Fixed domain transformations and split-step finite difference schemes for nonlinear Black-Scholes equations for American options

Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing

Volumn , Issue , 2008, Pages 243-272

  Ankudinova, Julia ^a   Ehrhardt, Matthias ^b  

^a TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

^b LEIBNIZ INSTITUT FÜR KATALYSE E V AN DER UNIVERSITÄT ROSTOCK   (Germany)

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EID: 84860400979     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter

References (68)

1. G. Alobaidi, American options and their strategies, Ph.D. thesis, University of Western Ontario, London, Canada, 2000.
2. J. Ankudinova and M. Ehrhardt, The numerical solution of nonlinear Black-Scholes equations, Comput. Math. Appl. 56 (2008), 799-812.
3. M. Avellaneda and A. Parás, Dynamic hedging portfolios for derivative securities in the presence of large transaction costs, Appl. Math. Finance 1 (1994), 165-193.
4. G. Barles and H.M. Soner, Option pricing with transaction costs and a nonlinear Black-Scholes equation, Finance Stoch. 2 (1998), 369-397.
5. G. Barone-Adesi and R. Whaley, Efficient Analytic Approximation of American Option Values, J. Finance 42 (1987), 302-320.
6. F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ. 81 (1973), 637-659.
7. L. Bordag and A. Chmakova, Explicit solutions for a nonlinear model of financial derivatives, Int. J. Theor. Appl. Finance 10 (2007), 1-21.
8. P. Boyle and T. Vorst, Option Replication in Discrete Time with Transaction Costs, J. Finance 47 (1992), 271-293.
9. M. Brennan and E. Schwartz, The valuation of American put options, J. Finance 32 (1977), 449-462.
10. M. Broadie and J. Detemple, American Option valuation: New bounds, approximations, and a comparison of existing methods, Rev. Fin. Stud. 9 (1996), 1211-1250.
11. P. Carr, R. Jarrow and R. Myneni, Alternative Characterizations of American Put Options, Math. Finance 2 (1992), 87-105.
12. A. Chmakova, Symmetriereduktionen und explizite Lösungen für ein nichtlineares Modell eines Preisbildungsprozesses in illiquiden Märkten, Dissertation, Technische Universität Cottbus, 2005 (in german).
13. G. Courtadon, AMore Accurate Finite Difference Approximation for the Valuation of Options, J. Fin. Quant. Anal. 15 (1982), 697-703.
14. J. Cox and S. Ross and M. Rubinstein, Option pricing: A simplified approach, J. Fin. Econ. 7 (1979), 229-263.
15. E. Cryer, The solution of a quadratic programming problem using systematic overrelaxation, SIAM J. Control 9 (1971), 385-392.
16. B. Dupire, Pricing with a Smile, Risk 7 (1994), 18-20.
17. M. Ehrhardt, Discrete Transparent Boundary Conditions for Parabolic Equations, Z. Angew. Math. Mech. 77 (1997) S2, 543-544.
18. M. Ehrhardt, Discrete Artificial Boundary Conditions, Ph.D. Thesis, Technische Universit ät Berlin, 2001.
19. M. Ehrhardt and R.E. Mickens, A fast, stable and accurate numerical method for the Black-Scholes equation of American options, Int. J. Theoret. Appl. Finance 11 (2008), 471-501.
20. L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.
21. G. Ferreyra, The Mathematics Behind the 1997 Nobel Prize in Economics, e-MATH, What's New in Mathematics, (electr. magazine of the Amer. Math. Soc.), December 1997.
22. P.A. Forsyth and K.R. Vetzal, Quadratic Convergence for Valuing American Options using a Penalty Method, SIAM J. Sci. Comput. 23 (2002), 2095-2122.
23. R. Frey and A. Stremme,Market Volatility and Feedback Effects from Dynamic Hedging, Math. Finance 4 (1997), 351-374.
24. R. Frey and P. Patie, Risk Management for Derivatives in Illiquid Markets: A Simulation Study, Springer, Berlin, 2002.
25. R. Geske and R. Roll, On valuing American call options with the Black-Scholes European formula, J. Finance 89 (1984), 443-455.
26. M. Günther and A. Jüngel, Finanzderivate mit MATLAB, Vieweg, 2003 (in german).
27. H. Han and X.Wu, A fast numerical method for the Black-Scholes equation of American options, SIAM J. Numer. Anal. 41 (2003), 2081-2095.
28. S. Heston, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Rev. Fin. Studies 6 (1993), 327-343.
29. S. Hodges and A. Neuberger, Optimal replication of contingent claims under transaction costs, Review of Futures Markets 8 (1989), 222-239.
30. T. Hoggard and A. E. Whalley and P. Wilmott, Hedging option portfolios in the presence of transaction costs, Adv. Fut. Opt. Res. 7 (1994), 21-35.
31. J.-Z. Huang, M.G. Subrahmanyam and G.G. Yu, Pricing and Hedging American Options: A Recursive Integration Method, Rev. Fin. Studies 9 (1996), 277-300.
