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Consistent stable difference schemes for nonlinear Black-Scholes equations modelling option pricingwith transaction costs

Published online by Cambridge University Press:  12 June 2009

Rafael Company ,
Lucas Jódar  and
José-Ramón Pintos
Rafael Company
Affiliation:
Instituto Universitario de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Edificio 8G, piso 2, P.O. Box 46022, Valencia, Spain. [email protected]; [email protected]; [email protected]
Lucas Jódar
Affiliation:
Instituto Universitario de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Edificio 8G, piso 2, P.O. Box 46022, Valencia, Spain. [email protected]; [email protected]; [email protected]
José-Ramón Pintos
Affiliation:
Instituto Universitario de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Edificio 8G, piso 2, P.O. Box 46022, Valencia, Spain. [email protected]; [email protected]; [email protected]
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Abstract

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.

Keywords

Nonlinear Black-Scholes equationoption pricingnumerical analysistransaction costs.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 6 , November 2009 , pp. 1045 - 1061
Copyright
© EDP Sciences, SMAI, 2009

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References

Avellaneda, M. and Parás, A., Dynamic hedging portfolios for derivative securities in the presence of large transaction costs. Appl. Math. Finance 1 (1994) 165–193.
G. Barles and H.M. Soner, Option pricing with transaction costs and a nonlinear Black–Scholes equation. Finance Stochast. 2 (1998) 369–397.
Boyle, P. and Vorst, T., Option replication in discrete time with transaction costs. J. Finance 47 (1973) 271–293. CrossRef
Company, R., Navarro, E., Pintos, J.R. and Ponsoda, E., Numerical solution of linear and nonlinear Black–Scholes option pricing equations. Comput. Math. Appl. 56 (2008) 813–821. CrossRef
Davis, M., Panis, V. and Zariphopoulou, T., European option pricing with transaction fees. SIAM J. Contr. Optim. 31 (1993) 470–493. CrossRef
J. Dewynne, S. Howinson and P. Wilmott, Option pricing: mathematical models and computations. Oxford Financial Press, Oxford (2000).
Düring, B., Fournier, M. and Jungel, A., Convergence of a high order compact finite difference scheme for a nonlinear Black–Scholes equation. ESAIM: M2AN 38 (2004) 359–369. CrossRef
Forsyth, P., Vetzal, K. and Zvan, R., A finite element approach to the pricing of discrete lookbacks with stochastic volatility. Appl. Math. Finance 6 (1999) 87–106. CrossRef
Harrison, J.M. and Pliska, S.R., Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes Appl. 11 (1981) 215–260. CrossRef
Hodges, S.D. and Neuberger, A., Optimal replication of contingent claims under transaction costs. Review of Futures Markets 8 (1989) 222–239.
Hoggard, T., Whalley, A.E. and Wilmott, P., Hedging option portfolios in the presence of transaction costs. Adv. Futures Options Research 7 (1994) 217–35.
Kangro, R. and Nicolaides, R., Far field boundary conditions for Black–Scholes equations. SIAM J. Numer. Anal. 38 (2000) 1357–1368. CrossRef
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).
Leland, H.E., Option pricing and replication with transactions costs. J. Finance 40 (1985) 1283–1301. CrossRef
Pironneau, O. and Hecht, F., Mesh adaption for the Black and Scholes equations. East-West J. Numer. Math. 8 (2000) 25–35.
Rigal, A., Numerical analisys of three-time-level finite difference schemes for unsteady diffusion-convection problems. J. Num. Meth. Engineering 30 (1990) 307–330. CrossRef
G.D. Smith, Numerical solution of partial differential equations: finite difference methods. Third Edition, Clarendon Press, Oxford (1985).
Soner, H.M., Shreve, S.E. and Cvitanic, J., There is no non-trivial hedging portfolio for option pricing with transaction costs. Ann. Appl. Probab. 5 (1995) 327–355. CrossRef
J.C. Strikwerda, Finite difference schemes and partial differential equations. Wadsworth & Brooks/Cole Mathematics Series (1989) 32–52.
D. Tavella and C. Randall, Pricing financial instruments – The finite difference method. John Wiley & Sons, Inc., New York (2000).
Whalley, A.E. and Wilmott, P., An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Math. Finance 7 (1997) 307–324. CrossRef