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A Numerical Discussion for the European Put Option Model

Year 2021, Volume: 14 Issue: 1, 132 - 140, 31.03.2021
https://doi.org/10.18185/erzifbed.758426

Abstract

The Black-Scholes equations have been increasingly popular over the last three decades since they provide more practical information for optional behaviours. Therefore, effective methods have been needed to analyze these models. This study will focus mainly on investigating the behavior of the Black-Scholes equation for the European put option pricing model. To achieve this, numerical solutions of the Black-Scholes European option pricing model are produced by three combined methods. Spatial discretization of the Black-Scholes model is performed using a fourth-order finite difference (FD4) scheme that allows a highly accurate approximation of the solutions. For the time discretization, three numerical techniques are proposed: a strong-stability preserving Runge Kutta (SSPRK3), a fourth-order Runge Kutta (RK4) and a one-step method. The results produced by the combined methods have been compared with available literature and the exact solution.

Thanks

I would like to thank Prof. Dr. Murat Sari for his valuable suggestions to improve this article.

References

  • Ankudinova, J. and Ehrhardt, M. 2008. “On the Numerical Solution of Nonlinear Black-Scholes Equation”, Computers and Mathematics with Applications, 56, 799-812.
  • Ascher, U. M., Mattheij, R.M.M. and Russell, R. D. (1995). “Numerical Solution of Boundary Value Problems for Ordinary Differential Equations”, Prentice-Hall, Englewood Cliffs, NU, 208-209.
  • Barles, G. and Soner, H. M. 1998. “Option Pricing with Transaction Costs and a Nonlinear Black- Scholes Equation”, Finance and Stochastic, 2, 369-397.
  • Black, F. and Scholes, M. 1973. “The Pricing of Options and Other Corporate Liabilities”, Journal of Economics and Political Economy, 81, 637-654.
  • Boyle, P. P. and Worst, T. 1992. “Option Replication in Discrete Time with Transaction Costs”, Journal of Finance, XLVII, 271-293.
  • Company, R., Navarro, E., Pintos, J.R. and Ponsoda, E. 2008. “Numerical Solution of Linear and Nonlinear Black-Scholes Option Pricing Equations”, Computers&Mathematics with Applications, 56, 813-821.
  • Company, R., Jodar, L. and Pintos, J. R. 2009. “A Numerical Method for European Option Pricing with Transaction Costs Nonlinear Equation”, Mathematical and Computer Modelling, 50, 910-920.
  • Courtadon, G. A. 1982. “A More Accurate Finite Difference Approximations for the Valuation of Options”, Journal of Financial and Quantitative Analysis, XVII, 697-703.
  • Cox, C. and Ross, S.A. 1976. “The Valuation of Options for Alternative Stochastic Process”, Journal of Financial Economics, 145-146.
  • Duffy, D., 1976. Finite Difference Methods in Financial Engineering- A Partial Differential Equation Approach. John Wiley&Sons Ltd.
  • Dura, G. and Moşneagu, A. M. 2010. “Numerical Approximation of Black-Scholes Equation”, Analele Ştiintifice Ale Universitatii AI.I Cuza Din Iaşi (S.N) Matematica, 5, 39-64.
  • Düring, B., Fournie, M. and Jüngel, A. 2003. “High Order Compact Finite Difference Schemes for A Nonlinear Black-Scholes Equation”, International Journal of Theoretical and Applied Finance, 6, 767-789.
  • Düring, B., Fournie, M. and Heuer, C. 2014. “High Order Compact Finite Difference Schemes for Option Pricing in Stochastic Volatility Models on Non-uniform Grids”, Journal of Computational and Applied Mathematics, 247-266.
  • Gottlieb, C., Shu,W. and Tadmor, E. 2001. “Strong Stability-Preserving High Order Time Discretization Methods”, SIAM Review, 43, 89-112.
  • Gulen S., Popescu, C. and Sari, M. 2019. “A New Approach for the Black-Scholes Model with Linear and Nonlinear Volatilities”, Mathematics, 7, 760.
  • Heston, S. 1993. “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options”, The Review of Financial Studies, 6, 327-343.
  • Jeong, D., Yoo, M. and Kim, J. 2018. “Finite Difference Method for the Black-Scholes Equation Without Boundary Conditions”, Computational Economics, 51, 961-972.
  • Koleva, M., Mudzimbabwe, W. and Vulkov, L. 2016. “Fourth Order Compact Finite Difference Schemes for a Parabolic Ordinary System of European Option Pricing Liquidity Shocks Model”, Numerical Algorithms, 74, 59-75.
  • Kusuoka, S. 1995. “Limit Theorem on Option Replication with Transaction Costs”, Annals of Applied Probability, 5, 198-221.
  • Leland, H.E. 1985. “Option Pricing and Replication with Transaction Costs”, Journal of Finance, 40, 1283-1301.
  • Lesmana, D.C. and Wang, S. 2013. “An Upwind Finite Difference Method for A Nonlinear Black-Scholes Equation Governing European Option Valuation Under Transaction Cost”, Applied Mathematics and Computation, 219, 8811-8828.
  • Liao, W. and Khaliq, A. M. Q. 2009. “High Order Compact Scheme for Solving Nonlinear Black-Scholes Equation with Transaction Cost”, International Journal of Computer Mathematics, 86, 1009-1023.
  • Mashayekhi, S. and Fugger, J. 2015. “Finite Difference Schemes for a Nonlinear Black-Scholes Model with Transaction Cost and Volatility Risk”, Acta Mathematica Universitasis Comenianae, 84, 255-266.
  • Merton, R.C. 1973. “Theory of Optional Option Pricing”, The Bell Journal of Economics and Management Science, 1, 141-183.
  • Rigal, A. 1994. “High Order Compact Finite Difference Schemes for Unsteady One-Dimensional Diffusion- Convection Problems”, Journal of Computational Physics, 114, 59-76.
  • Schwartz, E. 1977. “The Valuation of Warrants: Implementing a New Approach”, Journal of Financial Economics, 4, 79-93.
  • Tangman, D.Y., Gopaul, S. and Bhuruth, M. 2008. “A Fast High Order Finite Difference Algorithm for Pricing American Options”, Journal of Computational and Applied Mathematics, 222, 17-29.
  • Tavella, D. and Randall, C. (2000). “Pricing Financial Instruments- The Finite Difference Method, John Wiley & Sons, Inc.,New York.
  • Wilmott, P., Howison, S. and Dewynne, J. (1995). “The Mathematics of Financial Derivatives”, Cambridge University Press, New York.
  • Zhao, J. Davison, M. and Corless, R. M. 2007. “Compact Finite Difference Method for American Option Pricing”, Journal of Computational and Applied Mathematics, 206, 306-321.

