Efficient \(L\)-stable method for parabolic problems with application to pricing American options under stochastic volatility. (English) Zbl 1173.91022

The paper proposes an efficient \(L\)-stable numerical method for a semilinear parabolic problem with non-smooth initial data. The method presented here is based on a Padé approximation of the matrix exponential function, which is used to modify the existent exponential time differencing Runge-Kutta schemes for nonlinear parabolic problems. By using splitting techniques, computationally efficient parallel versions of the schemes are constructed. Based on this modified scheme, the author develops and implements an algorithm for solving two problems from financial mathematics: pricing American options under stochastic volatility and the two asset American options pricing problem. The numerical experiments presented here show a good agreement with the results obtained previously in the literature.


91B28 Finance etc. (MSC2000)
Full Text: DOI


[1] N. Clarke, K. Parrott, The multigrid solution of two-factor American put options, Technical Report 96-16, Oxford Computing Laboratory, Oxford, 1996.
[2] Cox, S.M.; Matthews, P.C., Exponential time differencing for stiff systems, Journal of computational physics, 176, 430-455, (2002) · Zbl 1005.65069
[3] Forsyth, P.A.; Vetzal, K.R., Quadratic convergence for valuing American options using a penalty method, SIAM journal on scientific computing, 23, 2095-2122, (2002) · Zbl 1020.91017
[4] E. Gallopoulos, Y. Saad, On the parallel solution of parabolic equations, CSRD Report, 854 (1988), University of Illinois, Urbana, Champaign, USA, Preprint. · Zbl 0676.65098
[5] Golub, G.; Ortega, J.M., Scientific computing, An introduction with parallel computing, (1993), Academic Press San Diego, CA · Zbl 0790.65001
[6] Heston, S., A closed-form solution for options with stochastic volatility with applications to bond and currency options, The review of financial studies, 6, 327-343, (1993) · Zbl 1384.35131
[7] Higham, D.J., An introduction to financial option valuation, (2004), Cambridge University Press · Zbl 1122.91001
[8] S. Ikonen, Efficient numerical solution of Black-Scholes equation by finite difference method, Licentiate Thesis, Department of Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland, 2003.
[9] Kassam, A.K.; Trefethen, L.N., Fourth-order time stepping for stiff pdes, SIAM journal on scientific computing, 26, 4, 1214-1233, (2005) · Zbl 1077.65105
[10] Khaliq, A.Q.M.; Martin, J.; Wade, B.A.; Yousuf, M., Smoothing schemes for reaction – diffusion systems with nonsmooth data, Journal of computational and applied mathematics, 223, 374-386, (2009) · Zbl 1155.65062
[11] Khaliq, A.Q.M.; Twizell, E.H.; Voss, D.A., On parallel algorithms for semidiscretized parabolic partial differential equations based on subdiagonal Padé approximations, Numerical methods for partial differential equations, 9, 107-116, (1993) · Zbl 0768.65059
[12] Khaliq, A.Q.M.; Voss, D.A.; Kazmi, S.H.K., A linearly implicit predictor – corrector scheme for pricing American options using a penalty method approach, Journal of banking and finance, 30, 489-502, (2006)
[13] Khaliq, A.Q.M.; Voss, D.A.; Yousuf, M., Pricing exotic options using L-stable Padé schemes, Journal of banking and finance, 31, 3438-3461, (2007)
[14] Khaliq, A.Q.M.; Wade, B.A.; Yousuf, M.; Vigo-Aguiar, J., Higher order smoothing schemes for inhomogeneous parabolic problems with applications to nonsmooth payoff in option pricing, Numerical methods partial differential equations, 23, 1249-1276, (2007) · Zbl 1130.91338
[15] Lambert, J.D., Numerical methods for ordinary differential systems, (2000), John Wiley & Sons Chichester · Zbl 0745.65049
[16] Livemore, P.W., An implementation of the exponential time differencing scheme to the magnetohydrodynamic equations in a spherical shell, Journal of computational physics, 220, 2, 824-834, (2007) · Zbl 1235.76093
[17] Nielsen, B.; Skavhaug, O.; Tveito, A., Penalty and front-fixing methods for the numerical solution of American option problems, Journal of computational finance, 5, 69-97, (2002)
[18] Oosterlee, C.W., On multigrid for linear complementarity problems with application to American-style options, Electronic transactions on numerical analysis, 15, 165-185, (2003) · Zbl 1031.65072
[19] Pazy, A., Semigroups of linear operators and applications to partial differential equations, Applied mathematics series, vol. 44, (1983), Springer-Verlag Berlin · Zbl 0516.47023
[20] Smith, G.D., Numerical solution of partial differential equations finite difference methods, Oxford applied mathematics and computing science series, (1985), Clarendon Press Oxford
[21] Thomee, V., Galerkin finite element methods for parabolic problems, Series of computational mathematics, vol. 25, (1997), Springer-Verlag Berlin · Zbl 0884.65097
[22] Voss, D.; Khaliq, A.Q.M., Time-stepping algorithms for semidiscretized linear parabolic PDEs based on rational approximations with distinct real poles, Advances in computational mathematics, 6, 353-363, (1996) · Zbl 0872.65091
[23] Wilmott, P., Paul wilmott on quantitative finance, (2000), John Wiley & Sons New York · Zbl 1127.91002
[24] Zvan, R.; Forsyth, P.A.; Vetzal, K.R., Penalty methods for American options with stochastic volatility, Journal of computational and applied mathematics, 91, 199-218, (1998) · Zbl 0945.65005
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.