Weak Galerkin finite element method for valuation of American options

Abstract

We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one-dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.

Keywords

American option / optimal exercise boundary / weak Galerkin finite element method

References

[1] AllegrettoW, LinY, YangH. Finite element error estimates for a nonlocal problem in American option valuation. SIAM J Numer Anal, 2001, 39: 834-857

[2] AminK, KhannaA. Convergence of American option values from discrete-to continuous-time financial models. Math Finance, 1994, 4: 289-304

[3] BadeaL, WangJ. A new formulation for the valuation of American options, I: Solution uniqueness. In: ParkE-J, LeeJ, eds. Analysis and Scientific Computing. Proceedings of the 19th Daewoo Workshop in Pure Mathematics, Vol 19, Part II, 1999. 2000, 3-16

[4] BadeaL, WangJ. A newformulation for the valuation of American options, II: Solution existence. In: ParkE-J, LeeJ, eds. Analysis and Scientific Computing. Proceedings of the 19th Daewoo Workshop in Pure Mathematics, Vol 19, Part II, 1999. 2000, 17-33

[5] BerengerJ P. A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys, 1994, 114: 185-200

[6] BerengerJ P. Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys, 1996, 127: 363-379

[7] BlackF, ScholesM. The pricing of options and corporate liabilities. J Pol Econ, 1973, 81: 637-659

[8] BrennanM, SchwartzE. The valuation of American put options. J Finance, 1977, 32: 449-462

[9] BrennanM, SchwartzE. Finite difference methods and jump processes arising in the pricing of contingent claims: A synthesis. J Financ Quant Anal, 1978, 13: 461-474

[10] BrunnerH. The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes. Math Comp, 1985, 45: 417-437

[11] BrunnerH. Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge: Cambridge University Press, 2004

[12] CarrP, JarrowR, MyneniR. Alternative characterizations of American put options. Math Finance, 1992, 2: 87-106

[13] CoxJ C, RossS A, RubinsteinM. Option pricing: A simplified approach. J Financ Econ, 1979, 7: 229-263

[14] EvansJ D, KuskeR, KellerJ B. American options on assets with dividends near expiry. Math Finance, 2002, 12: 219-237

[15] HanH, WuX. A fast numerical method for the Black-Scholes equation of American options. SIAM J Numer Anal, 2003, 41: 2081-2095

[16] HolmesA D, YangH. A front-fixing finite element method for the valuation of American options. SIAM J Sci Comput, 2008, 30: 2158-2180

[17] HullJ. Fundamentals of Futures and Options Markets. Upper Saddle River: Prentice Hall, 2007

[18] JailletP, LambertonD, LapeyreB. Variational inequalities and the pricing of American options. Acta Appl Math, 1990, 21: 263-289

[19] JiangL. Mathematical Modeling and Methods of Option Pricing. Singapore: World Scientific Publishing Company, 2005

[20] KimI J. The analytic valuation of American puts. Rev Financ Stud, 1990, 3: 547-572

[21] KwokY K. Mathematical Models of Financial Derivatives. 2nd ed. Berlin: Springer, 2008

[22] LantosN, NatafF. Perfectly matched layers for the heat and advection-diffusion equations. J Comput Phys, 2010, 229: 9042-9052

[23] LinY P, ZhangK, ZouJ. Studies on some perfectly matched layers for one-dimensional time-dependent systems. Adv Comput Math, 2009, 30: 1-35

[24] MaJ, XiangK, JiangY. An integral equation method with high-order collocation implementations for pricing American put options. Int J Econ Finance, 2010, 2: 102-112

[25] MuL, WangJ, WangY, YeX. A computational study of the weak Galerkin method for second-order elliptic equations. Numer Algorithms, 2012,

[26] MuL, WangJ, WeiG, YeX, ZhaoS. Weak Galerkin method for second order elliptic interface problem. J Comput Phys, 2013, 250: 106-125

[27] SchwartzE S. The valuation of warrants: Implementing a new approach. J Financ Econ, 1977, 4: 79-93

[28] WangJ, YeX. A weak Galerkin finite element method for second-order elliptic problems. J Comput Appl Math, 2013, 241: 103-115

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