# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Pricing options under jump diffusion processes with fitted finite volume method. (English) Zbl 1142.91576
Summary: This paper develops a numerical method for a partial integro-differential equation and a partial integro-differential complementarity problem arising from European and American options valuations respectively when the underlying assets are driven by a jump diffusion process. The method is based on a fitted finite volume scheme for the spatial discretization and the Crank-Nicolson scheme for the time discretization. The fully discretized system is solved by an iterative method coupled with an FFT for the evaluation of the discretized integral term, while the constraint in the American option model is imposed by adding a penalty term to the original partial integro-differential complementarity problem. We show that the system matrix of the discretized system is an $M$-matrix and propose an algorithm for solving the discretized system. Numerical experiments are implemented to show the efficiency and robustness of this method.

##### MSC:
 91B28 Finance etc. (MSC2000) 60J60 Diffusion processes 60J75 Jump processes 60H15 Stochastic partial differential equations 91B24 Price theory and market structure
Full Text:
##### References:
 [1] Almendral, A.; Oosterlee, C. W.: Numerical valuation of options with jumps in the underlying. Appl. math. Comput. 53, 1-18 (2005) · Zbl 1117.91028 [2] Amin, K.: Jump diffusion option valuation in discrete time. J. finance 48, 1863-1883 (1993) [3] Anderson, A.; Andresen, J.: Jump diffusion process: volatility smile Fitting and numerical methods for option pricing. Rev. derivat. Res. 4, 231-262 (2000) [4] Angermann, L.; Wang, S.: Convergence of a fitted finite volume method for European and American option valuation. Numer. math. 106, 1-40 (2007) · Zbl 1131.65301 [5] Black, F.; Scholes, M.: The pricing of options and corporate liabilities. J. political econ. 81, 637-659 (1973) · Zbl 1092.91524 [6] Cont, R.; Tankov, P.: Financial modelling with jump processes. (2004) · Zbl 1052.91043 [7] D’halluin, Y.; Forsyth, P. A.; Vetzal, K. R.: Robust numerical methods for contingent claims under jump diffusion processes. IMA J. Numer. anal. 25, 87-112 (2005) · Zbl 1134.91405 [8] Duffy, D. J.: Finite difference methods in financial engineering: A partial differential equation approach. (2006) · Zbl 1141.91002 [9] Dupire, B.: Pricing with a smile. Risk 7, 18-20 (1994) [10] Forsyth, P. A.; Vetzal, K. R.: Quadratic convergence for valuing American options using a penalty method. SIAM J. Sci. comput. 23, 2095-2122 (2002) · Zbl 1020.91017 [11] Heston, S. L.: A closed-form solution for options with stochastic volatility with applications to Bond and currency options. Rev. financial stud. 6, 327-343 (1993) [12] Huang, C. -S.; Hung, C. -H.; Wang, S.: A fitted finite volume method for the valuation of options on assets with stochastic volatilities. Computing 77, 297-320 (2006) · Zbl 1136.91441 [13] Hull, J.: Options, futures, and other derivatives. (2005) · Zbl 1087.91025 [14] Hull, J.; White, A.: The pricing of options on assets with stochastic volatilities. J. finance 42, 281-300 (1987) · Zbl 1126.91369 [15] Van Loan, C.: Computational frameworks for the fast Fourier transform. Frontiers in applied mathematics 10 (1992) · Zbl 0757.65154 [16] Merton, R. C.: Option pricing when underlying stock return are discontinuous. J. financial econ. 3, 125-144 (1976) · Zbl 1131.91344 [17] Rannacher, R.: Finite element solution of diffusion problems with irregular data. Numer. math. 43, 309-327 (1984) · Zbl 0512.65082 [18] Wang, S.: A novel fitted finite volume method for the black -- Scholes equation governing option pricing. IMA J. Numer. anal. 24, 699-720 (2004) · Zbl 1147.91332 [19] Wang, S.; Yang, X. Q.; Teo, K. L.: Power penalty method for a linear complementarity problem arising from American option valuation. J. optim. Theory appl. 129, 227-254 (2006) · Zbl 1139.91020 [20] Wilmott, P.: Derivatives. (1998) [21] Young, D. M.: Iterative solution of large linear systems. (1971) · Zbl 0231.65034 [22] K. Zhang, S. Wang, A computational scheme for options under jump diffusion processes, Int. J. Numer. Anal. Mod., in press. · Zbl 1159.91402 [23] K. Zhang, S. Wang, X.Q. Yang, K.L. Teo, A power penalty approach to numerical solutions of two-asset American Options, Working Paper, 2007. [24] Zhang, X. L.: Numerical analysis of American option pricing in a jump diffusion model. Math. operat. Res. 22, 668-690 (1997) · Zbl 0883.90021