zbMATH — the first resource for mathematics

A penalty method for American options with jump diffusion processes. (English) Zbl 1126.91036
Summary: The fair price for an American option where the underlying asset follows a jump diffusion process can be formulated as a partial integral differential linear complementarity problem. We develop an implicit discretization method for pricing such American options. The jump diffusion correlation integral term is computed using an iterative method coupled with an FFT while the American constraint is imposed by using a penalty method. We derive sufficient conditions for global convergence of the discrete penalized equations at each timestep. Finally, we present numerical tests which illustrate such convergence.

91B28 Finance etc. (MSC2000)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
45K05 Integro-partial differential equations
Full Text: DOI
[1] Amadori, A.L.: The obstacle problem for nonlinear integro-differential equations arising in option pricing. Working paper, Istituto pre le Applicazione del Calcolo ”M. Picone”, Rome, www.iac.rm.cnr.it/madori
[2] Amin, K.: Jump diffusion option valuation in discrete time. J. Finance 48, 1833–1863 (1993)
[3] Andersen, L., Andreasen, J.: Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing. Rev. Derivatives Res. 4, 231–262 (2000) · Zbl 1274.91398
[4] Ayache, E., Forsyth, P.A., Vetzal, K.R.: Next generation models for convertible bonds with credit risk. Wilmott Magazine, December 2002, pp. 68–77
[5] Barles, G.: Convergence of numerical schemes for degenerate parabolic equations arising in finance. In L.C.G. Rogers and D. Talay, (eds.), Numerical Methods in Finance, Cambridge University Press, Cambridge, 1997, pp. 1–21 · Zbl 0898.90015
[6] Briani, M., La Chioma, C., Natalini, R.: Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory. Working paper, Istituto pre le Applicazione del Calcolo ”M. Picone”, Rome, www.iac.rm.cnr.it/atalini, to appear in Numerische Mathematik · Zbl 1065.65145
[7] Broadie, M., Yamamoto, Y.: Application of the Fast Gauss transform to option pricing. Working paper, Columbia School of Business, 2002
[8] Coleman, T.F., Li, Y., Verma, A.: Reconstructing the unknown local volatility function. J. Comput. Finance 2, 77–102 (1999) · Zbl 0976.91019
[9] Cottle, R.W., Pang, J.-S., Stone, R.E: The Linear Complementarity Problem. Academic Press, 1992
[10] Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bulletin of the Am. Math. Soc. 27, 1–67 July 1992 · Zbl 0755.35015
[11] Cryer, C.W.: The efficient solution of linear complementarity problems for tridiagonal Minkowski matrices. ACM Trans. Math. Softw. 9, 199–214 (1983) · Zbl 0518.90085
[12] d’Halluin, Y., Forsyth, P.A., Vetzal, K.R.: Robust numerical methods for contingent claims under jump diffusion processes. www.scicom.uwaterloo.ca/aforsyt/jump.pdf, submitted to IMA J. Numer. Anal. · Zbl 1134.91405
[13] d’Halluin, Y., Forsyth, P.A., Vetzal, K.R., Labahn, G.: A numerical PDE approach for pricing callable bonds. Appl. Math. Finance 8, 49–77 (2001) · Zbl 1026.91046
[14] Dutt, A., Rokhlin, V.: Fast Fourier transforms for nonequispaced data. SIAM J. Sci. Comput. 14, 1368–1393 November 1993 · Zbl 0791.65108
[15] Elliot, E.M., Ockendon, J.R.: Weak and Variational Methods for Moving Boundary Problems. Pitman, 1982 · Zbl 0476.35080
[16] Forsyth, P.A., Vetzal, K.R.: Quadratic convergence of a penalty method for valuing American options. SIAM J. Sci. Comput. 23, 2096–2123 (2002) · Zbl 1020.91017
[17] Greengard, L., Strain, J.: The fast Gauss transform. SIAM J. Sci. Comput. 12, 79–94 (1991) · Zbl 0721.65089
