A fast preconditioned policy iteration method for solving the tempered fractional HJB equation governing American options valuation. (English) Zbl 1372.91115

Summary: A fast preconditioned policy iteration method is proposed for the Hamilton-Jacobi-Bellman (HJB) equation involving tempered fractional order partial derivatives, governing the valuation of American options whose underlying asset follows exponential Lévy processes. An unconditionally stable upwind finite difference scheme with shifted Grünwald approximation is first developed to discretize the established HJB equation under the tempered fractional diffusion models. Next, the policy iteration method as an outer iterative method is utilized to solve the discretized HJB equation and proven to be convergent in finite steps to its numerical solution. Given the Toeplitz-like structure of the coefficient matrix in each policy iteration, the resulting linear system can be fast solved by the Krylov subspace method as an inner iterative method via fast Fourier transform (FFT). Furthermore, a novel preconditioner is proposed to speed up the convergence rate of the inner Krylov subspace iteration with theoretical analysis to ensure the linear system can be solved in \(\mathcal O(N\log N)\) operations under some mild conditions, where \(N\) is the number of spatial node points. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed fast preconditioned policy method.


91G60 Numerical methods (including Monte Carlo methods)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
91G20 Derivative securities (option pricing, hedging, etc.)
60G40 Stopping times; optimal stopping problems; gambling theory


