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Convergence of a high-order compact finite difference scheme for a nonlinear Black–Scholes equation

Published online by Cambridge University Press:  15 March 2004

Bertram Düring ,
Michel Fournié  and
Ansgar Jüngel
Bertram Düring
Affiliation:
Fachbereich Mathematik und Informatik, Johannes Gutenberg-Universität Mainz, Germany, [email protected].
Michel Fournié
Affiliation:
UMR-CNRS 5640, Laboratoire MIP, Université Paul Sabatier, Toulouse, France.
Ansgar Jüngel
Affiliation:
Fachbereich Mathematik und Informatik, Johannes Gutenberg-Universität Mainz, Germany, [email protected].
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Abstract

A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.

Keywords

High-order compact finite differencesnumerical convergenceviscosity solutionfinancial derivatives.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 2 , March 2004 , pp. 359 - 369
Copyright
© EDP Sciences, SMAI, 2004

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References

Barles, G. and Perthame, B., Discontinuous solutions of deterministic optimal stopping time problems. RAIRO Modél. Math. Anal. Numér. 21 (1987) 557–579.
Barles, G. and Perthame, B., Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 (1988) 1133–1148.
Barles, G. and Soner, H.M., Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance Stoch. 2 (1998) 369–397. CrossRef
Barles, G. and Souganides, P.E., Convergence of approximation schemes for fully nonlinear second order equations. Asympt. Anal. 4 (1991) 271–283.
Barles, G., Daher, Ch. and Romano, M., Convergence of numerical schemes for parabolic equations arising in finance theory. Math. Models Meth. Appl. Sci. 5 (1995) 125–143.
Black, F. and Scholes, M., The pricing of options and corporate liabilities. J. Polit. Econ. 81 (1973) 637–659. CrossRef
Bodenmann, R. and Schroll, H.J., Compact difference methods applied to initial-boundary value problems for mixed systems. Numer. Math. 73 (1996) 291–309. CrossRef
Boyle, P. and Vorst, T., Option replication in discrete time with transaction costs. J. Finance 47 (1973) 271–293. CrossRef
Constantinides, G.M. and Zariphopoulou, T., Bounds on process of contingent claims in an intertemporal economy with proportional transaction costs and general preferences. Finance Stoch. 3 (1999) 345–369. CrossRef
Crandall, M. and Lions, P.L., Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983) 1–42. CrossRef
Crandall, M., Ishii, H. and Lions, P.L., User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1–67.
Davis, M., Panis, V. and Zariphopoulou, T., European option pricing with transaction fees. SIAM J. Control Optim. 31 (1993) 470–493. CrossRef
Düring, B., Fournié, M. and Jüngel, A., High order compact finite difference schemes for a nonlinear Black-Scholes equation. Int. J. Appl. Theor. Finance 6 (2003) 767–789.
Frey, R., Perfect option hedging for a large trader. Finance Stoch. 2 (1998) 115–141. CrossRef
R. Frey, Market illiquidity as a source of model risk in dynamic hedging, in Model Risk, R. Gibson Ed., RISK Publications, London (2000).
Genotte, G. and Leland, H., Market liquidity, hedging and crashes. Amer. Econ. Rev. 80 (1990) 999–1021.
Hodges, S.D. and Neuberger, A., Optimal replication of contingent claims under transaction costs. Rev. Future Markets 8 (1989) 222–239.
Il'in, V.P., On high-order compact difference schemes. Russ. J. Numer. Anal. Math. Model. 15 (2000) 29–46.
Ishii, H., A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations. Ann. Scuola Norm. Sup. Pisa 16 (1989) 105–135.
Jarrow, R., Market manipulation, bubbles, corners and short squeezes. J. Financial Quant. Anal. 27 (1992) 311–336. CrossRef
Kangro, P. and Nicolaides, R., Far field boundary conditions for Black-Scholes equations. SIAM J. Numer. Anal. 38 (2000) 1357–1368. CrossRef
D. Lamberton and B. Lapeyre, Introduction au calcul stochastique appliqué à la finance. 2e édn., Ellipses, Paris (1997).
J. Leitner, Continuous time CAPM, price for risk and utility maximization, in Mathematical Finance. Workshop of the Mathematical Finance Research Project, Konstanz, Germany, M. Kohlmann et al. Eds., Birkhäuser, Basel (2001).
Merton, R.C., Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4 (1973) 141–183.
Michelson, D., Convergence theorem for difference approximations of hyperbolic quasi-linear initial-boundary value problems. Math. Comput. 49 (1987) 445–459.
Platen, E. and Schweizer, M., On feedback effects from hedging derivatives. Math. Finance 8 (1998) 67–84. CrossRef
Rigal, A., High order difference schemes for unsteady one-dimensional diffusion-convection problems. J. Comp. Phys. 114 (1994) 59–76. CrossRef
Schönbucher, P. and Wilmott, P., The feedback effect of hedging in illiquid markets. SIAM J. Appl. Math. 61 (2000) 232–272.
Soner, H.M., Shreve, S.E. and Cvitanic, J., There is no nontrivial hedging portfolio for option pricing with transaction costs. Ann. Appl. Probab. 5 (1995) 327–355.
Strang, G., Accurate partial difference methods. II: Non-linear problems. Numer. Math. 6 (1964) 37–46. CrossRef
Wang, C. and Liu, J., Fourth order convergence of compact finite difference solver for 2D incompressible flow. Commun. Appl. Anal. 7 (2003) 171–191.
Whalley, A. and Wilmott, P., An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Math. Finance 7 (1997) 307–324. CrossRef