Avrupa Tipi Satış Opsiyonu Modeli İçin Nümerik Bir Tartışma

Year 2021, Volume: 14 Issue: 1, 132 - 140, 31.03.2021
https://doi.org/10.18185/erzifbed.758426

Abstract

Black-Scholes denklemleri opsiyon davranışlarında pratik bilgiler sağladığından son otuz yılda daha popüler hale gelmiştir. Bu nedenle, bu modelleri analiz etmek için etkili yöntemlere ihtiyaç duyulmaktadır. Bu çalışma temel olarak Avrupa tipi satış opsiyonu fiyatlama modeli için Black-Scholes denkleminin davranışını araştırmaya odaklanmıştır. Bunun için, Black-Scholes Avrupa tipi opsiyon fiyatlama modelinin sayısal çözümleri üç birleştirilmiş yöntem ile üretilmiştir. Black-Scholes modelinin uzaysal ayrıklaştırması, çözümlerin yüksek hassasiyeli yaklaşımlaşımlarına izin veren dördüncü mertebeden bir sonlu fark (FD4) kullanılarak yapılmıştır. Zaman ayrıklaştırması için üç sayısal teknik kullanılmıştır: Güçlü kararlılık koruyan Runge Kutta (SSPRK3), dördüncü mertebe Runge Kutta (RK4) ve tek adımlı yöntem. Birleştirilmiş yöntemlerle üretilen sonuçlar mevcut literatür çözümü ve tam çözüm ile karşılaştırılmıştır.

