Abstract
This paper aims to analyze the convergence rates of the iterative Laplace transform methods for solving the coupled PDEs arising in the regime-switching option pricing. The so-called iterative Laplace transform methods are described as follows. The semi-discretization of the coupled PDEs with respect to the space variable using the finite difference methods (FDMs) gives the coupled ODE systems. The coupled ODE systems are solved by the Laplace transform methods among which an iteration algorithm is used in the computational process. Finally, the numerical contour integral method is used as the Laplace inversion to restore the solutions to the original coupled PDEs from the Laplace space. This Laplace approach is regarded as a better alternative to the traditional time-stepping method. The errors of the approach are caused by the FDM semi-discretization, the iteration algorithm and the Laplace inversion using the numerical contour integral. This paper provides the rigorous error analysis for the iterative Laplace transform methods by proving that the method has a second-order convergence rate in space and exponential-order convergence rate with respect to the number of the quadrature nodes for the Laplace inversion.
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References
Ahna, J., Kang, S., Kwonc, Y.: A Laplace transform finite difference method for the Black–Scholes equation. Math. Comput. Model. 51, 247–255 (2010)
Cai, N., Kou, S.G.: Option pricing under a mixed-exponential jump diffusion model. Manag. Sci. 57, 2067–2081 (2011)
Cai, N., Kou, S.G.: Pricing Asian options under a hyper-exponential jump diffusion model. Oper. Res. 60, 64–77 (2012)
Elliott, R.J., Aggoun, L., Moore, J.B.: Hidden Markov Models: Estimation and Control. Springer, New York (1995)
Horn, R., Johnson, C.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)
In’t Hout, K.J., Weideman, J.A.C.: A contour integral method for the Black–Scholes and Heston equations. SIAM J. Sci. Comput. 33, 763–785 (2011)
Lee, H., Sheen, D.: Laplace transformation method for the Black–Scholes equation. Int. J. Numer. Anal. Model. 6, 642–658 (2009)
Lee, H., Sheen, D.: Numerical approximation of option pricing model under jump diffusion using the Laplace transformation method. Int. J. Numer. Anal. Model. 8, 566–583 (2011)
Lee, H., Lee, J., Sheen, D.: Laplace transform method for parabolic problems with time dependent coefficients. SIAM J. Numer. Anal. 51, 112–125 (2013)
Lee, Y.: Financial options pricing with regime-switching jump-diffusions. Comput. Math. Appl. 68, 392–404 (2014)
Lee, S., Pang, H., Sun, H.: Shift-invert Arnoldi approximation to the Toeplitz matrix exponential. SIAM J. Sci. Comput. 32, 774–792 (2010)
López-Fernández, M., Palencia, C.: On the numerical inversion of the Laplace transform of certain holomorphic mappings. Appl. Numer. Math. 51, 289–303 (2004)
López-Fernández, M., Palencia, C., Schädle, A.: A spectral order method for inverting sectorial Laplace transforms. SIAM J. Numer. Anal. 44, 1332–1350 (2006)
McLean, W., Thomée, V.: Time discretization of an evolution equation via Laplace transformation. IMA J. Numer. Anal. 24, 439–463 (2004)
McLean, W., Sloan, I.H., Thomée, V.: Time discretization via Laplace transformation of an integro-differential equation of parabolic type. Numer. Math. 102, 497–522 (2006)
McLean, W., Thomée, V.: Numerical solution via Laplace transformation of a fractional order evolution equation. J. Integral Equ. Appl. 22, 57–94 (2010)
McLean, W., Thomée, V.: Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional-order evolution equation. IMA J. Numer. Anal. 30, 208–230 (2010)
Pang, H., Sun, H.: Fast numerical contour integral method for fractional diffusion equations. J. Sci. Comput. 66, 41–66 (2016)
Sheen, D., Sloan, I.H., Thomée, V.: A parallel method for time discretization of parabolic problems based on contour integral representation and quadrature. Math. Comput. 69, 177–195 (1999)
Sheen, D., Sloan, I.H., Thomée, V.: A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature. IMA J. Numer. Anal. 23, 269–299 (2003)
Talbot, A.: The accurate numerical inversion of Laplace transforms. J. Inst. Math. Appl. 23, 97–120 (1979)
Tilli, P.: Singular values and eigenvalues of non-Hermitian block Toeplitz matrices. Linear Algebra Appl. 272, 59–89 (1998)
Weideman, J.A.C.: Improved contour integral methods for parabolic PDEs. IMA J. Numer. Anal. 30, 334–350 (2010)
Weideman, J.A.C., Trefethen, L.N.: Parabolic and hyperbolic contours for computing the Bromwich integral. Math. Comput. 76, 1341–1356 (2007)
Yao, D.D., Zhang, Q., Zhou, X.Y.: A Regime-Switching Model for European Options. In: Yan, H.M., Yin, G., Zhang, Q. (eds.) Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems, pp. 281–300. Springer, Berlin (2006)
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The work was supported by National Natural Science Foundation of China (Grant No. 11671323) and Program for New Century Excellent Talents in University (Grant No. NCET-12-0922).
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Ma, J., Zhou, Z. Convergence Analysis of Iterative Laplace Transform Methods for the Coupled PDEs from Regime-Switching Option Pricing. J Sci Comput 75, 1656–1674 (2018). https://doi.org/10.1007/s10915-017-0604-7
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DOI: https://doi.org/10.1007/s10915-017-0604-7
Keywords
- Numerical contour integral
- Convergence rates
- Option pricing
- Regime-switching
- Laplace transform
- Iteration algorithm