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An Operator Splitting Method for Pricing American Options 289 Table 3. Results for different grids with Kou’s model l m error ratio iter time 10 20 0.01050 3.1 0.10 18 40 0.00231 4.5 3.0 0.29 34 80 0.00056 4.1 3.0 0.97 66 160 0.00022 2.6 2.3 2.95 130 320 0.00006 3.7 2.0 10.17 Table 4. Results for different grids with Heston’s model l m n error ratio iter time 10 20 8 0.02576 1.0 0.7 18 40 16 0.00574 4.5 1.3 5.7 34 80 32 0.00420 1.4 2.0 59.4 66 160 64 0.00049 8.5 2.0 487.5 130 320 128 0.00012 4.1 2.0 4373.7 α 1 =3, α 2 =3, p = 1 , and µ =0.1. (25) 3 We have used the same space-time grids as with the Black–Scholes model. Table 3 reports the errors, their ratios and CPU times in milliseconds. The column iter in the table gives the average number of the iterations (20). The stopping criterion for the iterations was that the norm of the residual vector is less than 10 −11 times the norm of the right-hand side vector. 7.3 Heston’s Stochastic Volatility Model In Heston’s model the behavior of the stochastic volatility and its correlation with the value of the asset are described by the parameters α =5, β =0.16, γ =0.9, and ρ =0.1. (26) The values of these parameters are the same as in many previous studies including [CP99, IT07, Oos03, ZFV98]. The computational domain is truncated at X =20andY = 1 like also in [Oos03, IT07], for example. We use the same non-uniform grids as in [IT05] and the parameter w in the discretization of the cross derivative (not discussed in this paper) is chosen using the formula in [IT07]. For the time stepping we use uniform time steps. Table 4 reports the errors, their ratios, the average number of multigrid iterations, and CPU times in milliseconds. The stopping criterion for the multigrid iterations was that the norm of the residual vector is less than 10 −6 times the norm of the right-hand side vector.
290 S. Ikonen and J. Toivanen 8 Conclusions We described an operator splitting method for solving linear complementarity problems (LCPs) resulting from American option pricing problems. We considered it in the case of the Black–Scholes model, Kou’s jump-diffusion model, and Heston’s stochastic volatility model for the value of the underlying asset. The numerical results demonstrated that with all these models the prices can be computed in a few milliseconds on a PC. As future research one could consider the construction of adaptive discretization; see [AP05, LPvST07], for example. Also the robustness and accuracy of discretizations for Heston’s model with higher correlations could be studied. A natural generalization would be to extent the methods for stochastic volatility models including jumps like the ones in [Bat96, DPS00]. References [AA00] L. Andersen and J. Andreasen. Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing. Rev. Deriv. Res., 4:231–262, 2000. [AO05] A. Almendral and C. W. Oosterlee. Numerical valuation of options with jumps in the underlying. Appl. Numer. Math., 53:1–18, 2005. [AP05] Y. Achdou and O. Pironneau. Computational methods for option pricing, volume 30 of Frontiers in Applied Mathematics. SIAM, Philadelphia, PA, 2005. [Bat96] D. S. Bates. Jumps and stochastic volatility: Exchange rate processes implicit Deutsche mark options. Review Financial Stud., 9:69–107, 1996. [BC83] A. Brandt and C. W. Cryer. Multigrid algorithms for the solution of linear complementarity problems arising from free boundary problems. SIAM J. Sci. Statist. Comput., 4:655–684, 1983. [BS73] F. Black and M. Scholes. The pricing of options and corporate liabilities. J. Polit. Econ., 81:637–654, 1973. [BS77] M. J. Brennan and E. S. Schwartz. The valuation of American put options. J. Finance, 32:449–462, 1977. [CP99] N. Clarke and K. Parrott. Multigrid for American option pricing with stochastic volatility. Appl. Math. Finance, 6:177–195, 1999. [Cry71] C. W. Cryer. The solution of a quadratic programming problem using systematic overrelaxation. SIAM J. Control, 9:385–392, 1971. [CT04] R. Cont and P. Tankov. Financial modelling with jump processes. Chapman & Hall/CRC, Boca Raton, FL, 2004. [CV05] R. Cont and E. Voltchkova. A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM J. Numer. Anal., 43:1596–1626, 2005. [dFL04] Y. d’Halluin, P. A. Forsyth, and G. Labahn. A penalty method for American options with jump diffusion processes. Numer. Math., 97:321– 352, 2004.
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Partial Differential Equations
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Partial Differential Equations Mode
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Dedicated to Olivier Pironneau
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VIII Preface computers has been at
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Contents List of Contributors .....
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List of Contributors Yves Achdou UF
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List of Contributors XV Claude Le B
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Discontinuous Galerkin Methods Vive
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Discontinuous Galerkin Methods 5 2
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Discontinuous Galerkin Methods 9 (
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Discontinuous Galerkin Methods 15 W
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Let a h and b h denote the bilinear
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Discontinuous Galerkin Methods 19 t
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Discontinuous Galerkin Methods 21 l
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Table 1. Primal DG for transport Di
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Discontinuous Galerkin Methods 25 [
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Mixed Finite Element Methods on Pol
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Mixed FE Methods on Polyhedral Mesh
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4 Hybridization and Condensation Mi
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is symmetric and positive definite,
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with some coefficient α ∈ R wher
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Domain Decomposition and Electronic
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Computing the Eigenvalues of the La
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