# AN ADAPTIVE MULTIGRID TECHNIQUE FOR OPTION PRICING UNDER THE BLACK-SCHOLES MODEL

• Jeong, Darae (Department of Mathematics, Korea University) ;
• Li, Yibao (Department of Mathematics, Korea University) ;
• Choi, Yongho (Department of Mathematics, Korea University) ;
• Moon, Kyoung-Sook (Department of Mathematics and Information, Gachon University) ;
• Kim, Junseok (Department of Mathematics, Korea University)
• Accepted : 2013.11.05
• Published : 2013.12.25

#### Abstract

In this paper, we consider the adaptive multigrid method for solving the Black-Scholes equation to improve the efficiency of the option pricing. Adaptive meshing is generally regarded as an indispensable tool because of reduction of the computational costs. The Black-Scholes equation is discretized using a Crank-Nicolson scheme on block-structured adaptively refined rectangular meshes. And the resulting discrete equations are solved by a fast solver such as a multigrid method. Numerical simulations are performed to confirm the efficiency of the adaptive multigrid technique. In particular, through the comparison of computational results on adaptively refined mesh and uniform mesh, we show that adaptively refined mesh solver is superior to a standard method.

#### References

1. F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81, (1973), 637-654. https://doi.org/10.1086/260062
2. R. C. Merton, Theory of rational option pricing, The Bell Journal of Economics and Management Science, 4, (1973), 141-183. https://doi.org/10.2307/3003143
3. M. Broadie and J. B. Detemple, Option pricing: valuation models and applications, Management Sciences, 50, (2004), 1145-1177. https://doi.org/10.1287/mnsc.1040.0275
4. H. Han and X. Wu, A Fast numerical method for the Black-Scholes equation of American options, SIAM Journal on Numerical Analysis, 41, (2003), 2081-2095. https://doi.org/10.1137/S0036142901390238
5. G. Cortazar, Simulation and Numerical Methods in Real Options Valuation: Real Options and Investment Under Uncertainty, The MIT Press, Cambridge, 601-620, 2004.
6. J. C. Cox, S. A Ross, and M. Rubinstein, Option pricing: a simplified approach, Journal of Financial and Economics, 7, (1979), 229-263. https://doi.org/10.1016/0304-405X(79)90015-1
7. R. Geske and K. Shastri, Valuation by approximation: a comparison of alternative option valuation techniques, Journal of Financial and Quantitative Analysis, 20, (1985), 45-71. https://doi.org/10.2307/2330677
8. K. Moon and H. Kim, A cost-effective modification of the trinomial method for option pricing, Journal of Korean Society for Industrial and Applied Mathematics, 15, (2011), 1-17.
9. M. J. Brennan and E. S. Schwartz, Finite difference methods and jump processes arising in the pricing of contingent claims: asynthesis, Journal of Financial and Quantitative Analysis, 13, (1978), 461-474. https://doi.org/10.2307/2330152
10. D. Tavella and C. Randall, Pricing Financial Instruments: The Finite Difference Method, JohnWilley & Sons, New York, 2000.
11. M. M. Chawla and D. J. Evans, Numerical volatility in option valuation from Black-Scholes equation by finite differences, International Journal of Computer Mathematics, 81, (2004), 1039-1041. https://doi.org/10.1080/03057920412331272234
12. P. Wilmott, J. Dewynne, and S. Howison, Option Pricing : Mathematical Models and Computation, Oxford Financia Press, Oxford, 1993.
13. D. J. Duffy, Finite Difference Methods in Financial Engineering: a Partial Differential Equation Approach, John Wiley and Sons, New York, 2006.
14. R. Seydel, Tools for Computational Finance, Springer, Berlin, 2012.
15. J. Topper, Financial Engineering with Finite Elements, John Wiley and Sons, New York, 2005.
16. B. A. Wade, A. Q. M. Khaliq, M. Yousuf, J. Vigo-Aguiar, and R. Deininger, On smoothing of the Crank- Nicolson scheme and higher order schemes for pricing barrier options, Journal of Computational and Applied Mathematics, 204, (2007), 144-158. https://doi.org/10.1016/j.cam.2006.04.034
17. A. Q. M. Khaliq, D. A. Voss, and K. Kazmi, Adaptive $\theta$-methods for pricing American options, Journal of Computational and Applied Mathematics, 222, (2008), 210-227. https://doi.org/10.1016/j.cam.2007.10.035
18. D. Jeong, I. S. Wee, and J. Kim, An operator splitting method for pricing the ELS option, Journal of Korean Society for Industrial and Applied Mathematics, 14, (2010), 175-187.
19. Y. Achdou and N. Tchou, Variational analysis for the Black and Scholes equation with stochastic volatility, Mathematical Models and Numerical Analysis, 36, (2002), 373-395. https://doi.org/10.1051/m2an:2002018
20. A. Ern, S. Villeneuve, and A. Zanette, Adaptive finite element methods for local volatility European option pricing, International Journal of Theoretical and Applied Finance, 7, (2004), 659-684. https://doi.org/10.1142/S0219024904002669
21. C. Zhang, Pricing American Options by Adaptive Finite Element Method, Mathematics Department University of Maryland, 2005.
22. Z. Zhu and N. Stokes, A Finite Element Platform for Pricing Path-dependent Exotic Options, CSIRO Mathematical & Information Sciences, Australia, 1998.
