Abstract
In this paper, we consider the fast numerical valuation of European and American options under Merton’s jump-diffusion model, which is given by a partial integro-differential equations. Due to the singularities and discontinuities of the model, the time-space grids are nonuniform with refinement near the strike price and expiry. On such nonuniform grids, the spatial differential operators are discretized by finite difference methods, and time stepping is performed using the discontinuous Galerkin finite element method. Owing to the nonuniform grids, algebraic multigrid method is used for solving the dense algebraical system resulting from the discretization of the integral term associated with jumps in models, which is more challenging. Numerical comparison of algebraic multigrid, the generalized minimal residual method, and the incomplete LU preconditioner shows that algebraic multigrid method is superior to and more effective than the other two methods in solving such dense algebraical system.
Similar content being viewed by others
References
Achdou, Y., & Pironneau, O. (2005). Computational methods for option pricing (Vol. 30)., Frontiers in Applied Mathematics Philadelphia, PA: SIAM.
Almendral, A., & Oosterlee, C. W. (2005). Numerical valuation of options with jumps in the underlying. Applied Numerical Mathematics, 53, 1–18.
Andersen, L., & Andreasen, J. (2000). Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing. Review of Derivatives Research, 4(3), 231–262.
Bank, R., & Falgout, R. (2015). Algebraic multigrid domain and range decomposition (AMG-DD/AMG-RD). SIAM Journal on Scientific Computing, 5(37), S113–S136.
Barles, G., & Soner, H. M. (1998). Option pricing with transaction costs and a nonlinear Black–Scholes equation. Finance and Stochastics, 2(4), 369–397.
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. The Journal of Political Economy, 81, 637–654.
Bramble, J., Pasciak, J., & Xu, J. (1991). The analysis of multigrid algorithm with nonnestedspaces or noninherited quadratic forms. Mathematics of Computation, 56, 1–34.
Briani, M., Chioma, C. L., & Natalini, R. (2004). Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory. Numerische Mathematik, 98, 607–646.
Briani, M., Natalini, R., & Russo, G. (2007). Implicit–explict numerical schemes for jump-diffusion processes. Calcolo, 44, 33–57.
Briggs, W. L., Henson, V. E., & McCormick, S. F. (2000). A multigrid tutorial. Philadelphia, PA: SIAM.
Chan, R. T. L. (2016). Adaptive radial basis function methods for pricing options under jump-diffusion models. Computational Economics, 47, 623–843.
Chow, E., & Patel, A. (2015). Fine-grained parallel incomplete LU factorization. SIAM Journal on Scientific Computing, 37(2), C169–C193.
Cont, R., & Tankov, P. (2004). Financial modelling with jump processes. Boca Raton, FL: Chapman & Hall/CRC.
Cont, R., & Voltchkova, E. (2005). A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM Journal on Numerical Analysis, 43, 1596–1624.
d’ Halluin, Y., Forsyth, P. A., & Labahn, G. (2004). A penalty method for American options with jump diffusion processes. Numerische Mathematik, 97(2), 321–352.
d’ Halluin, Y. P., Forsyth, A., & Vetzal, K. R. (2005). Robust numerical methods for contingent claims under jump diffusion processes. IMA Journal of Numerical Analysis, 25(1), 87–112.
Frey, R., & Patie, P. (2002). Risk management for derivatives in illiquid markets: A simulation study. In K. Sandmann & P. Schönbucher (Eds.), Advances in finance and stochastics (pp. 137–159). Berlin: Springer.
Guo, J. Q., & Wang, W. S. (2014). An unconditonally stable, positivity-preserving splitting scheme for nonlinear Black–Scholes equation with transaction costs. Scientific World Journal, 1–11. ID 525207.
Guo, J. Q., & Wang, W. S. (2015). On the numerical solution of nonlinear option pricing equation in illiquid markets. Computers & Mathematics with Applications, 69, 117–133.
in’t Hout, K. J., & Volders, K. (2009). Stability of central finite difference schemes on non-uniform grids for the Black-Scholes equation. Applied Numerical Mathematics, 59, 2593–2609.
Kou, S. G. (2002). A jump diffusion model for option pricing. Management Science, 48, 1086–1101.
