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.
@article{M2AN_2004__38_2_359_0,
author = {D\"uring, Bertram and Fourni\'e, Michel and J\"ungel, Ansgar},
title = {Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {38},
year = {2004},
pages = {359-369},
doi = {10.1051/m2an:2004018},
zbl = {1124.91031},
mrnumber = {2069151},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2004__38_2_359_0}
}
Düring, Bertram; Fournié, Michel; Jüngel, Ansgar. Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) pp. 359-369. doi : 10.1051/m2an:2004018. http://gdmltest.u-ga.fr/item/M2AN_2004__38_2_359_0/
[1] and, Discontinuous solutions of deterministic optimal stopping time problems. RAIRO Modél. Math. Anal. Numér. 21 (1987) 557-579. | Numdam | MR 921827 | Zbl 0629.49017
[2] and, Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 (1988) 1133-1148. | MR 957658 | Zbl 0674.49027
[3] and, Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance Stoch. 2 (1998) 369-397. | MR 1809526 | Zbl 0915.35051
[4] and, Convergence of approximation schemes for fully nonlinear second order equations. Asympt. Anal. 4 (1991) 271-283. | MR 1115933 | Zbl 0729.65077
[5] , and, Convergence of numerical schemes for parabolic equations arising in finance theory. Math. Models Meth. Appl. Sci. 5 (1995) 125-143. | MR 1315000 | Zbl 0822.65056
[6] and, The pricing of options and corporate liabilities. J. Polit. Econ. 81 (1973) 637-659. | MR 3363443 | Zbl 1092.91524
[7] and, Compact difference methods applied to initial-boundary value problems for mixed systems. Numer. Math. 73 (1996) 291-309. | MR 1389489 | Zbl 0856.65099
[8] and, Option replication in discrete time with transaction costs. J. Finance 47 (1973) 271-293.
[9] and, Bounds on process of contingent claims in an intertemporal economy with proportional transaction costs and general preferences. Finance Stoch. 3 (1999) 345-369. | MR 1842289 | Zbl 0935.91014
[10] and, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983) 1-42. | Zbl 0599.35024
[11] , and, User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1-67. | Zbl 0755.35015
[12] , and, European option pricing with transaction fees. SIAM J. Control Optim. 31 (1993) 470-493. | Zbl 0779.90011
[13] , and, High order compact finite difference schemes for a nonlinear Black-Scholes equation. Int. J. Appl. Theor. Finance 6 (2003) 767-789. | Zbl 1070.91024
[14] , Perfect option hedging for a large trader. Finance Stoch. 2 (1998) 115-141. | Zbl 0894.90017
[15] , Market illiquidity as a source of model risk in dynamic hedging, in Model Risk, R. Gibson Ed., RISK Publications, London (2000).
[16] and, Market liquidity, hedging and crashes. Amer. Econ. Rev. 80 (1990) 999-1021.
[17] and, Optimal replication of contingent claims under transaction costs. Rev. Future Markets 8 (1989) 222-239.
[18] , On high-order compact difference schemes. Russ. J. Numer. Anal. Math. Model. 15 (2000) 29-46. | Zbl 0951.65104
[19] , A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations. Ann. Scuola Norm. Sup. Pisa 16 (1989) 105-135. | Numdam | Zbl 0701.35052
[20] , Market manipulation, bubbles, corners and short squeezes. J. Financial Quant. Anal. 27 (1992) 311-336.
[21] and, Far field boundary conditions for Black-Scholes equations. SIAM J. Numer. Anal. 38 (2000) 1357-1368. | Zbl 0990.35013
[22] and, Introduction au calcul stochastique appliqué à la finance1997). | MR 1607509
[23] , 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). | MR 1882836 | Zbl 1004.91040
[24] , Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4 (1973) 141-183.
[25] , Convergence theorem for difference approximations of hyperbolic quasi-linear initial-boundary value problems. Math. Comput. 49 (1987) 445-459. | Zbl 0632.65106
[26] and, On feedback effects from hedging derivatives. Math. Finance 8 (1998) 67-84. | Zbl 0908.90016
[27] , High order difference schemes for unsteady one-dimensional diffusion-convection problems. J. Comp. Phys. 114 (1994) 59-76. | Zbl 0807.65096
[28] and, The feedback effect of hedging in illiquid markets. SIAM J. Appl. Math. 61 (2000) 232-272. | Zbl 1136.91407
[29] , and, There is no nontrivial hedging portfolio for option pricing with transaction costs. Ann. Appl. Probab. 5 (1995) 327-355. | Zbl 0837.90012
[30] , Accurate partial difference methods. II: Non-linear problems. Numer. Math. 6 (1964) 37-46. | Zbl 0143.38204
[31] and, Fourth order convergence of compact finite difference solver for 2D incompressible flow. Commun. Appl. Anal. 7 (2003) 171-191. | Zbl 1085.65515
[32] and, An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Math. Finance 7 (1997) 307-324. | Zbl 0885.90019