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High Order Compact Finite Difference Schemes for a Nonlinear Black-Scholes Equation

Bertram Düring

Fachbereich Mathematik und Informatik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany

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Michel Fournié

UMR-CNRS 5640, Laboratoire MIP, Université Paul Sabatier, Toulouse 3, 31062 Toulouse Cedex, France

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, and

Ansgar Jüngel

Fachbereich Mathematik und Informatik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany

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https://doi.org/10.1142/S0219024903002183Cited by:63 (Source: Crossref)

Abstract

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 Rigal [29], 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.

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References

G. Barles and H. M. Soner, Finance Stochast. 2, 369 (1998), DOI: 10.1007/s007800050046. Crossref, Google Scholar
B. Bensaidet al., Math. Finance 2, 63 (1992), DOI: 10.1111/j.1467-9965.1992.tb00039.x. Crossref, Google Scholar
G. Ben-Yu, Contemp. Math. 163 (American Mathematical Society, 1994) pp. 33–54. Google Scholar
A. E. Bergeret al., Math. of Comp. 35(151), 695 (1980), DOI: 10.1090/S0025-5718-1980-0572850-8. Crossref, Web of Science, Google Scholar
F. Black and M. Scholes, J. Polit. Econ. 81, 637 (1973), DOI: 10.1086/260062. Crossref, Web of Science, Google Scholar
P. Boyle and T. Vorst, J. Finance 47, 271 (1973), DOI: 10.2307/2329098. Crossref, Web of Science, Google Scholar
G. H. Bruceet al., Trans. Am. Inst. Min. Engrs. (Petrol Div.) 198, 79 (1953). Web of Science, Google Scholar
G. M. Constantinides and T. Zariphopoulou, Finance Stoch. 3(3), 345 (1999), DOI: 10.1007/s007800050066. Crossref, Google Scholar
M. Davis, V. Panis and T. Zariphopoulou, SIAM J. Contr. Optim. 31, 470 (1993), DOI: 10.1137/0331022. Crossref, Web of Science, Google Scholar
J. Dewynne , S. Howison and P. Wilmott , Option Pricing: Mathematical Models and Computation ( Financial Press , Oxford , 1995 ) . Google Scholar
P. Forsyth, K. Vetzal and R. Zvan, Appl. Math. Finance 6, 87 (1999), DOI: 10.1080/135048699334564. Crossref, Google Scholar
R. Frey, Finance Stochast. 2, 115 (1998), DOI: 10.1007/s007800050035. Crossref, Google Scholar
R. Frey , Model Risk , ed. R. Gibson ( RISK Publications , London , 2000 ) . Google Scholar
G. Genotte and H. Leland, Amer. Econ. Rev. 80, 999 (1990). Web of Science, Google Scholar
S. D. Hodges and A. Neuberger, Rev. Future Markets 8, 222 (1989). Google Scholar
R. Jarrow, J. Financial Quant. Anal. 27, 311 (1992), DOI: 10.2307/2331322. Crossref, Web of Science, Google Scholar
P. Kangro and R. Nicolaides, SIAM J. Numer. Anal. 38, 1357 (2000), DOI: 10.1137/S0036142999355921. Crossref, Web of Science, Google Scholar
H. B. Keller, Math. Comp. 29, 464 (1975), DOI: 10.1090/S0025-5718-1975-0371058-7. Crossref, Web of Science, Google Scholar
S. Kusuoka, Option replication cost with transaction costs, in: Nonlinear and Convex Analysis in Economic Theory, Proceedings of the T.I. Tech/K.E.S. conference on nonlinear and convex analysis in economic theory, Lect. Notes Econ. Math. Syst. 419, ed. T. Maruyama (1995) pp. 197–202. Google Scholar
Y. Lai and J. Spanier , Applications of Monte Carlo/quasi-Monte Carlo methods in finance: Option pricing , Monte Carlo/Quasi-Monte Carlo Methods, Proceedings of a conference , ed. H. Niederreiter ( Springer , Berlin , 2000 ) . Google Scholar
D. Lamberton, Ann. Appl. Probab. 8, 206 (1998). Web of Science, Google Scholar
J. Leitner , Continuous time CAPM, price for risk and utility maximization, in Mathematical Finance , Workshop of the Mathematical Finance Research Project , eds. M. Kohlmann et al. ( Birkhäuser , Basel , 2001 ) . Google Scholar
H. Leland, J. Finance 40, 1283 (1985), DOI: 10.2307/2328113. Crossref, Web of Science, Google Scholar
R. C. Merton, Bell J. Econ. Manag. Sci. 4, 141 (1973), DOI: 10.2307/3003143. Crossref, Web of Science, Google Scholar
A. Parás and M. Avellaneda, Appl. Math. Finance 1, 165 (1994). Google Scholar
E. Platen and M. Schweizer, Math. Finance 8, 67 (1998), DOI: 10.1111/1467-9965.00045. Crossref, Web of Science, Google Scholar
O. Pironneau and F. Hecht, East-West J. Numer. Math. 8, 25 (2000). Web of Science, Google Scholar
R. D. Richtmyer and K. W. Morton , Difference Methods for Initial Value Problems ( Interscience , New York , 1967 ) . Google Scholar
A. Rigal, J. Comp. Phys. 114(1), 59 (1994), DOI: 10.1006/jcph.1994.1149. Crossref, Web of Science, Google Scholar
A. Rigal, J. Num. Meth. Engineering 30, 307 (1990), DOI: 10.1002/nme.1620300207. Crossref, Web of Science, Google Scholar
P. Schönbucher and P. Wilmott, Z. Angew. Math. Mech. 76(Suppl. 3), 81 (1996). Google Scholar
P. Schönbucher and P. Wilmott, SIAM J. Appl. Math. 61, 232 (2000). Crossref, Web of Science, Google Scholar
K. R. Sircar and G. Papanicolaou, Appl. Math. Finance 5(1), 45 (1998), DOI: 10.1080/135048698334727. Crossref, Google Scholar
H. J. Stetter , Analysis of Discretization Methods for Ordinary Differential Equations , Springer Tracts in Natural Philosophy 23 ( Springer , 1973 ) . Crossref, Google Scholar
J. C. Strikwerda , Finite Difference Schemes and Partial Differential Equations , Mathematics series ( Wadsworth & Brooks/Cole , 1989 ) . Google Scholar
D. Tavella and C. Randall , Pricing Financial Instruments: The Finite Difference Method ( John Wiley & Sons , 2000 ) . Google Scholar
R. F. Warming and B. J. Hyett, J. Comput. Phys. 14, 159 (1974), DOI: 10.1016/0021-9991(74)90011-4. Crossref, Web of Science, Google Scholar
A. Whalley and P. Wilmott, Math. Finance 7, 307 (1997), DOI: 10.1111/1467-9965.00034. Crossref, Web of Science, Google Scholar