23. J. Topper, Option Pricing with Finite Elements, Wilmott Magazine, 84-90, 2005.
24. R. Zvan, P. A. Forsyth, and K. R. Vetzal, A General Finite Element Approach for PDE Option Pricing Models, University of Waterloo, Canada, 1998.
25. P. A. Forsyth and K. R. Vetzal, Quadratic convergence for valuing American options using a penalty method, SIAM Journal on Scientific Computing, 23, (2002), 2095-2122. https://doi.org/10.1137/S1064827500382324
26. R. Zvan, P. A. Forsyth, and K. R. Vetzal, Penalty methods for American options with stochastic volatility, Journal of Computational and Applied Mathematics, 91, (1998), 199-218. https://doi.org/10.1016/S0377-0427(98)00037-5
27. Z. Zhou and H. Wu, Finite element multigrid method for the boundary value problem of fractional advection dispersion equation, Journal of Applied Mathematics, 2013, (2013), 1-8.
28. W. Hackbusch, Multi-grid Methods and Applications, Springer-Verlag, New York, 1985.
29. P. Wesseling, An Introduction to Multigrid Methods, John Wiley and Sons, Chichester, 1995.
30. U. Trottenberg, C. W. Oosterlee, and A. Schuller, Multigrid, Academic press, London, 2000.
31. D. Jeong, J. Kim, and I. S. Wee, An accurate and efficient numerical method for Black-Scholes equations, Communications of the Korean Mathematical Society, 24, (2009), 617-628. https://doi.org/10.4134/CKMS.2009.24.4.617
32. R. Zvan, K. R. Vetzal, and P. A. Forsyth, PDE methods for pricing barrier options, Journal of Economic Dynamics & Control, 24, (2000), 1563-1590. https://doi.org/10.1016/S0165-1889(00)00002-6
33. Y. Achdou and O. Pironneau, Computational Methods for Option Pricing, SIAM, Philadelphia, 2005.
34. O. Pironneau and F. Hecht, Mesh adaption for the Black and Scholes equations, East-West Journal of Numerical Mathematics, bf 8, (2000), 25-36.
35. J. Persson and L. von Persson, Pricing European multi-asset options using a space-time adaptive FD-method, Computing and Visualization in Science, 10, (2007), 173-183. https://doi.org/10.1007/s00791-007-0072-y
36. H. Ji, F. S. Lien, and E. Yee, Parallel adaptive mesh refinement combined with additive multigrid for the efficient solution of the poisson equation, ISRN Applied Mathematics, 2012, (2012), 1-24.
37. A. S. Almgren, J. B. Bell, P. Colella, L. H. Howell, and M. L. Welcome, A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations, Journal of Computational Physics, 142, (1998), 1-46. https://doi.org/10.1006/jcph.1998.5890
38. C. J. Garcia-Cervera and A. M. Roma, Adaptive mesh refinement for micromagnetics simulations, IEEE Transactions on Magnetics, 42, (2006), 1648-1654. https://doi.org/10.1109/TMAG.2006.872199
39. S. Wise, J. Kim, and J. Lowengrub, Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method, Journal of Computational Physics, 226, (2007), 414-446. https://doi.org/10.1016/j.jcp.2007.04.020
40. M. J. Berger and P. Colella, Local adaptive mesh refinement for shock hydrodynamics, Journal of Computational Physics, 82, (1989), 64-84. https://doi.org/10.1016/0021-9991(89)90035-1
41. M. J. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations, Journal of Computational Physics, 53, (1984), 484-512. https://doi.org/10.1016/0021-9991(84)90073-1
42. Applied numerical algorithms group. The chombo framework for block-structured adaptive mesh refinement, Technical report, Lawrence Berkeley National Laboratory, 2005 (Available online at http://seesar.lbl.gov/ANAG/chombo).
43. A. Brandt, Multi-level adaptive solutions to boundary-value problems, Mathematics of Computation, 31, (1977), 333-390. https://doi.org/10.1090/S0025-5718-1977-0431719-X
44. R. Kangro and R. Nicolaides, Far field boundary conditions for Black-Scholes equations, SIAM Journal on Numerical Analysis, 38 (2000), 1357-1368. https://doi.org/10.1137/S0036142999355921
45. C. A. Rendleman, V. E. Beckner, M. Lijewski, W. Crutchfield, and J. B. Bell, Parallelization of structured, hierarchical adaptive mesh refinement algorithms, Computing and Visualization in Science, 3, (2000), 147-157. https://doi.org/10.1007/PL00013544
46. M. Berger and I. Rigoutsos, An algorithm for point clustering and grid generation, Systems, Man and Cybernetics, IEEE Transactions on, 21, (1991), 1278-1286. https://doi.org/10.1109/21.120081
47. D. Bai and A. Brandt, Local mesh refinement multilevel techniques, SIAM Journal on Scientific and Statistical Computing, 8, (1987), 109-134. https://doi.org/10.1137/0908025
48. G. W. Buetow and J. S. Sochacki, The trade-offs between alternative finite difference techniques used to price derivative securities, Applied Mathematics and Computation, 115, (2000), 177-190. https://doi.org/10.1016/S0096-3003(99)00141-1
49. S. Figlewski and B. Gao, The adaptive mesh model: a new approach to efficient option pricing, Journal of Financial Economics, 53, (1999), 313-351. https://doi.org/10.1016/S0304-405X(99)00024-0
50. D. Jeong, Mathematical model and numerical simulation in computational finance, Ph.D. Thesis, Department of Mathematics, Korea University, Korea, December, 2012.