References

  • Ankudinova, J. and Ehrhardt, M. 2008. “On the Numerical Solution of Nonlinear Black-Scholes Equation”, Computers and Mathematics with Applications, 56, 799-812.
  • Ascher, U. M., Mattheij, R.M.M. and Russell, R. D. (1995). “Numerical Solution of Boundary Value Problems for Ordinary Differential Equations”, Prentice-Hall, Englewood Cliffs, NU, 208-209.
  • Barles, G. and Soner, H. M. 1998. “Option Pricing with Transaction Costs and a Nonlinear Black- Scholes Equation”, Finance and Stochastic, 2, 369-397.
  • Black, F. and Scholes, M. 1973. “The Pricing of Options and Other Corporate Liabilities”, Journal of Economics and Political Economy, 81, 637-654.
  • Boyle, P. P. and Worst, T. 1992. “Option Replication in Discrete Time with Transaction Costs”, Journal of Finance, XLVII, 271-293.
  • Company, R., Navarro, E., Pintos, J.R. and Ponsoda, E. 2008. “Numerical Solution of Linear and Nonlinear Black-Scholes Option Pricing Equations”, Computers&Mathematics with Applications, 56, 813-821.
  • Company, R., Jodar, L. and Pintos, J. R. 2009. “A Numerical Method for European Option Pricing with Transaction Costs Nonlinear Equation”, Mathematical and Computer Modelling, 50, 910-920.
  • Courtadon, G. A. 1982. “A More Accurate Finite Difference Approximations for the Valuation of Options”, Journal of Financial and Quantitative Analysis, XVII, 697-703.
  • Cox, C. and Ross, S.A. 1976. “The Valuation of Options for Alternative Stochastic Process”, Journal of Financial Economics, 145-146.
  • Duffy, D., 1976. Finite Difference Methods in Financial Engineering- A Partial Differential Equation Approach. John Wiley&Sons Ltd.
  • Dura, G. and Moşneagu, A. M. 2010. “Numerical Approximation of Black-Scholes Equation”, Analele Ştiintifice Ale Universitatii AI.I Cuza Din Iaşi (S.N) Matematica, 5, 39-64.
  • Düring, B., Fournie, M. and Jüngel, A. 2003. “High Order Compact Finite Difference Schemes for A Nonlinear Black-Scholes Equation”, International Journal of Theoretical and Applied Finance, 6, 767-789.
  • Düring, B., Fournie, M. and Heuer, C. 2014. “High Order Compact Finite Difference Schemes for Option Pricing in Stochastic Volatility Models on Non-uniform Grids”, Journal of Computational and Applied Mathematics, 247-266.
  • Gottlieb, C., Shu,W. and Tadmor, E. 2001. “Strong Stability-Preserving High Order Time Discretization Methods”, SIAM Review, 43, 89-112.
  • Gulen S., Popescu, C. and Sari, M. 2019. “A New Approach for the Black-Scholes Model with Linear and Nonlinear Volatilities”, Mathematics, 7, 760.
  • Heston, S. 1993. “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options”, The Review of Financial Studies, 6, 327-343.
  • Jeong, D., Yoo, M. and Kim, J. 2018. “Finite Difference Method for the Black-Scholes Equation Without Boundary Conditions”, Computational Economics, 51, 961-972.
  • Koleva, M., Mudzimbabwe, W. and Vulkov, L. 2016. “Fourth Order Compact Finite Difference Schemes for a Parabolic Ordinary System of European Option Pricing Liquidity Shocks Model”, Numerical Algorithms, 74, 59-75.
  • Kusuoka, S. 1995. “Limit Theorem on Option Replication with Transaction Costs”, Annals of Applied Probability, 5, 198-221.
  • Leland, H.E. 1985. “Option Pricing and Replication with Transaction Costs”, Journal of Finance, 40, 1283-1301.
  • Lesmana, D.C. and Wang, S. 2013. “An Upwind Finite Difference Method for A Nonlinear Black-Scholes Equation Governing European Option Valuation Under Transaction Cost”, Applied Mathematics and Computation, 219, 8811-8828.
  • Liao, W. and Khaliq, A. M. Q. 2009. “High Order Compact Scheme for Solving Nonlinear Black-Scholes Equation with Transaction Cost”, International Journal of Computer Mathematics, 86, 1009-1023.
  • Mashayekhi, S. and Fugger, J. 2015. “Finite Difference Schemes for a Nonlinear Black-Scholes Model with Transaction Cost and Volatility Risk”, Acta Mathematica Universitasis Comenianae, 84, 255-266.
  • Merton, R.C. 1973. “Theory of Optional Option Pricing”, The Bell Journal of Economics and Management Science, 1, 141-183.
  • Rigal, A. 1994. “High Order Compact Finite Difference Schemes for Unsteady One-Dimensional Diffusion- Convection Problems”, Journal of Computational Physics, 114, 59-76.
  • Schwartz, E. 1977. “The Valuation of Warrants: Implementing a New Approach”, Journal of Financial Economics, 4, 79-93.
  • Tangman, D.Y., Gopaul, S. and Bhuruth, M. 2008. “A Fast High Order Finite Difference Algorithm for Pricing American Options”, Journal of Computational and Applied Mathematics, 222, 17-29.
  • Tavella, D. and Randall, C. (2000). “Pricing Financial Instruments- The Finite Difference Method, John Wiley & Sons, Inc.,New York.
  • Wilmott, P., Howison, S. and Dewynne, J. (1995). “The Mathematics of Financial Derivatives”, Cambridge University Press, New York.
  • Zhao, J. Davison, M. and Corless, R. M. 2007. “Compact Finite Difference Method for American Option Pricing”, Journal of Computational and Applied Mathematics, 206, 306-321.
There are 30 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Makaleler
Authors

Seda Gülen 0000-0001-7092-0628

Publication Date March 31, 2021
Published in Issue Year 2021 Volume: 14 Issue: 1

Cite

APA Gülen, S. (2021). A Numerical Discussion for the European Put Option Model. Erzincan University Journal of Science and Technology, 14(1), 132-140. https://doi.org/10.18185/erzifbed.758426