Leland, H. E. (1985). Option pricing and replication with transaction costs. Journal of Finance, 40, 1283–1301.
Lesaint, P., & Raviart, P. A. (1974). On a finite element method for solving the neutron transport equation. In C. de Boor (Ed.), Mathematical aspets of finite elements in partial differential equations (pp. 89–123). New York, NY: Academic Prss.
Liu, J., & Sun, H. W. (2014). A fast high-order sinc-based algorithm for pricing options under jump-diffusion processes. International Journal of Computer Mathematics, 91(10), 2163–2184.
Lyons, T. J. (1995). Uncertain volatility and the risk-free synthesis of derivatives uncertain volatility and the risk-free synthesis of derivatives. Applied Mathematical Finance, 2(2), 117–133.
Matache, A. M., Schwab, C., & Wihler, T. P. (2004). Fast numerial solution Of parabolic integro-differential equations with applications in finance. IMA preprint series 1954, University of Minnesota.
Matache, A. M., von Petersdorff, T., & Schwab, C. (2004). Fast deterministic pricing of options on Lévy driven assets. ESAIM. Mathematical Modelling and Numerical Analysis, 38(1), 37–71.
Merton, R. C. (1973). Theory of rational option pricing. Rand Jouranl of Economics, 4, 141–183.
Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3, 125–144.
Myneni, R. (1992). The pricing of the American option. Annals of Applied Probability, 2(1), 1–23.
Saad, Y. (1996). Iterative methods for sparse linear systems (1st ed.). Boston, MA: PWS.
Salmi, S., & Toivanen, J. (2011). An iterative method for pricing American options under jump-diffusion models. Applied Numerical Mathematics, 61, 821–831.
Sarra, S. A. (2012). A linear system-free Gaussian RBF method for the Gross-Pitaevskii equation on unbounded domains. Numerical Methods for Partial Differential Equations, 28(2), 389–401.
Schotzau, D., & Schwab, C. (2000). Time discretization of parabolic problems by the hp-vertion of discontinuous Galerkin finite element method. SIAM Journal on Numerical Analysis, 38(3), 837–875.
Smith, B., Bjorstad, P., & Gropp, W. (1996). Domain decomposition: Parallel multilevel methods for elliptic partial differential equations. Cambridge: Cambridge University Press.
Spike, L., & Sun, H. W. (2009). Fourth order compact boundary value method for option pricing with jumps. Advances in Applied Mathematics and Mechanics, 1(6), 845–861.
Tavella, D., & Randall, C. (2000). Pricing financial instruments: The finite difference method. Chichester: Wiley.
Toivanen, J. (2008). Numerical valuation of European and American options under Kou’s jump-diffusion model. SIAM Journal on Scientific Computing, 4, 1949–1970.
Wang, W., & Chen, Y. (2017). Fast numerical valuation of options with jump under Merton’s model. Journal of Computational and Applied Mathematics, 318, 79–92.
Werder, T., Gerdes, K., Schötzau, D., & Schwab, C. (2001). \(hp\)-discontinuous Galerkin time stepping for parabolic problems. Computer Methods in Applied Mechanics and Engineering, 190, 6685–6708.
Zecchin, A. C., & Thum, P. (2012). Steady-state behavior of large water distribution systems: Algebraic multigrid method for the fast solution of the linear step. Journal of Water Resources Planning and Management, 138(6), 639–650.
Zhang, X. L. (1997). Numerical analysis of American option pricing in a jump-diffusion model. Mathematics of Operations Research, 22(3), 668–690.
Acknowledgements
The authors would like to thank Editor-in-Chief, Professor Hans Amman, and the four referees for comments and suggestions that led to improvements in the presentation of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the Natural Science Foundation of China (Grant Nos. 11771060, 11671343, 11371074).
Rights and permissions
About this article
Cite this article
Chen, Y., Wang, W. & Xiao, A. An Efficient Algorithm for Options Under Merton’s Jump-Diffusion Model on Nonuniform Grids. Comput Econ 53, 1565–1591 (2019). https://doi.org/10.1007/s10614-018-9823-8
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10614-018-9823-8