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.

Classification: 35K55, 65M12, 39A10, 90A09

Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {43}, number = {6}, year = {2009}, pages = {1045-1061}, doi = {10.1051/m2an/2009014}, zbl = {1175.91071}, mrnumber = {2588432}, language = {en}, url = {http://www.numdam.org/item/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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 6, pp. 1045-1061. doi : 10.1051/m2an/2009014. http://www.numdam.org/item/M2AN_2009__43_6_1045_0/

[1] Dynamic hedging portfolios for derivative securities in the presence of large transaction costs. Appl. Math. Finance 1 (1994) 165-193.

and ,[2] Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance Stochast. 2 (1998) 369-397. | MR 1809526 | Zbl 0915.35051

and ,[3] Option replication in discrete time with transaction costs. J. Finance 47 (1973) 271-293.

and ,[4] Numerical solution of linear and nonlinear Black-Scholes option pricing equations. Comput. Math. Appl. 56 (2008) 813-821. | MR 2435587 | Zbl 1155.65370

, , and ,[5] European option pricing with transaction fees. SIAM J. Contr. Optim. 31 (1993) 470-493. | MR 1205985 | Zbl 0779.90011

, and ,[6] Option pricing: mathematical models and computations. Oxford Financial Press, Oxford (2000). | Zbl 0844.90011

, and ,[7] Convergence of a high order compact finite difference scheme for a nonlinear Black-Scholes equation. ESAIM: M2AN 38 (2004) 359-369. | Numdam | MR 2069151 | Zbl 1124.91031

, and ,[8] A finite element approach to the pricing of discrete lookbacks with stochastic volatility. Appl. Math. Finance 6 (1999) 87-106. | Zbl 1009.91030

, and ,[9] Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes Appl. 11 (1981) 215-260. | MR 622165 | Zbl 0482.60097

and ,[10] Optimal replication of contingent claims under transaction costs. Review of Futures Markets 8 (1989) 222-239.

and ,[11] Hedging option portfolios in the presence of transaction costs. Adv. Futures Options Research 7 (1994) 217-35.

, and ,[12] Far field boundary conditions for Black-Scholes equations. SIAM J. Numer. Anal. 38 (2000) 1357-1368. | MR 1790037 | Zbl 0990.35013

and ,[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 1882836 | Zbl 1004.91040

,[14] Option pricing and replication with transactions costs. J. Finance 40 (1985) 1283-1301.

,[15] Mesh adaption for the Black and Scholes equations. East-West J. Numer. Math. 8 (2000) 25-35. | MR 1757144 | Zbl 0995.91026

and ,[16] Numerical analisys of three-time-level finite difference schemes for unsteady diffusion-convection problems. J. Num. Meth. Engineering 30 (1990) 307-330. | MR 1064008 | Zbl 0714.76072

,[17] Numerical solution of partial differential equations: finite difference methods. Third Edition, Clarendon Press, Oxford (1985). | MR 827497 | Zbl 0576.65089

,[18] There is no non-trivial hedging portfolio for option pricing with transaction costs. Ann. Appl. Probab. 5 (1995) 327-355. | MR 1336872 | Zbl 0837.90012

, and ,[19] Finite difference schemes and partial differential equations. Wadsworth & Brooks/Cole Mathematics Series (1989) 32-52. | MR 1005330 | Zbl 0681.65064

,[20] Pricing financial instruments - The finite difference method. John Wiley & Sons, Inc., New York (2000).

and ,[21] An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Math. Finance 7 (1997) 307-324. | MR 1459062 | Zbl 0885.90019

and ,