Full Text: DOI


[1] Wilmott, P.; Dewynne, J.; Howison, S., Option pricing: mathematical models and computation, (1993), Oxford Financial Press Oxford · Zbl 0797.60051
[2] Black, F.; Scholes, M., The pricing of options and corporate liabilities, J. Polit. Econ., 81, 3, 637-654, (1973) · Zbl 1092.91524
[3] Cont, R.; Tankov, P., Financial modelling with jump processes, vol. 2, (2003), CRC Press London
[4] Merton, R., Option pricing when underlying stock returns are discontinuous, J. Finan. Econ., 3, 1-2, 125-144, (1976) · Zbl 1131.91344
[5] Kou, S., A jump diffusion model for option pricing, Manage. Sci., 48, 8, 1086-1101, (2002) · Zbl 1216.91039
[6] Carr, P.; Wu, L., The finite moment log stable process and option pricing, J. Financ., 58, 2, 753-777, (2003)
[7] Carr, P.; Geman, H.; Madan, D. B.; Yor, M., The fine structure of asset returns: an empirical investigation, J. Bus., 75, 2, 305-333, (2002)
[8] Boyarchenko, S.; Levendorskii, S., Non-Gaussian Merton-Black-Scholes theory, vol. 9, (2002), World Scientific Singapore · Zbl 0997.91031
[9] Salmi, S.; Toivanen, J., Comparison and survey of finite difference methods for pricing American options under finite activity jump-diffusion models, Int. J. Comput. Math., 89, 9, 1112-1134, (2012) · Zbl 1255.91410
[10] Almendral, A.; Oosterlee, C. W., Accurate evaluation of European and American options under the CGMY process, SIAM J. Sci. Comput., 29, 1, 93-117, (2007) · Zbl 1151.91473
[11] Wang, I.; Wan, J.; Forsyth, P., Robust numerical valuation of European and American options under the CGMY process, J. Comput. Fin., 10, 4, 31, (2007)
[12] Cartea, A.; del-Castillo-Negrete, D., Fractional diffusion models of option prices in markets with jumps, Physica A, 374, 2, 749-763, (2007)
[13] Chen, W.; Wang, S., A penalty method for a fractional order parabolic variational inequality governing American put option valuation, Comput. Math. Appl., 67, 1, 77-90, (2014) · Zbl 1386.91160
[14] Chen, W.; Xu, X.; Zhu, S., A predictor-corrector approach for pricing American options under the finite moment log-stable model, Appl. Numer. Math., 97, 15-29, (2015) · Zbl 1329.91136
[15] S. Lei, W. Wang, X. Chen, D. Ding, A fast preconditioned penalty method for American options pricing under regime-switching tempered fractional diffusion models, submitted for publication. · Zbl 1406.91484
[16] Forsyth, P. A.; Vetzal, K., Quadratic convergence for valuing American options using a penalty method, SIAM J. Sci. Comput., 23, 6, 2095-2122, (2002) · Zbl 1020.91017
[17] Dai, M.; Zhong, Y., Penalty methods for continuous time portfolio selection with transaction costs, J. Comput. Fin., 13, 3, 1-31, (2010) · Zbl 1284.91515
[18] Forsyth, P. A.; Labahn, G., Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance, J. Comput. Fin., 11, 2, 1-44, (2007)
[19] Reisinger, C.; Witte, J. H., On the use of policy iteration as an easy way of pricing American options, SIAM J. Financ. Math., 3, 1, 459-478, (2012) · Zbl 1257.91051
[20] Sabzikar, F.; Meerschaert, M. M.; Chen, J., Tempered fractional calculus, J. Comput. Phys., 293, 14-28, (2015) · Zbl 1349.26017
[21] Zhang, H.; Liu, F.; Turner, I.; Chen, S., The numerical simulation of the tempered fractional black Scholes equation for European double barrier option, Appl. Math. Model., 40, 11, 5819-5834, (2016)
[22] Wang, W.; Chen, X.; Ding, D.; Lei, S., Circulant preconditioning technique for barrier options pricing under fractional diffusion models, Int. J. Comput. Math., 92, 2596-2614, (2015) · Zbl 1337.91130
[23] Meng, Q.; Ding, D.; Sheng, Q., Preconditioned iterative methods for fractional diffusion models in finance, Numer. Methods Partial Differential Equations, 31, 5, 1382-1395, (2015) · Zbl 1329.91141
[24] Podlubny, I., Fractional differential equations, vol. 198, (1999), Academic Press New York
[25] Chen, M.; Deng, W., High order algorithms for the fractional substantial diffusion equation with truncated Lévy flights, SIAM J. Sci. Comput., 37, 2, A890-A917, (2015) · Zbl 1317.65198
[26] Lesmana, D.; Wang, S., An upwind finite difference method for a nonlinear Black-Scholes equation governing European option valuation under transaction costs, Appl. Math. Comput., 219, 16, 8811-8828, (2013) · Zbl 1288.91193
[27] Plemmons, R. J., M-matrix characterizations.I-nonsingular M-matrices, Linear Algebra Appl., 18, 2, 175-188, (1977) · Zbl 0359.15005
[28] Chan, R.; Jin, X., An introduction to iterative Toeplitz solvers, vol. 5, (2007), SIAM Philadelphia
[29] Ng, M., Iterative methods for Toeplitz systems, (2004), Oxford University Press New York · Zbl 1059.65031
[30] Wang, H.; Wang, K.; Sircar, T., A direct \(\mathcal{O}(n \log^2 n)\) finite difference method for fractional diffusion equations, J. Comput. Phys., 229, 21, 8095-8104, (2010) · Zbl 1198.65176
[31] Lei, S.; Sun, H., A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242, 715-725, (2013) · Zbl 1297.65095
[32] S. Lei, D. Fan, X. Chen, Fast solution algorithms for exponentially tempered fractional diffusion equations, (submitted for publication). · Zbl 1407.65112
[33] Saad, Y.; Schultz, M., GMRES: A generalized minimal residual algorithm for solving non-symmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 856-869, (1986) · Zbl 0599.65018
[34] Yousuf, M.; Khaliq, A. Q.M.; Liu, R. H., Pricing American options under multi-state regime switching with an efficient L-stable method, Int. J. Comput. Math., 92, 12, 2530-2550, (2015) · Zbl 1386.91168
[35] Amarala, S.; Wan, J. W., Numerical methods for dynamic bertrand oligopoly and American options under regime switching, Quant. Finance, 16, 11, 1741-1762, (2016) · Zbl 1400.91575
[36] Zheng, M.; Karniadakis, G. E., Numerical methods for SPDEs with tempered stable processes, SIAM J. Sci. Comput., 37, 3, A1197-A1217, (2015) · Zbl 1320.65020
[37] Lin, F.; Yang, S.; Jin, X., Preconditioned iterative methods for fractional diffusion equation, J. Comput. Phys., 256, 109-117, (2014) · Zbl 1349.65314
[38] Pang, H.; Sun, H., Multigrid method for fractional diffusion equations, J. Comput. Phys., 231, 693-703, (2012) · Zbl 1243.65117
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.