Abstract
In this paper, we consider the partial differential equations approach for valuing American style options on multiple assets. We develop and analyze a penalty method to solve the parabolic variational inequality that characterizes the American style option. Such a method is obtained by adding a non-linear penalty term to the Black–Scholes equation. This approach gives a fixed solution domain, removing the free and moving boundary imposed by the early exercise feature of the contract. The proposed penalty method is based on the reformulation of the original inequality in the form of a variational inequality without constraints, which includes a non-differentiable functional that has a different form for convex and non-convex payoff functions. The penalty operator is defined as the gradient of a differentiable approximation of this functional. It is shown that various well-known penalty operators can be obtained in a similar way. Under realistic assumptions about the regularity of the payoff function, error estimates of a penalty method in uniform and energy norm are established.
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REFERENCES
A. Bensoussan and J.-L. Lions, Applications of Variational Inequalities in Stochastic Control (North-Holland, Amsterdam, 1982).
A. Bensoussan, ‘‘On the theory of option pricing,’’ Acta Appl. Math. 2, 139–158 (1984).
I. Karatzas, ‘‘On the pricing of american options,’’ Appl. Math. Optim. 17, 37–60 (1988).
P. Jaillet, D. Lamberton, and B. Laperyre, ‘‘Variational inequalities and the pricing of american options,’’ Acta Appl. Math. 21, 263–289 (1990).
A. Friedman, Variational Principles and Free Boundary Problems (Wiley-Interscience, New York, 1990).
R. Glowinski, J.-L. Lions, and R. Tremolieres, Numerical Analysis of Variational Inequalities (North-Holland, Amsterdam, 1981).
M. Marcozzi, ‘‘On the approximation of optimal stop** problems with application to financial mathematics,’’ SIAM J. Sci. Comput. 22, 1865–1884 (2001).
P. A. Forsyth and K. R. Vetzal, ‘‘Quadratic convergence for valuing American options using a penalty method,’’ SIAM J. Sci. Comput. 23, 2095–2122 (2002).
Y. d’Halluin, P. A. Forsyth, and G. Labah, ‘‘A penalty method for American options with jump-diffusion processes,’’ Numer. Math. 97, 321–352 (2004).
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, 227–254 (2006).
A. Q. 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, 489–502 (2006).
P. Kovalov, V. Linetsky, and M. Marcozzi, ‘‘Pricing multi-asset American options: A finite element method-of-lines with smooth penalty,’’ J. Sci. Comput. 33, 209–237 (2007).
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, 3–16 (2008).
R. Zhang, Q. Zhang, and H. Song, ‘‘An efficient finite element method for pricing American multi-asset put options,’’ Commun. Nonlin. Sci. Numer. Simul. 29, 25–36 (2015).
Q. Zhang, H. Song, C. Yang, and F. Wu, ‘‘An efficient numerical method for the valuation of American multi-asset options,’’ Comput. Appl. Math. 39, 240 (2020).
R. Z. Dautov, ‘‘Penalty methods for one-sided parabolic problems with piecewise smooth obstacles,’’ Lobachevskii J. Math. 42, 1643–1651 (2021).
R. Z. Dautov and A. V. Lapin, ‘‘Three new weak formulations of the problem of American call options valuation,’’ J. Phys.: Conf. Ser. 1158, 022033 (2019).
R. Z. Dautov and A. V. Lapin, ‘‘Approximations of evolutionary inequality with Lipschitz-continuous functional and minimally regular input data,’’ Lobachevskii J. Math. 40, 425–438 (2019).
R. Scholz, ‘‘Numerical solution of the obstacle problem by the penalty method. Part II. Time-dependent problems,’’ Numer. Math. 49, 255–268 (1986).
M. Boman, A Posteriori Error Analysis in the Maximum Norm for a Penalty Finite Element Method for the Time-Dependent Obstacle Problem (Finite Element Center, Chalmers, 2000).
R. Z. Dautov and A. I. Mikheeva, ‘‘Exact penalty operators and regularization of parabolic variational inequalities with in obstacle inside a domain,’’ Differ. Equat. 44, 75–81 (2008).
P. Jaillet, D. Lamberton, and B. Laperyre, ‘‘Variational inequalities and the pricing of american options,’’ Acta Appl. Math. 21, 263–289 (1990).
P. Wilmott, Derivatives, the Theory and Practice of Financial Engineering (Wiley, Chichester, 1998).
L. Jiang, Mathematical Modeling and Methods of Option Pricing (Tongji Univ., China, 2005).
Y. Achdou and O. Pironneau, Computational Methods for Option Pricing (SIAM, Philadelphia, 2005).
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology (Springer, Berlin, 1992), Vol. 5.
T. Roubiček, Nonlinear Partial Differential Equations with Applications (Birkhäuser, Basel, 2005).
L. C. Evans, Partial Differential Equations (AMS, Providence, RI, 1998).
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications (Soc. Ind. Appl. Math., Philadelphia, 1980).
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This paper has been supported by the Kazan Federal University Strategic Academic Leadership Program (PRIORITY-2030).
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Dautov, R.Z. A Penalty Method for American Multi-Asset Option Problems. Lobachevskii J Math 44, 269–281 (2023). https://doi.org/10.1134/S1995080223010092
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DOI: https://doi.org/10.1134/S1995080223010092