This paper deals with the numerical solution of nonlinear Black-Scholes equation modeling European vanilla call option pricing under transaction costs. Using an explicit finite difference scheme consistent with the partial differential equation valuation problem, a sufficient condition for the stability of the solution is given in terms of the stepsize discretization variables and the parameter measuring the transaction costs. This stability condition is linked to some properties of the numerical approximation of the Gamma of the option, previously obtained. Results are illustrated with numerical examples.
Mots-clés : nonlinear Black-Scholes equation, option pricing, numerical analysis, transaction costs
@article{M2AN_2009__43_6_1045_0,
author = {Company, Rafael and J\'odar, Lucas and Pintos, Jos\'e-Ram\'on},
title = {Consistent stable difference schemes for nonlinear {Black-Scholes} equations modelling option pricing with transaction costs},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {1045--1061},
publisher = {EDP-Sciences},
volume = {43},
number = {6},
year = {2009},
doi = {10.1051/m2an/2009014},
mrnumber = {2588432},
zbl = {1175.91071},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2009014/}
}
TY - JOUR AU - Company, Rafael AU - Jódar, Lucas AU - Pintos, José-Ramón TI - Consistent stable difference schemes for nonlinear Black-Scholes equations modelling option pricing with transaction costs JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2009 SP - 1045 EP - 1061 VL - 43 IS - 6 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2009014/ DO - 10.1051/m2an/2009014 LA - en ID - M2AN_2009__43_6_1045_0 ER -
%0 Journal Article %A Company, Rafael %A Jódar, Lucas %A Pintos, José-Ramón %T Consistent stable difference schemes for nonlinear Black-Scholes equations modelling option pricing with transaction costs %J ESAIM: Modélisation mathématique et analyse numérique %D 2009 %P 1045-1061 %V 43 %N 6 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2009014/ %R 10.1051/m2an/2009014 %G en %F M2AN_2009__43_6_1045_0
Company, Rafael; Jódar, Lucas; Pintos, José-Ramón. Consistent stable difference schemes for nonlinear Black-Scholes equations modelling option pricing with transaction costs. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 6, pp. 1045-1061. doi : 10.1051/m2an/2009014. http://archive.numdam.org/articles/10.1051/m2an/2009014/
[1] and , Dynamic hedging portfolios for derivative securities in the presence of large transaction costs. Appl. Math. Finance 1 (1994) 165-193.
[2] and , Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance Stochast. 2 (1998) 369-397. | MR | Zbl
[3] and , Option replication in discrete time with transaction costs. J. Finance 47 (1973) 271-293.
[4] , , and , Numerical solution of linear and nonlinear Black-Scholes option pricing equations. Comput. Math. Appl. 56 (2008) 813-821. | MR | Zbl
[5] , and , European option pricing with transaction fees. SIAM J. Contr. Optim. 31 (1993) 470-493. | MR | Zbl
[6] , and , Option pricing: mathematical models and computations. Oxford Financial Press, Oxford (2000). | Zbl
[7] , and , Convergence of a high order compact finite difference scheme for a nonlinear Black-Scholes equation. ESAIM: M2AN 38 (2004) 359-369. | Numdam | MR | Zbl
[8] , and , A finite element approach to the pricing of discrete lookbacks with stochastic volatility. Appl. Math. Finance 6 (1999) 87-106. | Zbl
[9] and , Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes Appl. 11 (1981) 215-260. | MR | Zbl
[10] and , Optimal replication of contingent claims under transaction costs. Review of Futures Markets 8 (1989) 222-239.
[11] , and , Hedging option portfolios in the presence of transaction costs. Adv. Futures Options Research 7 (1994) 217-35.
[12] and , Far field boundary conditions for Black-Scholes equations. SIAM J. Numer. Anal. 38 (2000) 1357-1368. | MR | Zbl
[13] , Continuous time CAPM, price for risk and utility maximization, in Mathematical Finance - Workshop of the Mathematical Finance Research Project, Konstanz, Germany, M. Kohlmann and S. Tang Eds., Birkhäuser, Basel (2001). | MR | Zbl
[14] , Option pricing and replication with transactions costs. J. Finance 40 (1985) 1283-1301.
[15] and , Mesh adaption for the Black and Scholes equations. East-West J. Numer. Math. 8 (2000) 25-35. | MR | Zbl
[16] , Numerical analisys of three-time-level finite difference schemes for unsteady diffusion-convection problems. J. Num. Meth. Engineering 30 (1990) 307-330. | MR | Zbl
[17] , Numerical solution of partial differential equations: finite difference methods. Third Edition, Clarendon Press, Oxford (1985). | MR | Zbl
[18] , and , There is no non-trivial hedging portfolio for option pricing with transaction costs. Ann. Appl. Probab. 5 (1995) 327-355. | MR | Zbl
[19] , Finite difference schemes and partial differential equations. Wadsworth & Brooks/Cole Mathematics Series (1989) 32-52. | MR | Zbl
[20] and , Pricing financial instruments - The finite difference method. John Wiley & Sons, Inc., New York (2000).
[21] and , An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Math. Finance 7 (1997) 307-324. | MR | Zbl
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