32. J. Hull and A. White, The Pricing of Options on Assets with Stochastic Volatilities, J. Finance 42 (1987), 281-300.
33. J. Hull, Options, Futures and Other Derivatives, 5th Edition, Prentice Hall, 2002.
34. P. Jäckel, Monte Carlo methods in finance, John Wiley & Sons, Inc., London, 2002.
35. M. Jandačka and D. Šev̌covič, On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile, J. Appl. Math. 3 (2005), 235-258.
36. L. Jódar, P. Sevilla-Peris, J.C. Cortés, and R. Sala, A new direct method for solving the Black-Scholes equation, Appl. Math. Lett. 18 (2005), 29-32.
37. Y. Kabanov and M. Safarian, On Leland's Strategy of Option Pricing with Transactions Costs, Finance Stoch. 1 (1997), 239-250.
38. T. Kim and Y. Kwon, Finite-Difference Bisection Algorithms for free boundaries of American Call and Put Options, Preprint Pohang University of Science and Technology, 2002.
39. M. Kratka, No Mystery Behind the Smile, Risk 9 (1998), 67-71.
40. Y.-K. Kwok, Mathematical Models of Financial Derivatives, Springer Finance, 1998.
41. H.E. Leland, Option pricing and replication with transactions costs, J. Finance 40 (1985), 1283-1301.
42. L. MacMillan, Analytic Approximation for the American Put Option, Adv. Fut. Opt. Res. 1 (1986), 119-139.
43. B.J. Mccartin and S.M. Labadie, Accurate and efficient pricing of vanilla stock options via the Crandall-Douglas Scheme, Appl. Math. Comp. 143 (2003), 39-60.
44. R.C. Merton, Theory of rational option pricing, Bell J. Econ. Manag. Sci. 4 (1973), 141-183.
45. G. Meyer and J. van der Hoek, The valuation of American options with the method of lines, Adv. Fut. Opt. Res. 9 (1997), 265-286.
46. B.F. Nielsen, O. Skavhaug and A. Tveito, Penalty and front-fixing methods for the numerical solution of American option problems, J. Comp. Finance 5 (2002), 69-97.
47. R. Panini and R. P. Srivastav, Option Pricing with Mellin Transforms, Math. Comput. Modelling 40 (2004), 43-56.
48. O. Pauly, Numerical simulation of American Options, Diploma thesis, Universität Ulm, 2004.
49. J. Persson, Accurate Finite Difference Methods for Option Pricing, Dissertation, Uppsala University, 2006.
50. P. Schönbucher and P. Wilmott, The feedback-effect of hedging in illiquid markets, SIAM J. Appl.Math. 61 (2000), 232-272.
51. D. Šev̌covič, Analysis of the free boundary for the pricing of an American call option, Euro J. Appl. Math 12 (2001), 25-37.
52. D. Šev̌covič, An iterative algorithm for evaluating approximations to the optimal exercise boundary for a nonlinear Black-Scholes equation, Canad. Appl. Math. Quarterly 15 (2007), 77-79.
53. R. Seydel, Tools for Computational Finance, Second ed., Springer, 2004.
54. H.M. Soner, S.E. Shreve and J. Cvitanič, There is No Nontrivial Hedging Portfolio for Option Pricing with Transaction Costs, Ann. Appl. Probab. 5 (1995), 327-355.
55. H.M. Soner and N. Touzi, Superreplication Under Gamma Constraints, SIAM J. Contr. Optim. 39 (2001), 73-96.
56. G. Strang, On the Construction and Comparison of Difference Schemes, SIAM J. Numer. Anal. 5 (1968), 506-517.
57. J.C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Wadsworth & Brooks/Cole, 1989.
58. H. Sun and J. Zhang, A high order compact boundary value method for solving one dimensional heat equations, Tech. Rep. No. 333-02, University of Kentucky, 2002.
59. D.Y. Tangman, A. Gopaul and M. Bhuruth, Numerical pricing of options using high- order compact finite difference schemes, in press: J. Comput. Appl. Math. (2007).
60. D.Y. Tangman, A. Gopaul and M. Bhuruth, A Fast High-Order Finite Difference Algorithm for Pricing American Options, submitted to: J. Comput. Appl. Math.
61. D. Tavella and C. Randall, Pricing financial instruments: The finite difference method, John Wiley & Sons, 2000.
62. P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford, Financial Press, 1993.
63. P. Wilmott, S. Howison, and J. Dewynne, The Mathematics of Financial Derivatives, A Student Introduction, Cambridge University Press, 2002.
64. L.Wu and Y. Kwok, A front-fixing finite difference method for the valuation of American options, J. Fin. Engin. 6 (1997), 83-97.
65. X. Wu and Z.-Z. Sun, Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions, Appl. Numer. Math. 50 (2004), 261-277.
66. J. Zhao, M. Davison and R.M. Corless, Compact finite difference method for American option pricing, J. Comput. Appl. Math. 206 (2007), 306-321.
67. Y. Zhu, X. Wu and I. Chern, Derivative securities and difference methods, Springer Finance, New York, 2004.
68. R. Zvan, P. Forsyth and K. Vetzal, A penalty method for American options with stochastic volatility, J. Comp. Appl. Math. 91 (1998), 199-218.