American Institute of Mathematical Sciences

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April  2011, 7(2): 435-447. doi: 10.3934/jimo.2011.7.435

Convergence property of an interior penalty approach to pricing American option

Kai Zhang 1, and  Song Wang 2,

1. 

Department of Finance, Business School, Shenzhen University, Nanhai Ave 3688, Shenzhen, Guangdong 518060
2. 

School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009

Received  February 2010 Revised  February 2011 Published  April 2011

This paper establishes a convergence theory for an interior penalty method for a linear complementarity problem governing American option valuation. By introducing an interior penalty term, we first transform the complementarity problem into a nonlinear degenerated Black-Scholes PDE. We then prove that the PDE is uniquely solvable and its solution converges to that of the original complementarity problem. Furthermore, we demonstrate the advantages of the interior penalty method over exterior penalty methods by comparing it with an existing exterior penalty method.
Keywords: Complementarity problem, penalty method., option pricing, variational inequalities.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Kai Zhang, Song Wang. Convergence property of an interior penalty approach to pricing American option. Journal of Industrial & Management Optimization, 2011, 7 (2) : 435-447. doi: 10.3934/jimo.2011.7.435
References:
[1]

A. Bensoussan and J. L. Lions, "Applications of Variational Inequalities in Stochastic Control," North-Holland, Amsterdam-New York-Oxford, 1982.  Google Scholar

[2]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Economy, 81 (1973), 637-659. doi: 10.1086/260062.  Google Scholar

[3]

M. Brennan and E. Schwartz, The valuation of American put options, J. Finance, 32 (1977), 449-462. doi: 10.2307/2326779.  Google Scholar

[4]

J. C. Cox, S. A. Ross and M. Rubinstein, Option pricing: A simplified approach, J. Financial Econom., 7 (1979), 229-263. doi: 10.1016/0304-405X(79)90015-1.  Google Scholar

[5]

G. Dauvaut and L. J. Lions, "Inequalities in Mechanics and Physics," Springer-Verlag, Berlin-Heidelberg-New York, 1976.  Google Scholar

[6]

M. A. H. Dempster, J. P. Hutton and D. G. Richards, LP valuation of exotic American options exploiting structure, J. Comp. Fin., 2 (1998), 61-84. Google Scholar

[7]

E. M. Elliot and J. R. Ockendon, "Weak and Variational Methods for Moving Boundary Problems," Pitman, 1982.  Google Scholar

[8]

P. A. Forsyth and K. R. Vetzal, Quadratic convergence for valuing American options using a penalty method, SIAM J. on Sci. Comput., 23 (2002), 2095-2122. doi: 10.1137/S1064827500382324.  Google Scholar

[9]

R. Glowinski, "Numerical Methods for Nonlinear Variational Problems," Springer-Verlag, New York, 1984.  Google Scholar

[10]

J. Haslinger, M. Miettinen and D. P. Panagiotopoulos, "Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications," Kluwer Academic Publishers, Dordrecht, 1999.  Google Scholar

[11]

J. Huang and J.-S. Pang, Option pricing and linear complementarity, J. Comp. Fin., 2 (1998), 31-60. Google Scholar

[12]

A. Q. M. Khaliq, D. A. Voss and S. H. K. Kazmi, A linearly implicit predictor-corrector scheme for pricing American options using a penalty method approach, J. Banking Finance, 30 (2006), 489-502. doi: 10.1016/j.jbankfin.2005.04.017.  Google Scholar

[13]

B. F. Nielsen, O. Skavhaug and A. Tveito, Penalty and front-fixing methods for the numerical solution of American option problems, J. Comp. Fin., 5 (2001), 69-97. Google Scholar

[14]

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), 3-16. doi: 10.1016/j.cam.2007.10.041.  Google Scholar

[15]

S. Wang, X. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, J. Optim. Theory Appl., 129 (2006), 227-254. doi: 10.1007/s10957-006-9062-3.  Google Scholar

[16]

P. Wilmott, J. Dewynne and S. Howison, "Option Pricing: Mathematical Models and Computation," Oxford Financial Press, Oxford, 1994. Google Scholar

[17]

