Abstract
Fractional Black-Scholes (FBS) equation has been widely applied in options pricing problems. However, most of the literatures focus on pricing the single-asset option using the FBS model, and research on time-space FBS with multiple assets has remained vacant. Option pricings are always governed by multiple assets. Under the assumption that fluctuation of asset price is regarded as a fractal transmission system and follows two independent geometric Lévy processes, 2D time-space fractional Black-Scholes equation (TDTSFBSE) is proposed to describe the instantaneous price in the financial market. In this work, we discrete the TDTSFBSE with implicit finite difference and present a fast parallel all-at-once iterative method for the resulting linear system. The proposed method can greatly reduce the storage requirements and computational cost. Theoretically, we discuss the convergence property of the fast iterative method. Numerical examples are given to illustrate the effectiveness and efficiency of our method.
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References
Bai, Z.-Z., Lu, K.-Y.: Fast matrix splitting preconditioners for higher dimensional spatial fractional diffusion equations. J. Comput. Phys. 404, 109117 (2020)
Bai, Z.-Z., Lu, K.-Y., Pan, J.-Y. : Diagonal and Toeplitz splitting iteration methods for diagonal-plus-Toeplitz linear systems from spatial fractional diffusion equations. Numer Linear Algber. Appl 24(4), 1–15 (2017)
Boyarchenko, S., Levendorskiǐ, S.: Non-Gaussian Merton-Black-Scholes Theory, vol. 9. World Scientific, Singapore (2002)
Carr, P., Wu, L.: The finite moment log stable process and option pricing. J. Finance LVIII(2), 753–777 (2003)
Carr, P., Geman, H., Madan, D.B., Yor, M.: Stochastic volatility for levý processes. Math. Finance 13, 345–382 (2003)
Cartea, A., del Castillo-Negrete, D.: Fractional diffusion models of option prices in markets with jumps. Physica A 374(2), 749–763 (2007)
Chan, R., Jin, X.-Q.: An Introduction to Iterative Toeplitz Solvers. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2007)
Chen, W., Wang, S.: A 2nd-order FDM for a 2D fractional Black-Scholes equation. In: Dimov, I., Faragó, I., Vulkov, L. (eds.) Numerical Analysis and Its Applications. NAA 2016. Lecture Notes in Computer Science, vol. 10187. https://doi.org/10.1007/978-3-319-57099-0_5. Springer, Cham (2017)
Chen, W., Wang, S.: A 2nd-order ADI finite difference method for a 2D fractional Black-Scholes equation governing European two asset option pricing. Math. Comput. Simul. 79, 440–456 (2020)
Chen, X., Ding, D., Lei, S.-L., Wang, W.: A fast preconditioned iterative method for two-dimensional options pricing under fractional differential models. Comput. Math. Appl. 171(1), 279–293 (2020)
Chen, X., Ding, D., Lei, S.-L., Wang, W.: An implicit-explicit preconditioned direct method for pricing options under regime-switching tempered fractional partial differential models. Numer Algor. https://doi.org/10.1007/s11075-020-00994-7 (2020)
Golbabai, A, Nikan, O., Nikazad, T.: Numerical analysis of time fractional Black-Scholes European option pricing model arising in financial market. Comput. Appl. Math. 1, 177–183 (2019)
Jumarie, G.: Derivation and solutions of some fractional Black -Scholes equations in coarse-grained space and time. Application to Mertons optimal portfolio. Comput. Math. Appl. 3(59), 1142–1164 (2010)
Koponen, I.: Analytic approach to the problem of convergence of truncated Levy flights towards the gaussian stochastic process. Phys. Rev. E 52, 1197–1199 (1995)
Liang, J.-R., Wang, J., Zhang, W.-J.: The solutions to a bi-fractional Black-Scholes-Merton differential equation. Int. J. Pure Appl. Math. 58 (1), 99–112 (2010)
Lin, X.-L., Ng, M.: An all-at-once preconditioner for evolutionary partial differential equations. arXiv:https://arxiv.org/abs/2002.01108
Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)
McDonald, E., Pestana, J., Wathen, A.: Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations. SIAM J. Sci. Comput. 40(2), 1012–1033 (2018)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Prathumwan, D., Trachoo, K.: On the solution of two-dimensional fractional Black-Scholes equation for European put option. Adv. Differ. Equ. 146. https://doi.org/10.1186/s13662-020-02554-8 (2020)
Song, L.: A space-time fractional derivative model for European option pricing with transaction costs in fractal market. Chaos Soliton Fract. 103, 123–130 (2017)
Strang, G.: A proposal for Toeplitz matrix calculations. Stud. Appl. Math. 74, 171–176 (1986)
Wyss, W.: The fractional Black-Scholes equation. Fract. Calc. Appl. Anal. 3(1), 51–61 (2000)
Zhang, H., Liu, F., Turner, I., Yang, Q.: Numerical solution of the time fractional Black-Scholes model governing European options. Comput. Math. Appl. 71(1), 1772–1783 (2016)
Zhang, H., Liu, F., Chen, S., Anh, V., Chen, J.: Fast numerical simulation of a new time-space fractional option pricing model governing European call option. Appl. Math. Comput. 339(3), 186–198 (2018)
Zhao, H., Tian, H.: Finite difference methods of the spatial fractional Black-Schloes equation for a European call option. IMA. J. Appl. Math. 82, 836–848 (2017)
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This work was supported by the National Natural Science Foundation of China (11771193).
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Zhang, M., Zhang, GF. Fast solution method and simulation for the 2D time-space fractional Black-Scholes equation governing European two-asset option pricing. Numer Algor 91, 1559–1575 (2022). https://doi.org/10.1007/s11075-022-01314-x
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DOI: https://doi.org/10.1007/s11075-022-01314-x
Keywords
- 2D time-space fractional Black-Scholes equation
- European two-asset option pricing
- All-at-once preconditioners
- Toeplitz matrix
- Low rank approximation