The Operator Splitting Method for Black-Scholes Equation

ABSTRACT

The Operator Splitting method is applied to differential equations occurring as mathematical models in financial models. This paper provides various operator splitting methods to obtain an effective and accurate solution to the Black-Scholes equation with appropriate boundary conditions for a European option pricing problem. Finally brief comparisons of option prices are given by different models.

The Operator Splitting method is applied to differential equations occurring as mathematical models in financial models. This paper provides various operator splitting methods to obtain an effective and accurate solution to the Black-Scholes equation with appropriate boundary conditions for a European option pricing problem. Finally brief comparisons of option prices are given by different models.

Cite this paper

nullY. Daoud and T. Öziş, "The Operator Splitting Method for Black-Scholes Equation,"*Applied Mathematics*, Vol. 2 No. 6, 2011, pp. 771-778. doi: 10.4236/am.2011.26103.

nullY. Daoud and T. Öziş, "The Operator Splitting Method for Black-Scholes Equation,"

References

[1] R. C. Merton, “Optimum Consumption and Portfolio Rules in a Continuous Time Model,” Journal of Economic Theory, Vol. 3, No. 4, 1971, pp. 373-413. doi:10.1016/0022-0531(71)90038-X

[2] R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Sciences, Vol. 4, No. 1, 1973, pp.141-183. doi:10.2307/3003143

[3] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-654. doi:10.1086/260062

[4] P. Wilmott, S. Howison and J. Dewynne, “The Mathematics of Financial Derivatives,” Cambridge University Press, Cambridge, 1997.

[5] P. Wilmott, J. Dewynne and S. Howison, “Option Pricing,” Oxford Financial Press, Oxford, 1993.

[6] P. Wilmott, J. Dewynne and S. Howison, “The Mathematics of Financial Derivatives: A Student Introduction,” Cambridge University Press, Cambridge, 1997.

[7] P. Wilmott, J. Dewynne and S. Howison, “Option Pricing: Mathematical Models and Computation,” Oxford Financial Press, Oxford, 1998.

[8] P. Wilmott, S. Howison and J. Dewynne, “The Mathematics of Financial Derivatives,” Cambridge University Press, Cambridge, 1997.

[9] D. Duffy, “Numerical Methods for Instrument Pricing,” John Wiley & Sons, Chichester, 2004.

[10] D. J. Duffy, “Financial Instrument Pricing in C++,” John Wiley & Sons, Chichester, 2004.

[11] J. Geiser, “Decomposition Methods for Differential Equations Theory and Applications,” Chapman Hall/CRC Press, London, 2009. doi:10.1201/9781439810972

[12] I. Farago and J. Geiser, “Iterative Operator Splitting Methods for Linear Problems,” International Journal of Computational Science and Engineering, Vol. 3, No. 4, 2007, pp. 255-263.

[13] P. Csomo’s, I. Farago and A. Havasi, “Weighted Sequential Splitting and Their Analysis,” Computers & Mathematics with Applications, Vol. 50, 2005, pp. 1017-1031. doi:10.1016/j.camwa.2005.08.004

[14] J. Geiser, “Linear and Quasi-Linear Iterative Splitting Methods: Theory and Applications,” International Mathematical Forum, Vol. 2, No. 49, 2007, pp. 2391-2416.

[1] R. C. Merton, “Optimum Consumption and Portfolio Rules in a Continuous Time Model,” Journal of Economic Theory, Vol. 3, No. 4, 1971, pp. 373-413. doi:10.1016/0022-0531(71)90038-X

[2] R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Sciences, Vol. 4, No. 1, 1973, pp.141-183. doi:10.2307/3003143

[3] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-654. doi:10.1086/260062

[4] P. Wilmott, S. Howison and J. Dewynne, “The Mathematics of Financial Derivatives,” Cambridge University Press, Cambridge, 1997.

[5] P. Wilmott, J. Dewynne and S. Howison, “Option Pricing,” Oxford Financial Press, Oxford, 1993.

[6] P. Wilmott, J. Dewynne and S. Howison, “The Mathematics of Financial Derivatives: A Student Introduction,” Cambridge University Press, Cambridge, 1997.

[7] P. Wilmott, J. Dewynne and S. Howison, “Option Pricing: Mathematical Models and Computation,” Oxford Financial Press, Oxford, 1998.

[8] P. Wilmott, S. Howison and J. Dewynne, “The Mathematics of Financial Derivatives,” Cambridge University Press, Cambridge, 1997.

[9] D. Duffy, “Numerical Methods for Instrument Pricing,” John Wiley & Sons, Chichester, 2004.

[10] D. J. Duffy, “Financial Instrument Pricing in C++,” John Wiley & Sons, Chichester, 2004.

[11] J. Geiser, “Decomposition Methods for Differential Equations Theory and Applications,” Chapman Hall/CRC Press, London, 2009. doi:10.1201/9781439810972

[12] I. Farago and J. Geiser, “Iterative Operator Splitting Methods for Linear Problems,” International Journal of Computational Science and Engineering, Vol. 3, No. 4, 2007, pp. 255-263.

[13] P. Csomo’s, I. Farago and A. Havasi, “Weighted Sequential Splitting and Their Analysis,” Computers & Mathematics with Applications, Vol. 50, 2005, pp. 1017-1031. doi:10.1016/j.camwa.2005.08.004

[14] J. Geiser, “Linear and Quasi-Linear Iterative Splitting Methods: Theory and Applications,” International Mathematical Forum, Vol. 2, No. 49, 2007, pp. 2391-2416.