality. The inequality is discretised by finite differences and finite elements in the temporal and spatial directions, respectively. By enforcing inequality constraints related to the options using Lagrange multipliers, the discretised variational inequality is reformulated as a set of semi-smooth equations, which are solved by a primal-dual active set method. One of the major advantages of our algorithm is that we can obtain the option values and the free boundary simultaneously, and numerical simulations show that our approach is as efficient as some other methods.

Keywords

American lookback optionlinear complementarity problemvariational inequalityfinite element methodprimal-dual active set method

MSC classification

Secondary: 35A35: Theoretical approximation to solutions 65K10: Optimization and variational techniques 65M12: Stability and convergence of numerical methods 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

Information

Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 3 , August 2017 , pp. 603 - 614
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Babbs, S., Binomial valuation of lookback options, J. Econ. Dynam. Control 24, 1499–1525 (2000).CrossRefGoogle Scholar
[2] Bergounioux, M., Ito, K. and Kunisch, K., Primal-dual strategy for constrained optimal control problem, SIAM J. Control Optim. 37, 1176–1194 (1999).CrossRefGoogle Scholar
[3] Bergounioux, M., Haddou, M., Hintermüller, M. and Kunisch, K., A comparison of a Moreau-Yosida-based active set strategy and interior point methods for constrained optimal control problem, SIAM J. Optimiz. 11, 495–521 (2000).CrossRefGoogle Scholar
[4] Boyle, P.P., Lai, Y. and Tan, K.S., Pricing options using lattice rules, N. Am. Actuar. J. 9, 50–76 (2005).CrossRefGoogle Scholar
[5] Buchen, P. and Konstandatos, O., A new method of pricing lookback options, Math. Finance 15, 245–259 (2005).CrossRefGoogle Scholar
[6] Cox, J.C., Ross, S.A. and Rubinstein, M., Option pricing: A simplified approach, J. Fin. Econ. 7, 229–263 (1979).CrossRefGoogle Scholar
[7] Dai, M. and Kwok, Y.K., American options with lookback payoff, SIAM J. App. Math. 66, 206–227 (2005).CrossRefGoogle Scholar
[8] Duffy, D.J., Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, Wiley (2006).CrossRefGoogle Scholar
[9] Hager, C., Hüeber, S. and Wohlmuth, B., Numerical techniques for the valuation of basket options and their Greeks, J. Comput. Finance 13, 3–33 (2010).CrossRefGoogle Scholar
[10] He, B., A class of projection and contraction methods for monotone variational inequalities, Appl. Math. Opt. 35, 69–76 (1997).CrossRefGoogle Scholar
[11] Hintermüller, M., Ito, K. and Kunisch, K., The primal-dual active set strategy as a semi-smooth Newton method, SIAM J. Optimiz. 13, 865–888 (2003).CrossRefGoogle Scholar
[12] Huang, J. and Pang, J.S., Option pricing and linear complementarity, J. Comput. Finance 2, 31–60 (1998).CrossRefGoogle Scholar
[13] Hüeber, S. and Wohlmuth, B.I., A primal-dual active set strategy for non-linear multibody contact problems, Comput. Method. Appl. Math. 194, 3147–3166 (2005).Google Scholar
[14] Ito, K. and Kunisch, K., Optimal control of elliptic variational inequalities, Appl. Math. Opt. 41, 343–364 (2000).CrossRefGoogle Scholar
[15] Ito, K. and Toivanen, J., Lagrange multiplier approach with optimisedfinite difference stencils for pricing American options under stochastic volatility, SIAM J. Sci. Comput. 31, 2646–2664 (2009).CrossRefGoogle Scholar
[16] Ito, K. and Kunisch, K., Semi-smooth newton methods for variational inequality of the first kind, Math. Model. Numer. Anal. 41, 591–616 (2003).Google Scholar
[17] Jiang, L.S., Mathematical Modeling and Methods of Option Pricing, World Scientific Press, Singapore (2005).CrossRefGoogle Scholar
[18] Jiang, L.S. and Dai, M., Convergence of binomial tree methods for European/American path-dependent options, SIAM J. Numer. Anal. 42, 1094–1109 (2004).CrossRefGoogle Scholar
[19] Kim, K.I., Park, H.S. and Qian, X.S., A mathemathical modeling for the lookback option with jump-diffusion using binomial tree method, J. Comput. Appl. Math. 235, 5140–5154 (2011).CrossRefGoogle Scholar
[20] Kimura, T., American fractional lookback options: valuation and premium decomposition, SIAM J. Appl. Math. 71, 517–539 (2011).CrossRefGoogle Scholar
[21] Kunisch, K. and Rösch, A., Primal-dual active set strategy for general class of contrained control problems, SIAM J. Optimiz. 13, 321–334 (2002).CrossRefGoogle Scholar
[22] Lai, T.L. and Lim, T.W., Exercise regions and efficient valuation of American lookback options, Math. Finance 14, 249–269 (2004).CrossRefGoogle Scholar
[23] Nicholls, D.P. and Sward, A., A discontinuous Galerkin method for pricing American options under the constant elasticity of variance model, Commun. Comput. Phys. 17, 761–778 (2015).CrossRefGoogle Scholar
[24] Song, H. M. and Zhang, R., Projection and contraction method for the valuation of American options, E. Asian J. Appl. Math. 5, 48–60 (2015).CrossRefGoogle Scholar
[25] Song, H.M., Zhang, Q. and Zhang, R., A fast numerical method for the valuation of American lookback put options, Commun. Nonlinear Sci. 27, 302–313 (2015).CrossRefGoogle Scholar
[26] Song, H.M., Zhang, K. and Li, Y.T., Finite Element and Discontinuous Galerkin Methods with Perfect Matched Layers for American Options, Numer. Math. Theory Methods Appl. (to appear).Google Scholar
[27] Sun, Y., Shi, Y. and Gu, X., An integro-differential parabolic variational inequality arising from the valuation of double barrier American option, J. Sys. Sci. Complex. 27, 276–288 (2014).CrossRefGoogle Scholar
[28] Zhang, K., Yang, X.Q. and Teo, K.L., Augmented Lagrangian method applied to American option pricing, Automatica 42, 1407–1416 (2006).CrossRefGoogle Scholar
[29] Zhang, T., Numerical Methods for American Option Pricing, Acta Math. Appl. Sin. 25, 113–122 (2002).Google Scholar
[30] Zhang, T., Zhang, S., Zhu, D., Finite difference approximation for pricing the American lookback option, J. Comput. Math. 27, 484–494 (2009).CrossRefGoogle Scholar
[31] Zhang, R., Song, H.. and Luan, N., A weak Galerkin finite element method for the valuation of American options, Front. Math. China 9, 455–476 (2014).CrossRefGoogle Scholar
[32] Zhang, R., Zhang, Q. and Song, H.M., An efficient finite element method for pricing American multi-asset put options, Commun. Nonlinear Sci. 29, 25–36 (2015).CrossRefGoogle Scholar