Space-time adaptive finite difference method for European multi-asset options

Computers and Mathematics with Applications

Volumn 53, Issue 8, 2007, Pages 1159-1180

  Lötstedt, Per ^a   Persson, Jonas ^a   von Sydow, Lina ^a   Tysk, Johan ^a  

^a UPPSALA UNIVERSITY   (Sweden)

Author keywords

Black Scholes equation; Finite difference method; Maximum principle; Space adaptation; Time adaptation

Indexed keywords

COMPUTATIONAL GEOMETRY; ERROR ANALYSIS; MAXIMUM PRINCIPLE; MONTE CARLO METHODS; PROBLEM SOLVING;

BLACK SCHOLES EQUATION; DISCRETIZATION ERRORS; PARABOLIC EQUATIONS; SPACE TIME ADAPTIVE FINITE DIFFERENCE METHOD;

FINITE DIFFERENCE METHOD;

EID: 34248524462     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2006.09.014     Document Type: Article

References (40)

1. Black F., and Scholes M. The pricing of options and corporate liabilities. Journal of Political Economy 81 (1973) 637-659
2. Merton R.C. Theory of rational option pricing. Bell Journal of Economical Management Science 4 (1973) 141-183
3. Musiela M., and Rutkowski M. Martingale Methods in Financial Modelling (1997), Springer-Verlag, Berlin
4. Caflisch R.E. Monte Carlo and quasi-Monte Carlo methods. Acta Numerica 7 (1998) 1-49
5. Glasserman P. Monte Carlo Methods in Financial Engineering (2004), Springer-Verlag, New York, NY
6. J. Persson, L. von Sydow, Pricing European multi-asset options using a space-time adaptive FD-method, Technical report 2003-059, Department of Information Technology, Uppsala University, Uppsala, Sweden, 2003, Computing and Visualization in Science (in press). Available at http://www.it.uu.se/research/reports/
7. Tavella D., and Randall C. Pricing Financial Instruments - The Finite Difference Method (2000), John Wiley & Sons, Chichester
8. Wilmott P. Option Pricing: Mathematical Models and Computation (1993), Oxford Financial Press, Oxford
9. Gilli M., Këllezi E., and Pauletto G. Solving finite difference schemes arising in trivariate option pricing. Journal of Economic Dynamics & Control 26 (2002) 1499-1515
10. McCartin B.J., and Labadie S.M. Accurate and efficient pricing of vanilla stock options via the Crandall-Douglas scheme. Applied Mathematics and Computation 143 (2003) 39-60
11. Schwartz E.S. The valuation of warrants: Implementing a new approach. Journal of Financial Economy 4 (1977) 79-93
12. Engelmann B., and Schwender P. The pricing of multi-asset options using a Fourier grid method. Journal of Computational Finance 1 (1998) 53-61
13. Pironneau O., and Achdou Y. Computational Methods for Option Pricing. Frontiers in Applied Mathematics vol. 30 (2005), SIAM, Philadelphia, PA
14. Pironneau O., and Hecht F. Mesh adaption for the Black & Scholes equations. East-West Journal of Numerical Mathematics 8 (2000) 25-35
15. Bungartz H.-J., and Griebel M. Sparse grids. Acta Numerica 13 (2004) 147-269
16. C. Reisinger, Numerische Methoden für hochdimensionale parabolische Gleichungen am Beispiel von Optionspreisaufgaben, Ph.D. Thesis, Universität Heidelberg, Heidelberg, Germany, 2004
17. Lötstedt P., Söderberg S., Ramage A., and Hemmingsson-Frändén L. Implicit solution of hyperbolic equations with space-time adaptivity. BIT 42 (2002) 128-153
18. Vande Wouwer A., Saucez P., and Schiesser W.E. Adaptive Method of Lines (2001), Chapman & Hall, Boca Raton, FL
19. Figlewski S., and Gao B. The adaptive mesh model: A new approach to efficient option pricing. Journal of Financial Economics 53 (1999) 313-351
20. Boyle P.P., Broadie M., and Glasserman P. Monte Carlo methods for security pricing. Journal of Economic Dynamics & Control 21 (1997) 1267-1321
21. Joy C., Boyle P.P., and Tan K.S. Quasi-Monte Carlo methods in numerical finance. Management Science 42 (1996) 926-938
22. Bratley P., and Fox B.L. Algorithm 659, Implementing Sobol's quasirandom sequence generator. ACM Transactions on Mathematical Software 14 (1988) 88-100
23. Sobol I.M. On the distribution of points in a cube and the approximate evaluation of integrals. USSR Computational Mathematics and Mathematical Physics 7 (1967) 86-112
24. J. Burkardt, School of Computational Science, Florida State University, Tallahassee, FL. http://www.scs.fsu.edu/~burkardt/msrc/msrc.html, 2006
25. Björk T. Arbitrage Theory in Continuous Time (1998), Oxford University Press, New York, NY
26. Cox J.C., and Ross S.A. The valuation of options for alternative stochastic processes. Journal of Financial Economics 3 (1976) 145-166
27. Caflisch R.E., and Moskowitz B. Modified Monte Carlo methods using quasi-random sequences. In: Niederreiter H., and Shiue P.J.S. (Eds). Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing. Lecture Notes in Statistics vol. 106 (1995), Springer, New York 1-16
28. Morokoff W.J. Generating Quasi-Random paths for stochastic processes. SIAM Review 40 (1998) 765-788
29. Morokoff W.J., and Caflisch R.E. Quasi-Monte Carlo simulation of random walks in finance. In: Niederreiter H., Hellekalek P., Larcher G., and Zinterhof P. (Eds). Monte Carlo and Quasi Monte Carlo Methods 1996. Lecture Notes in Statistics vol. 127 (1998), Springer, New York 340-352
30. Clarke N., and Parrott K. Multigrid for American option pricing with stochastic volatility. Applied Mathematical Finance 6 (1999) 177-195
31. Ikonen S., and Toivanen J. Operator splitting methods for American option pricing. Applied Mathematical Letters 17 (2004) 809-814
32. Nielsen B.F., Skavhaug O., and Tveito A. Penalty and front-fixing methods for the numerical solution of American option problems. Journal of Computational Finance 5 4 (2002) 69-97
33. Oosterlee C.W. On multigrid for linear complementarity problems with application to American-style options. Electronic Transactions on Numerical Analysis 15 (2003) 165-185
34. Manteuffel T.A., and White Jr. A.B. The numerical solution of second-order boundary value problems on nonuniform meshes. Mathematics of Computation 47 (1986) 511-535
35. Hairer E., Nørsett S.P., and Wanner G. Solving Ordinary Differential Equations, Nonstiff Problems. 2nd ed. (1993), Springer-Verlag, Berlin
36. Becker R., and Rannacher R. An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica 10 (2001) 1-102
37. Eriksson K., Estep D., Hansbo P., and Johnson C. Introduction to adaptive methods for differential equations. Acta Numerica 4 (1995) 105-158
38. Friedman A. Partial Differential Equations of Parabolic Type (1964), Prentice-Hall, Englewood Cliffs, NJ
39. Saad Y., and Schultz M.H. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific Computing 7 (1986) 856-869
40. Greenbaum A. Iterative Methods for Solving Linear Systems (1997), SIAM, Philadelphia, PA