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Düring, Bertram; Fournié, Michel; Jüngel, Ansgar

High-order compact finite difference schemes for a nonlinear Black-Scholes equation. (English) Zbl 1070.91024

Int. J. Theor. Appl. Finance 6, No. 7, 767-789 (2003).
Summary: A nonlinear Black-Scholes equation which models transaction costs arising in the hedging of portfolios is discretized semi-implicitly using high order compact finite difference schemes. A new compact scheme, generalizing the compact schemes of A. Rigal [J. Comp. Phys. 114, No. 1, 59–76 (1994; Zbl 0807.65096)], is derived and proved to be unconditionally stable and non-oscillatory. The numerical results are compared to standard finite difference schemes. It turns out that the compact schemes have very satisfying stability and non-oscillatory properties and are generally more efficient than the considered classical schemes.

Cited in 33 Documents

MSC:

91B28 Finance etc. (MSC2000)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Keywords:

option pricing; parabolic equations; transaction costs

Citations:

Zbl 0807.65096
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\textit{B. Düring} et al., Int. J. Theor. Appl. Finance 6, No. 7, 767--789 (2003; Zbl 1070.91024)
Full Text: DOI

References:

[1] DOI: 10.1007/s007800050046 · Zbl 0915.35051
[2] DOI: 10.1111/j.1467-9965.1992.tb00039.x · Zbl 0900.90100
[3] Ben-Yu G., Contemp. Math. 163 pp 33– (1994)
[4] DOI: 10.1090/S0025-5718-1980-0572850-8
[5] DOI: 10.1086/260062 · Zbl 1092.91524
[6] DOI: 10.1111/j.1540-6261.1992.tb03986.x
[7] Bruce G. H., Trans. Am. Inst. Min. Engrs. (Petrol Div.) 198 pp 79–
[8] DOI: 10.1007/s007800050066 · Zbl 0935.91014
[9] DOI: 10.1137/0331022 · Zbl 0779.90011
[10] Dewynne J., Option Pricing: Mathematical Models and Computation (1995)
[11] DOI: 10.1080/135048699334564 · Zbl 1009.91030
[12] DOI: 10.1007/s007800050035 · Zbl 0894.90017
[13] Frey R., Model Risk (2000)
[14] Genotte G., Amer. Econ. Rev. 80 pp 999–
[15] Hodges S. D., Rev. Future Markets 8 pp 222–
[16] DOI: 10.2307/2331322
[17] DOI: 10.1137/S0036142999355921 · Zbl 0990.35013
[18] DOI: 10.1090/S0025-5718-1975-0371058-7
[19] Lamberton D., Ann. Appl. Probab. 8 pp 206–
[20] DOI: 10.1111/j.1540-6261.1985.tb02383.x
[21] DOI: 10.2307/3003143
[22] Parás A., Appl. Math. Finance 1 pp 165–
[23] DOI: 10.1111/1467-9965.00045 · Zbl 0908.90016
[24] Pironneau O., East-West J. Numer. Math. 8 pp 25–
[25] Richtmyer R. D., Difference Methods for Initial Value Problems (1967) · Zbl 0155.47502
[26] DOI: 10.1006/jcph.1994.1149 · Zbl 0807.65096
[27] DOI: 10.1002/nme.1620300207 · Zbl 0714.76072
[28] Schönbucher P., Z. Angew. Math. Mech. 76 pp 81–
[29] Schönbucher P., SIAM J. Appl. Math. 61 pp 232–
[30] DOI: 10.1080/135048698334727 · Zbl 1009.91023
[31] DOI: 10.1007/978-3-642-65471-8
[32] Strikwerda J. C., Mathematics series, in: Finite Difference Schemes and Partial Differential Equations (1989) · Zbl 0681.65064
[33] Tavella D., Pricing Financial Instruments: The Finite Difference Method (2000)
[34] DOI: 10.1016/0021-9991(74)90011-4 · Zbl 0291.65023
[35] DOI: 10.1111/1467-9965.00034 · Zbl 0885.90019
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.