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)
High-order compact finite difference scheme for option pricing in stochastic volatility models. (English) Zbl 1244.91100
Summary: We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility models. The scheme is fourth order accurate in space and second order accurate in time. Under some restrictions, theoretical results like unconditional stability in the sense of von Neumann are presented. Where the analysis becomes too involved we validate our findings by a numerical study. Numerical experiments for the European option pricing problem are presented. We observe fourth order convergence for non-smooth payoff.
MSC:
91G60Numerical methods in mathematical finance
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
91G20Derivative securities
References:
[1]Black, F.; Scholes, M.: The pricing of options and corporate liabilities, J. polit. Econ. 81, 637-659 (1973)
[2]Heston, S. L.: A closed-form solution for options with stochastic volatility with applications to Bond and currency options, Rev. finan. Stud. 6, No. 2, 327-343 (1993)
[3]Düring, B.: Asset pricing under information with stochastic volatility, Rev. deriv. Res. 12, No. 2, 141-167 (2009) · Zbl 1175.91072 · doi:10.1007/s11147-009-9031-8
[4]Benhamou, E.; Gobet, E.; Miri, M.: Time dependent Heston model, SIAM J. Financ. math. 1, 289-325 (2010) · Zbl 1198.91203 · doi:10.1137/090753814
[5]Tavella, D.; Randall, C.: Pricing financial instruments: the finite difference method, (2000)
[6]Tangman, D. Y.; Gopaul, A.; Bhuruth, M.: Numerical pricing of options using high-order compact finite difference schemes, J. comput. Appl. math. 218, No. 2, 270-280 (2008) · Zbl 1146.91338 · doi:10.1016/j.cam.2007.01.035
[7]Düring, B.; Fournié, M.; Jüngel, A.: Convergence of a high-order compact finite difference scheme for a nonlinear black–Scholes equation, Math. modelling numer. Anal. 38, No. 2, 359-369 (2004) · Zbl 1124.91031 · doi:10.1051/m2an:2004018 · doi:numdam:M2AN_2004__38_2_359_0
[8]Düring, B.; Fournié, M.; Jüngel, A.: High-order compact finite difference schemes for a nonlinear black–Scholes equation, Int. J. Theor. appl. Finance 6, No. 7, 767-789 (2003) · Zbl 1070.91024 · doi:10.1142/S0219024903002183
[9]Liao, W.; Khaliq, A. Q. M.: High-order compact scheme for solving nonlinear black–Scholes equation with transaction cost, Int. J. Comput. math. 86, No. 6, 1009-1023 (2009) · Zbl 1163.91411 · doi:10.1080/00207160802609829
[10]Zvan, R.; Forsyth, P. A.; Vetzal, K. R.: Penalty methods for American options with stochastic volatility, J. comput. Appl. math. 91, No. 2, 199-218 (1998) · Zbl 0945.65005 · doi:10.1016/S0377-0427(98)00037-5
[11]Clarke, N.; Parrott, K.: Multigrid for American option pricing with stochastic volatility, Appl. math. Finance 6, No. 3, 177-195 (1999) · Zbl 1009.91034 · doi:10.1080/135048699334528
[12]Hilber, N.; Matache, A.; Schwab, C.: Sparse wavelet methods for option pricing under stochastic volatility, J. comput. Finance 8, No. 4, 1-42 (2005)
[13]Zhu, W.; Kopriva, D. A.: A spectral element approximation to price European options with one asset and stochastic volatility, J. sci. Comput. 42, No. 3, 426-446 (2010) · Zbl 1203.91307 · doi:10.1007/s10915-009-9333-x
[14]Ikonen, S.; Toivanen, J.: Efficient numerical methods for pricing American options under stochastic volatility, Numer. methods partial differential equations 24, No. 1, 104-126 (2008) · Zbl 1152.91516 · doi:10.1002/num.20239
[15]Hout, K. J. In’t; Foulon, S.: ADI finite difference schemes for option pricing in the Heston model with correlation, Int. J. Numer. anal. Model. 7, 303-320 (2010)
[16]Kangro, P.; Nicolaides, R.: Far field boundary conditions for black–Scholes equations, SIAM J. Numer. anal. 38, 1357-1368 (2000) · Zbl 0990.35013 · doi:10.1137/S0036142999355921
[17]Gustafsson, B.: The convergence rate for difference approximation to general mixed initial-boundary value problems, SIAM J. Numer. anal. 18, No. 2, 179-190 (1981) · Zbl 0469.65068 · doi:10.1137/0718014
[18]Spotz, W. F.; Carey, C. F.: Extension of high-order compact schemes to time-dependent problems, Numer. methods partial differential equations 17, No. 6, 657-672 (2001) · Zbl 0998.65101 · doi:10.1002/num.1032
[19]Strikwerda, J. C.: Finite difference schemes and partial differential equations, (2004)
[20]Gustafsson, B.; Kreiss, H. -O.; Oliger, J.: Time dependent problems and difference methods, (1996)
[21]Fournié, M.; Rigal, A.: High order compact schemes in projection methods for incompressible viscous flows, Commun. comput. Phys. 9, No. 4, 994-1019 (2011)
[22]Mishra, S.; Svärd, M.: On stability of numerical schemes via frozen coefficients and the magnetic induction equations, BIT numer. Math. 50, 85-108 (2010) · Zbl 1205.65248 · doi:10.1007/s10543-010-0249-5
[23]Wade, B. A.: Symmetrizable finite difference operators, Math. comp. 54, 525-543 (1990) · Zbl 0697.65069 · doi:10.2307/2008500
[24]Strikwerda, J. C.; Wade, B. A.: An extension of the kreiss matrix theorem, SIAM J. Numer. anal. 25, No. 6, 1272-1278 (1988) · Zbl 0667.65074 · doi:10.1137/0725071
[25]Richtmyer, R. D.; Morton, K. W.: Difference methods for initial value problems, (1967) · Zbl 0155.47502
[26]Widlund, O. B.: Stability of parabolic difference schemes in the maximum norm, Numer. math. 8, 186-202 (1966) · Zbl 0173.44805 · doi:10.1007/BF02163187
[27]Düring, B.; Fournié, M.: On the stability of a compact finite difference scheme for option pricing, Progress in industrial mathematics at ECMI 2010, 215-221 (2012)
[28]Rannacher, R.: Finite element solution of diffusion problems with irregular data, Numer. math. 43, No. 2, 309-327 (1984) · Zbl 0512.65082 · doi:10.1007/BF01390130
[29]Fournié, M.: High order conservative difference methods for 2d drift-diffusion model on non-uniform grid, Appl. numer. Math. 33, No. 1–4, 381-392 (2000) · Zbl 0959.82033 · doi:10.1016/S0168-9274(99)00105-1