[18] Hull, J.: Options, Futures, and Other Derivatives. Prentice Hall, Inc., Upper Saddle River, NJ, 3rd edition, 1997 · Zbl 1087.91025
[19] Johnson, C.: Numerical Solutions of Partial Differential Equations By the Finite Element Method. Cambridge University Press, Cambridge, 1987 · Zbl 0628.65098
[20] Kangro, R., Nicolaides, R.: Far field boundary conditions for Black-Scholes equations. SIAM J. Numer. Anal. 38(4), 1357–1368 (2000) · Zbl 0990.35013
[21] Lewis, A.: Fear of jumps. Wilmott Magazine, December 2002, pp. 60–67
[22] Merton, R.C.: Option pricing when underlying stock returns are discontinuous. J. Financial Econ. 3, 125–144 (1976) · Zbl 1131.91344
[23] Meyer, G.H.: The numerical valuation of options with underlying jumps. Acta Math. Univ. Comenianae 67, 69–82 (1998) · Zbl 0936.91019
[24] Mulinacci, S.: An approximation of American option prices in a jump diffusion model. Stochastic Processes and their Applications 62, 1–17 (1996) · Zbl 0848.90005
[25] Pham, H.: Optimal stopping of controlled jump diffusion processes: a viscosity solution approach. J. Math. Syst. Estimation and Control 8, 1–27 (1998) · Zbl 0899.60039
[26] Pooley, D.M., Forsyth, P.A., Vetzal, K.R.: Numerical convergence properties of option pricing PDEs with uncertain volatility. IMA J. Numer. Anal. 23, 241–267 (2003) · Zbl 1040.91053
[27] Potts, D., Steidl, G., Tasche, M.: Fast Fourier transforms for nonequispaced data: A tutorial. In: Modern Sampling Theory: Mathematics and Application, J.J. Benedetto and P. Ferreira, (eds.), ch. 12, Birkhauser, 2000 , pp. 253–274
[28] Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge (UK) and New York, 2nd edition, 1992 · Zbl 0778.65002
[29] Rannacher, R.: Finite element solution of diffusion problems with irregular data. Numer. Math. 43, 309–327 (1984) · Zbl 0524.65072
[30] Tavella, D., Randall, C.: Pricing financial instruments: the finite difference method. John Wiley & Sons, Inc, 2000
[31] Vazquez, A.A., Oosterlee, C.W.: Numerical valuation of options with jumps in the underlying. Working paper, Delft University of Technology, 2003
[32] Ware, A.F.: Fast approximate Fourier transforms for irregularly spaced data. SIAM Rev. 40, 838–856 (1998) · Zbl 0917.65122
[33] Wilmott, P.: Derivatives. John Wiley and Sons Ltd, Chichester, 1998
[34] Windcliff, H., Forsyth, P.A., Vetzal, K.R.: Shout options: a framework for pricing contracts which can be modified by the investor. J. Comput. Appl. Math. 134, 213–241 (2001) · Zbl 1017.91060
[35] Windcliff, H., Forsyth, P.A., Vetzal, K.R.: Valuation of segregated funds: shout options with maturity extensions. Insurance: Math. Econ. 29, 1–21 (2001) · Zbl 1055.91036
[36] Windcliff, H., Forsyth, P.A., Vetzal, K.R.: Analysis of the stability of the linear boundary condition for the Black-Scholes equation, 2003. Submitted to the J. of Comput. Finance · Zbl 1178.91205
[37] Zhang, X.L.: Numerical analysis of American option pricing in a jump-diffusion model. Math. Oper. Res. 22, 668–690 (1997) · Zbl 0883.90021
[38] Zvan, R., Forsyth, P.A., Vetzal, K.R.: Penalty methods for American options with stochastic volatility. J. Comput. Appl. Math. 91, 199–218 (1998) · Zbl 0945.65005
[39] Zvan, R., Forsyth, P.A., Vetzal, K.R.: Discrete Asian barrier options. J. Comput. Finance 3(Fall), 41–67 (1999) · Zbl 1009.91030
[40] Zvan, R., Forsyth, P.A., Vetzal, K.R.: A finite element approach to the pricing of discrete lookbacks with stochastic volatility. Appl. Math. Finance 6, 87–106 (1999) · Zbl 1009.91030
[41] Zvan, R., Forsyth, P.A., Vetzal, K.R.: A finite volume approach for contingent claims valuation. IMA J. Numer. Anal. 21, 703–731 (2001) · Zbl 1004.91032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.