A finite volume approach for contingent claims valuation

IMA Journal of Numerical Analysis

Volumn 21, Issue 3, 2001, Pages 703-731

  Zvan, R ^a   Forsyth, P A ^b   Vetzal, K R ^c  

^a Bear Stearns & Co Inc   (United States)

^b UNIVERSITY OF WATERLOO   (Canada)

^c NONE

Author keywords

Contingent claims; Finite volume; Option pricing; Positive coefficient

Indexed keywords

CONSTRAINED OPTIMIZATION; ECONOMICS; FINITE DIFFERENCE METHOD;

CONTINGENT CLAIMS; CONVECTION-DOMINATED; FINANCE LITERATURES; FINITE DIFFERENCE SCHEME; FINITE VOLUME APPROACH; MODEL SPECIFICATIONS; OPTION PRICING; POSITIVE COEFFICIENTS;

FINITE VOLUME METHOD;

EID: 0035534539     PISSN: 02724979     EISSN: None     Source Type: Journal    
DOI: 10.1093/imanum/21.3.703     Document Type: Article

References (38)

1. ARMINJON, P. & DERVIEUX, A. 1993 Construction of TVD-like artificial viscosities on two-dimensional arbitrary FEM grids. J. Comput. Phys., 106, 176-198.
2. BARRAQUAND, J. & PUDET, T. 1996 Pricing of American path-dependent contingent claims. Math. Fin., 6, 17-51.
3. BARTH, T. J. 1994 Aspects of unstructured grids and finite-volume solvers for the Euler and Navier-Stokes equations. Lecture Series 1994-05: Computational Fluid Dynamics. Brussells: von Karman Institute for Fluid Dynamics.
4. BLACK, F. & SCHOLES, M. 1973 The pricing of options and corporate liabilities. J. Pol. Econ., 81, 637-654.
5. BLUNT, M. & RUBIN, B. 1992 Implicit flux limiting schemes for petroleum reservoir simulation. J. Comput. Phys., 102, 194-210.
6. BRENNAN, M. J. & SCHWARTZ, E. S. 1977 Convertible bonds: valuation and optimal strategies for call and conversion. J. Fin., 32, 1699-1715.
7. CHEUNG, W. & NELKEN, I. 1994 Costing the converts. RISK, 7, 47-49.
8. D'AZEVEDO, E. F., FORSYTH, P. A. & TANG, W. P. 1992 Towards a cost effective ILU preconditioner with high level fill. BIT, 32, 442-463.
9. DEMPSTER, M. A. H., HUTTON, J. P. & RICHARDS, D. G. 1998 LP valuation of exotic American options exploiting structure. J. Comput. Fin., 2, 61-84.
10. DEWYNNE, J. N. & WILMOTT, P. 1995 A note on average rate options with discrete sampling. SIAM J. Appl. Math., 55, 267-276.
11. D'HALLUIN, Y., FORSYTH, P. A., VETZAL, K. R. & LABAHN, G. 2000 A numerical PDE approach for pricing callable bonds. To appear in Appl. Math. Fin.
12. FORSYTH, P. A. & JIANG, H. 1997 Nonlinear iteration methods for high speed laminar compressible Navier-Stokes equations. Comput. Fluids, 26, 249-268.
13. FORSYTH, P. A., VETZAL, K. R. & ZVAN, R. 1999 A finite element approach to the pricing of discrete lookbacks with stochastic volatility. Appl. Math. Fin., 6, 87-106.
14. GOODMAN, J. B. & LEVEQUE, R. J. 1985 On the accuracy of stable schemes for 2D scalar conservation laws. Math. Comput., 45, 15-21.
15. HO, T. S. Y. & PFEFFER, D. M. 1996 Convertible bonds: model, value attribution, and analytics. Fin. Anal. J., 52, 35-44.
16. HOUSTON, P., SCHWAB, C. & SÜLI, E. 2000 Discontinuous hp-finite element methods for advection-diffusion problems. Working Paper. Oxford: Oxford University Computing Laboratory.
17. HULL, J. C. 1997 Options, Futures, and Other Derivatives, 3rd edn. Upper Saddle River, NJ: Prentice-Hall.
18. HULL, J. C. & WHITE, A. 1993 Efficient procedures for valuing European and American path-dependent options. J. Derivatives, 1, 21-31.
19. INGERSOLL JR., J. E. 1977 A contingent-claims valuation of convertible securities. J. Fin. Econ., 4, 289-322.
20. JAMESON, A. 1995 Positive schemes and shock modelling for compressible flows. Int. J. Numer. Meth. Fluids, 20, 743-776.
21. KEMNA, A. G. Z. & VORST, A. C. F. 1990 A pricing method for options based on average asset values. J. Bank. Fin., 14, 113-129.
22. 1 Surface Interpolation. Mathematical Software III. (J. R. Rice ed.). New York: Academic, pp. 161-193.
23. MERTON, R. C. 1973 Theory of rational option pricing. Bell J. Econ. Mgmt. Sci., 4, 141-183.
24. NICOLAIDES, R. A. 1992 Direct discretization of planar div-curl problems. SIAM J. Numer. Anal., 29, 32-56.
25. OLEǏNIK, O. A. & RADKEVIC, E. V. 1973 Second Order Equations with Nonnegative Characteristic Form. Providence, R.I: American Mathematical Society.
26. ROGERS, L. C. G. & SHI, Z. 1995 The value of an Asian option. J. Appl. Probab., 32, 1077-1088.
27. SPEKREIJSE, S. 1987 Multigrid solution of monotone second-order discretizations of hyperbolic conservation laws. Math. Comp., 49, 135-155.
28. STULZ, R. M. 1982 Options on the minimum or the maximum of two risky assets. J. Fin. Econ., 10, 161-185.
29. SWEBY, P. K. 1984 High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal., 21, 995-1011.
30. THUBURN, J. 1997 TVD schemes, positive schemes, and the universal limiter. Monthly Weather Rev., 125, 1990-1993.
31. VAN DER VORST, H. A. 1992 Bi-CGSTAB: a fast and smoothly converging variant of bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 13, 631-645.
32. VAN LEER, B. 1974 Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. J. Comput. Phys., 14, 361-370.
33. WILMOTT, P. 1998 Derivatives: The Theory and Practice of Financial Engineering. West Sussex, UK: Wiley.
34. WILMOTT, P., DEWYNNE, J. N. & HOWISON, S. 1993 Option Pricing: Mathematical Models and Computation. Oxford: Oxford Financial Press.
35. ZVAN, R. 2000 The numerical solution of two-factor PDE option pricing models, Ph. D. Thesis, University of Waterloo, Onterio.
36. ZVAN, R., FORSYTH, P. A. & VETZAL, K. R. 1998a Penalty methods for American options with stochastic volatility. J. Comput. Appl. Math., 91, 199-218.
37. ZVAN, R., FORSYTH, P. A. & VETZAL, K. R. 1998b Robust numerical methods for PDE models of Asian options. J. Comput. Fin., 1, 39-78.
38. ZVAN, R., FORSYTH, P. A. & VETZAL, K. R. 1999 Discrete Asian barrier options. J. Comput. Fin., 3, 41-67.