Numerical valuation of European and American options under Kou's jump-diffusion model

SIAM Journal on Scientific Computing

Volumn 30, Issue 4, 2007, Pages 1949-1970

  Toivanen, Jari ^a  

^a Stanford University   (United States)

Author keywords

Finite difference method; Jump diffusion model; Linear complementarity problem; Operator splitting method; Option pricing; Partial integro differential equation; Penalty method

Indexed keywords

BROWNIAN MOVEMENT; CONSTRAINED OPTIMIZATION; COSTS; DIFFUSION; ECONOMICS; FINANCIAL MARKETS; INTEGRODIFFERENTIAL EQUATIONS; ITERATIVE METHODS; LINEAR SYSTEMS; MATHEMATICAL OPERATORS; NUMERICAL METHODS;

AMERICAN OPTIONS; EUROPEAN OPTION; FINITE-DIFFERENCE METHODS; JUMP DIFFUSION MODELS; LINEAR COMPLEMENTARITY PROBLEMS; OPERATOR SPLITTING METHOD; OPTIONS PRICING; PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS; PENALTY METHODS; SPATIAL DIFFERENTIAL OPERATORS;

FINITE DIFFERENCE METHOD;

EID: 55549112874     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/060674697     Document Type: Article

References (41)

1. Y. ACHDOU AND O. PLRONNEAU, Computational Methods for Option. Pricing, Frontiers Appl. Math. 30, S1AM, Philadelphia, PA, 2005.
2. A. ALMENDRAL AND C. W. OOSTERLEE, Numerical valuation of options with jumps in the underlying, Appl. Numer. Math., 53 (2005), pp. 1-18.
3. A. ALMENDRAL AND C. W. OOSTERLEE, Accurate evaluation of European and American options under the CGMY process, SIAM J. Sci. Comput., 29 (2007), pp. 93-117.
4. L. ANDERSEN AND J. ANDREASEN, Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing, Rev. Deriv. Res., 4 (2000), pp. 231-262.
5. H. T. BANKS, V. A. BOKIL. D. CIORANESCU, N. L. GIBSON, G. GRISO, and B. MIARA, Homogenization of periodically varying coefficients in electromagnetic materials, J. Sci. Comput., 28 (2006), pp. 191-221.
6. D. S. BATES, Jumps and stochastic volatility: Exchange rate processes implicit Deutsche mark options. Rev. Financial Stud., 9 (1996), pp. 69-107.
7. F. BLACK AND M. SCHOLES, The pricing of options and corporate liabilities, J. Political Economy, 81 (1973), pp. 637-654.
8. M. BRIANI, C. LA CHIOMA, AND R. NATALINI, Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory, Numer. Math., 98 (2004), pp. 607-646.
9. P. CARR. H. GEMAN. D. B. MADAN, AND M. YOR, The fine structure of asset returns: An empirical investigation, J. Business, 75 (2002), pp. 305-332.
10. N. CLARKE AND K. PARROTT, Multigrid for American option pricing with stochastic volatility, Appl. Math. Finance, 6 (1999), pp. 177-195.
11. R. CONT AND P. TANKOV, Financial Modelling with Jump Processes, Chapman & Hall/CRC, Boca Raton, FL, 2004.
12. R. CONT AND E. VOLTCHKOVA, A finite difference scheme for option pricing in jump diffusion and exponential Lévy models, SIAM Numer. Anal., 43 (2005), pp. 1596-1626.
13. Y. D'HALLUIN, P. A. FORSYTH, AND G. LABAHN, A penalty method for American options with jump diffusion processes, Numer. Math., 97 (2004), pp. 321-352.
14. Y. D'HALLUIN, P. A. FORSYTH, AND K. R. VETZAL, Robust numerical methods for contingent claims under jump diffusion processes, IMA J. Numer. Anal., 25 (2005), pp. 87-112.
15. D. DUFFIE, J. PAN, AND K. SINGLETON, Transform analysis and asset pricing for affine jumpdiffusions, Econometrica, 68 (2000), pp. 1343-1376.
16. B. DUPIRE, Pricing with a smile, Risk, 7 (1994), pp. 18-20.
17. P. A. FORSYTH AND K. R. VETZAL, Quadratic convergence for valuing American options using a penalty method, SIAM J. Sci. Comput., 23 (2002), pp. 2095-2122.
18. M. FRIGO AND S. G. JOHNSON, The design and implementation of FFTWS, in Proceedings of the IEEE, 93 (2005), pp. 216-231.
