AM  Vol.5 No.8 , May 2014
L-Asymptotic Behavior of the Variational Inequality Related to American Options Problem
ABSTRACT

We study the approximation of variational inequality related to American options problem. A simple proof to asymptotic behavior is also given using the theta time scheme combined with a finite element spatial approximation in uniform norm, which enables us to locate free boundary in practice.



Cite this paper
Benchettah, D. and Haiour, M. (2014) L-Asymptotic Behavior of the Variational Inequality Related to American Options Problem. Applied Mathematics, 5, 1299-1309. doi: 10.4236/am.2014.58122.
References
[1]   Black, F. and Scholes, M. (1973) The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81, 637-654.
http://dx.doi.org/10.1086/260062

[2]   Jaillet, P., Lamberton, D. and Lapeyre, B. (1990) Variational Inequalities and the Pricing of American Options. Acta Applicandae Mathematicae, 213, 263-289.
http://dx.doi.org/10.1007/BF00047211

[3]   Wilmott, P., Dewyne, J. and Howison, S. (1993) Options Pricing: Mathematical Model and Computation. Oxford Financial Press.

[4]   Boulaaras, S. and Haiour, M. (2011) L∞-Asymptotic Bhavior for a Finite Element Approximation in Parabolic QuasiVariational Inequalities Related to Impulse Control Problem. Applied Mathematics and Computation, 217, 6443-6450.
http://dx.doi.org/10.1016/j.amc.2011.01.025

[5]   Boulaaras, S. and Haiour, M. (2013) The Finite Element Approximation in Parabolic Quasi-Variational Inequalities Related to Impulse Control Problem with Mixed Boundary Conditions. Journal of Taibah University for Science, 7, 105-113.
http://dx.doi.org/10.1016/j.jtusci.2013.05.005

[6]   Boulbrachene, M. (2005) Pointwise Error Estimates for a Class of Elliptic Quasi-Variational Inequalities with Nonlinear Source Terms. Applied Mathematics and Computation, 161, 129-138.
http://dx.doi.org/10.1016/j.amc.2003.12.015

[7]   Boulbrachene, M. (2002) Optimal L∞-Error Estimate for Variational Inequalities with Nonlinear Source Terms. Applied Mathematics Letters, 15, 1013-1017.
http://dx.doi.org/10.1016/S0893-9659(02)00078-2

[8]   Lions, J.-L. and Stampacchia, G. (1967) Variational Inequalities. Communications on Pure and Applied Mathematics, 20, 493-519.
http://dx.doi.org/10.1002/cpa.3160200302

[9]   Cortey-Dumont, P. (1985) Sur les inéquations variationnelles a opérateurs non coercifs. Modélisation Mathématique et Analyse Numérique, 19, 195-212.

[10]   Bensoussan, A. and Lions, J.-L. (1978) Applications des Inéquations Variationnelles en Controle Stochastique. Dunod, Paris, France.

[11]   Karatzas, I. (1988) On the Pricing of American Options. Applied Mathematics & Optimization, 17, 37-60.
http://dx.doi.org/10.1007/BF01448358

[12]   Lamberton, D. and Villeneuve, S. (2003) Critical Price near Maturity for an American Option on a Dividend-Paying Stock. The Annals of Applied Probability, 13, 800-815.
http://dx.doi.org/10.1214/aoap/1050689604

[13]   Nystrom, K. (2008) On the Behaviour near Expiry for Multi-Dimensional American Options. Journal of Mathematical Analysis and Applications, 339, 644-654.
http://dx.doi.org/10.1016/j.jmaa.2007.06.068

[14]   Lamberton, D. and Lapeyre, B. (1991) Introduction au calcul stochastique applique a la finance. Ellipses.

[15]   Achdou, Y. and Pironneau, O. (2005) Numerical Procedure for Calibration of Volatility with American Options. Applied Mathematical Finance, 12, 201-241.
http://dx.doi.org/10.1080/1350486042000297252

[16]   Oliver, P. (2004) Numerical Simulation of American Options. Thése Allemande.

[17]   Trabelsi, F. (2011) Asymptotic Behavior of Random Maturity American Options. IAENG International Journal of Applied Mathematics, 41, 112-121.

[18]   Ciarlet, P.G. and Raviart, P.A. (1973) Maximum Principle and Uniform Convergence for the Finite Element Method. Computer Methods in Applied Mechanics and Engineering, 2, 17-31.
http://dx.doi.org/10.1016/0045-7825(73)90019-4

[19]   Quarteroni, A. and Valli, A. (1994) Numerical Approximation of Partial Differential Equations. Springer, Berlin and Heidelberg.

[20]   Blanchet, A., Dolbeault, J. and Monneau, R. (2005) Formulation de monotonie appliquées à des problèmes à frontiére libre et de modélisation en biologie. Thése de Doctorat, Université Paris Dauphine.

 
 
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