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.

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.

DOI : 10.1051/m2an/2009014
Classification : 35K55, 65M12, 39A10, 90A09
Mots clés : nonlinear Black-Scholes equation, option pricing, numerical analysis, transaction costs
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     title = {Consistent stable difference schemes for nonlinear {Black-Scholes} equations modelling option pricing with transaction costs},
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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] M. Avellaneda and A. Parás, Dynamic hedging portfolios for derivative securities in the presence of large transaction costs. Appl. Math. Finance 1 (1994) 165-193.

[2] G. Barles and H.M. Soner, Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance Stochast. 2 (1998) 369-397. | MR | Zbl

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

[4] R. Company, E. Navarro, J.R. Pintos and E. Ponsoda, Numerical solution of linear and nonlinear Black-Scholes option pricing equations. Comput. Math. Appl. 56 (2008) 813-821. | MR | Zbl

[5] M. Davis, V. Panis and T. Zariphopoulou, European option pricing with transaction fees. SIAM J. Contr. Optim. 31 (1993) 470-493. | MR | Zbl

[6] J. Dewynne, S. Howinson and P. Wilmott, Option pricing: mathematical models and computations. Oxford Financial Press, Oxford (2000). | Zbl

[7] B. Düring, M. Fournier and A. Jungel, 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] P. Forsyth, K. Vetzal and R. Zvan, A finite element approach to the pricing of discrete lookbacks with stochastic volatility. Appl. Math. Finance 6 (1999) 87-106. | Zbl

[9] J.M. Harrison and S.R. Pliska, Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes Appl. 11 (1981) 215-260. | MR | Zbl

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

[11] T. Hoggard, A.E. Whalley and P. Wilmott, Hedging option portfolios in the presence of transaction costs. Adv. Futures Options Research 7 (1994) 217-35.

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

[13] 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 and S. Tang Eds., Birkhäuser, Basel (2001). | MR | Zbl

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

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

[16] A. Rigal, 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] G.D. Smith, Numerical solution of partial differential equations: finite difference methods. Third Edition, Clarendon Press, Oxford (1985). | MR | Zbl

[18] H.M. Soner, S.E. Shreve and J. Cvitanic, There is no non-trivial hedging portfolio for option pricing with transaction costs. Ann. Appl. Probab. 5 (1995) 327-355. | MR | Zbl

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

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

[21] A.E. Whalley and P. Wilmott, 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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