19. M. B. GILES AND R. CARTER, Convergence analysis of Grank-Nicolson and Rannacher timemarching, J. Comput. Finance, 9 (2006), pp. 89-112.
20. S. HESTON, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financial Stud., 6 (1993), pp. 327-343.
21. M. HINTERMÜLLER, K. ITO, AND K. KUNISCH, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim., 13 (2003), pp. 865-888.
22. S. IKONEN AND J. TOIVANEN, Operator splitting methods for American option pricing, Appl. Math. Lett., 17 (2004), pp. 809-814.
23. S. IKONEN AND J. TOIVANEN, Operator Splitting Methods for Pricing American Options with Stochastic Volatility, Technical Report B11/2004, Department of Mathematical Information Technology, University of Jyváskylá, Jyváskylá, Finland, 2004.
24. S. IKONEN AND J. TOIVANEN, Efficient numerical methods for pricing American options under stochastic volatility, Numer. Methods Partial Differential Equations, 10 (2007), pp. 331-361.
25. K. ITO AND K. KUNISCH, Parabolic variational inequalities: The Lagrange multiplier approach, J. Math. Pures Appl., 85 (2006), pp. 415-449.
26. K. ITO AND J. TOIVANEN, Lagrange Multiplier Approach with Optimized Finite Difference Stencils for Pricing American Options, Technical Report B6/2006, Department of Mathematical Information Technology, University of Jyváskylá, Jyváskylá, Finland, 2006.
27. S. G. Kou, A jump-diffusion model for option pricing, Management Sci., 48 (2002), pp. 1086-1101.
28. S. G. KOU AND HUI WANG, Option pricing under a double exponential jump diffusion model, Management Sci., 50 (2004), pp. 1178-1198.
29. T. A. MANTEUFFEL AND A. B. WHITE, Jr., The numerical solution of second-order boundary value problems on nonuniform meshes, Math. Comp., 47 (1986), pp. 511-535.
30. J. MARTIKAINEN, T. ROSSI, AND J. TOIVANEN, A fast direct solver for elliptic problems with a divergence constraint, Numer. Linear Algebra Appl., 9 (2002), pp. 629-652.
31. A.-M. MATACHE, P.-A. NLTSCHE, AND C. SCHWAB, Wavelet Galerkin pricing of American options on Levy driven assets, Quant. Finance, 5 (2005), pp. 403-424.
32. A.-M. MATACHE, C. SCHWAB, AND T. P. WLHLER, Fast numerical solution of parabolic integrodifferential equations with applications in finance, SIAM J. Sci. Comput., 27 (2005), pp. 369-393.
33. A.-M. MATACHE, T. VON PETERSDORFF, AND C. SCHWAB, Fast deterministic pricing of options on Levy driven assets, M2AN Math. Model. Numer. Anal., 38 (2004), pp. 37-71.
34. R. C. MERTON, Theory of rational option pricing, Bell J. Econom. Management Sci., 4 (1973), pp. 141-183.
35. R. C. MERTON, Option pricing when underlying stock returns are discontinuous, J. Financial Econ., 3 (1976), pp. 125-144.
36. M. PICKERING, An Introduction to Fast Fourier Transform Methods for Partial Differential Equations, with. Applications, Electronic & Electrical Engineering Research Studies: Applied and Engineering Mathematics Series 4, Research Studies Press, Chichester, UK, 1986.
37. D. M. POOLEY, K. VETZAL, AND P. A. FORSYTH, Remedies for non-smooth payoffs in option pricing, J. Comput. Finance, 6 (2003), pp. 25-40.
38. R. RANNACHER, Finite element solution of diffusion problems with irregular data, Numer. Math., 43 (1984), pp. 309-327.
39. D. TAVELLA AND C. RANDALL, Pricing financial instruments: The finite difference method, John Wiley & Sons, Chichester, UK, 2000.
40. I. R. WANG, J. W. L. WAN, AND P. A. FORSYTH, Robust numerical valuation of European and American options under the GGMY process, J. Comput. Finance, 10 (2007), pp. 31-69.
41. X. L. ZHANG, Numerical analysis of American option pricing in a jump-diffusion model, Math. Oper. Res., 22 (1997), pp. 668-690.