K. Zhang, S. Wang, X. Q. Yang and K. L. Teo, A power penalty approach to numerical solution of two-factor American option pricing, Numer. Math: TMA, 2 (2009), 202-223.  Google Scholar

[18]

K. Zhang, X. Q. Yang and K. L. Teo, A power penalty approach to American option pricing with jump diffusion processes, JIMO, 4 (2008), 767-782.  Google Scholar

[19]

R. Zvan, P. A. Forsyth and K. R. Vetzal, Penalty methods for American options with stochastic volatility, J. Comput. Appl. Math., 91 (1998), 199-218. doi: 10.1016/S0377-0427(98)00037-5.  Google Scholar

show all references

References:
[1]

A. Bensoussan and J. L. Lions, "Applications of Variational Inequalities in Stochastic Control," North-Holland, Amsterdam-New York-Oxford, 1982.  Google Scholar

[2]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Economy, 81 (1973), 637-659. doi: 10.1086/260062.  Google Scholar

[3]

M. Brennan and E. Schwartz, The valuation of American put options, J. Finance, 32 (1977), 449-462. doi: 10.2307/2326779.  Google Scholar

[4]

J. C. Cox, S. A. Ross and M. Rubinstein, Option pricing: A simplified approach, J. Financial Econom., 7 (1979), 229-263. doi: 10.1016/0304-405X(79)90015-1.  Google Scholar

[5]

G. Dauvaut and L. J. Lions, "Inequalities in Mechanics and Physics," Springer-Verlag, Berlin-Heidelberg-New York, 1976.  Google Scholar

[6]

M. A. H. Dempster, J. P. Hutton and D. G. Richards, LP valuation of exotic American options exploiting structure, J. Comp. Fin., 2 (1998), 61-84. Google Scholar

[7]

E. M. Elliot and J. R. Ockendon, "Weak and Variational Methods for Moving Boundary Problems," Pitman, 1982.  Google Scholar

[8]

P. A. Forsyth and K. R. Vetzal, Quadratic convergence for valuing American options using a penalty method, SIAM J. on Sci. Comput., 23 (2002), 2095-2122. doi: 10.1137/S1064827500382324.  Google Scholar

[9]

R. Glowinski, "Numerical Methods for Nonlinear Variational Problems," Springer-Verlag, New York, 1984.  Google Scholar

[10]

J. Haslinger, M. Miettinen and D. P. Panagiotopoulos, "Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications," Kluwer Academic Publishers, Dordrecht, 1999.  Google Scholar

[11]

J. Huang and J.-S. Pang, Option pricing and linear complementarity, J. Comp. Fin., 2 (1998), 31-60. Google Scholar

[12]

A. Q. M. Khaliq, D. A. Voss and S. H. K. Kazmi, A linearly implicit predictor-corrector scheme for pricing American options using a penalty method approach, J. Banking Finance, 30 (2006), 489-502. doi: 10.1016/j.jbankfin.2005.04.017.  Google Scholar

[13]

B. F. Nielsen, O. Skavhaug and A. Tveito, Penalty and front-fixing methods for the numerical solution of American option problems, J. Comp. Fin., 5 (2001), 69-97. Google Scholar

[14]

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), 3-16. doi: 10.1016/j.cam.2007.10.041.  Google Scholar

[15]

S. Wang, X. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, J. Optim. Theory Appl., 129 (2006), 227-254. doi: 10.1007/s10957-006-9062-3.  Google Scholar

[16]

P. Wilmott, J. Dewynne and S. Howison, "Option Pricing: Mathematical Models and Computation," Oxford Financial Press, Oxford, 1994. Google Scholar

[17]

K. Zhang, S. Wang, X. Q. Yang and K. L. Teo, A power penalty approach to numerical solution of two-factor American option pricing, Numer. Math: TMA, 2 (2009), 202-223.  Google Scholar

[18]

K. Zhang, X. Q. Yang and K. L. Teo, A power penalty approach to American option pricing with jump diffusion processes, JIMO, 4 (2008), 767-782.  Google Scholar

[19]

R. Zvan, P. A. Forsyth and K. R. Vetzal, Penalty methods for American options with stochastic volatility, J. Comput. Appl. Math., 91 (1998), 199-218. doi: 10.1016/S0377-0427(98)00037-5.  Google Scholar

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