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Robust Numerical Scheme for Pricing American Options under Jump Diffusion Models

Authors: Salah Alrabeei, Mohammad Yousuf


The goal of option pricing theory is to help the investors to manage their money, enhance returns and control their financial future by theoretically valuing their options. However, most of the option pricing models have no analytical solution. Furthermore, not all the numerical methods are efficient to solve these models because they have nonsmoothing payoffs or discontinuous derivatives at the exercise price. In this paper, we solve the American option under jump diffusion models by using efficient time-dependent numerical methods. several techniques are integrated to reduced the overcome the computational complexity. Fast Fourier Transform (FFT) algorithm is used as a matrix-vector multiplication solver, which reduces the complexity from O(M2) into O(M logM). Partial fraction decomposition technique is applied to rational approximation schemes to overcome the complexity of inverting polynomial of matrices. The proposed method is easy to implement on serial or parallel versions. Numerical results are presented to prove the accuracy and efficiency of the proposed method.

Keywords: Integral differential equations, American options, jump–diffusion model, rational approximation.

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[1] F. Black and M. Scholes, “The pricing of options and corporate liabilities,” The journal of political economy, pp. 637–654, 1973.
[2] R. C. Merton, “Option pricing when underlying stock returns are discontinuous,” Journal of financial economics, vol. 3, no. 1-2, pp. 125–144, 1976.
[3] S. G. Kou, “A jump-diffusion model for option pricing,” Management science, vol. 48, no. 8, pp. 1086–1101, 2002.
[4] L. Andersen and J. Andreasen, “Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing,” Review of Derivatives Research, vol. 4, no. 3, pp. 231–262, 2000.
[5] K. I. Amin, “Jump diffusion option valuation in discrete time,” The journal of finance, vol. 48, no. 5, pp. 1833–1863, 1993.
[6] S. Hussain and N. Rehman, “Regularity of the american option value function in jump-diffusion model.,” Journal of Computational Analysis and Applications, vol. 22, no. 2, pp. 286–297, 2017.
[7] Y. Chen, W. Wang, and A. Xiao, “An efficient algorithm for options under merton’s jump-diffusion model on nonuniform grids,” Computational Economics, vol. 53, no. 4, pp. 1565–1591, 2019.
[8] X. Gan, Y. Yang, and K. Zhang, “A robust numerical method for pricing american options under kou’s jump-diffusion models based on penalty method,” Journal of Applied Mathematics and Computing, vol. 62, no. 1-2, pp. 1–21, 2020.
[9] D. Tangman, A. Gopaul, and M. Bhuruth, “Exponential time integration and chebychev discretisation schemes for fast pricing of options,” Applied Numerical Mathematics, vol. 58, no. 9, pp. 1309–1319, 2008.
[10] L. Boen and K. J. in’t Hout, “Operator splitting schemes for american options under the two-asset merton jump-diffusion model,” Applied Numerical Mathematics, 2020.
[11] S. Salmi and J. Toivanen, “Imex schemes for pricing options under jump–diffusion models,” Applied Numerical Mathematics, vol. 84, pp. 33–45, 2014.
[12] M. K. Kadalbajoo, L. P. Tripathi, and A. Kumar, “Second order accurate imex methods for option pricing under merton and kou jump-diffusion models,” Journal of Scientific Computing, vol. 65, no. 3, pp. 979–1024, 2015.
[13] R. Mollapourasl, A. Fereshtian, H. Li, and X. Lu, “Rbf-pu method for pricing options under the jump–diffusion model with local volatility,” Journal of Computational and Applied Mathematics, vol. 337, pp. 98–118, 2018.
[14] K. Kazmi, “An imex predictor–corrector method for pricing options under regime-switching jump-diffusion models,” International Journal of Computer Mathematics, vol. 96, no. 6, pp. 1137–1157, 2019.
[15] A. Khaliq, B. Wade, M. Yousuf, and J. Vigo-Aguiar, “High order smoothing schemes for inhomogeneous parabolic problems with applications in option pricing,” Numerical Methods for Partial Differential Equations, vol. 23, no. 5, pp. 1249–1276, 2007.
[16] H. P. Bhatt and A.-Q. M. Khaliq, “Fourth-order compact schemes for the numerical simulation of coupled burgers’ equation,” Computer Physics Communications, vol. 200, pp. 117–138, 2016.
[17] H. P. Bhatt and A. Khaliq, “A compact fourth-order l-stable scheme for reaction–diffusion systems with nonsmooth data,” Journal of Computational and Applied Mathematics, vol. 299, pp. 176–193, 2016.
[18] R. Zvan, P. Forsyth, and K. Vetzal, “Penalty methods for american options with stochastic volatility,” Journal of Computational and Applied Mathematics, vol. 91, no. 2, pp. 199–218, 1998.
[19] Y. dHalluin, P. A. Forsyth, and G. Labahn, “A penalty method for american options with jump diffusion processes,” Numerische Mathematik, vol. 97, no. 2, pp. 321–352, 2004.
[20] L. N. Trefethen, Spectral methods in MATLAB, vol. 10. Siam, 2000.
[21] S. Cox and P. Matthews, “Exponential time differencing for stiff systems,” Journal of Computational Physics, vol. 176, no. 2, pp. 430–455, 2002.
[22] A.-K. Kassam and L. N. Trefethen, “Fourth-order time-stepping for stiff pdes,” SIAM Journal on Scientific Computing, vol. 26, no. 4, pp. 1214–1233, 2005.
[23] A. Khaliq, J. Martin-Vaquero, B. Wade, and M. Yousuf, “Smoothing schemes for reaction-diffusion systems with nonsmooth data,” Journal of Computational and Applied Mathematics, vol. 223, no. 1, pp. 374–386, 2009.
[24] V. Thom´ee, Galerkin finite element methods for parabolic problems, vol. 1054. Springer, 1984.
[25] E. Gallopoulos and Y. Saad, “On the parallel solution of parabolic equations,” in Proceedings of the 3rd international conference on Supercomputing, pp. 17–28, ACM, 1989.
[26] A. Khaliq, E. Twizell, and D. Voss, “On parallel algorithms for semidiscretized parabolic partial differential equations based on subdiagonal pad´e approximations,” Numerical Methods for Partial Differential Equations, vol. 9, no. 2, pp. 107–116, 1993.
[27] M. Yousuf, A. Khaliq, and B. Kleefeld, “The numerical approximation of nonlinear black–scholes model for exotic path-dependent american options with transaction cost,” International Journal of Computer Mathematics, vol. 89, no. 9, pp. 1239–1254, 2012.
[28] C. R. Vogel, Computational methods for inverse problems, vol. 23. Siam, 2002.
[29] A. Almendral and C. W. Oosterlee, “Numerical valuation of options with jumps in the underlying,” Applied Numerical Mathematics, vol. 53, no. 1, pp. 1–18, 2005.