A Numerical Discussion for the European Put Option Model
Yıl: 2021 Cilt: 14 Sayı: 1 Sayfa Aralığı: 132 - 140 Metin Dili: İngilizce DOI: 10.18185/erzifbed.758426 İndeks Tarihi: 02-03-2022
A Numerical Discussion for the European Put Option Model
Öz: 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 analyse these models. This study will mainly focus on investigating the behaviour 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. It has seen that the results with minimal computational effort are sufficiently accurate.
Anahtar Kelime: Avrupa Tipi Satış Opsiyonu Modeli için Nümerik bir Değerlendirme
Öz: 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 hassasiyetli yaklaşımlarına izin veren dördüncü mertebeden bir sonlu fark (FD4) şeması kullanılarak yapılmıştır. Zaman ayrıklaştırması için üç sayısal teknik kullanılmıştır: Kuvvetli kararlılık koruyan RungeKutta (SSPRK3), dördüncü mertebe Runge Kutta (RK4) ve tek adımlı bir yöntem. Birleştirilmiş yöntemlerle üretilen sonuçlar literatürde mevcut olan çözüm ve tam çözüm ile karşılaştırılmıştır. Sonuçların minimum hesaplama çabasıyla yeterince hassas olduğu görülmüştür.
Anahtar Kelime: Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık
- Ankudinova, J. and Ehrhardt, M. 2008. “On the Numerical Solution of Nonlinear Black Scholes Equations”, Computers and Mathematics with Applications, 56, 799-812.
- Ascher, U. M., Mattheij, R.M.M., Russell, R. D. (1995). “Numerical Solution of Boundary Value Problems for Ordinary Differential Equations”, Printice-Hall, Englewood Cliffs, NJ.,1-595.
- Barles, G., Soner, H.M. 1998. “Option Pricing with Transaction Costs and a Nonlinear Black-Scholes Equation”, Finance and Stochastic, 2, 369-397.
- Black, F., Scholes, M. 1973. “The Pricing of Options and Other Corporate Liabilities”, Journal of Economics and Political Economy, 81, 637-654.
- Boyle, P.P., Worst, T. 1992. “Option Replication in Discrete Time with Transaction Costs”, Journal of Finance, XLVII, 271-293.
- Chen, W., Wang, S. 2020. “A 2nd-Order ADI Finite Difference Method for a 2D Fractional Black-Scholes Equation governing European Two Asset Option Pricing”, Mathematics and Computers in Simulation, 171, 279-293.
- Company, R., Navarro, E., Pintos, J.R, Ponsoda, E. 2008. “Numerical Solution of Linear and Nonlinear Black-Scholes Option Pricing Equations”, Computers and Mathematics with Applications, 56, 813-821.
- Company, R., Jodar, L., 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., 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., England, 1-423.
- Dura, G., 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., 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., 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, 271, 247-266.
- Gottlieb, C., Shu, W., Tadmor, E. 2001. “Strong Stability-Preserving High Order Time Discretization Methods”, SIAM Review, 43, 89-112.
- Gulen S., Popescu, C., Sari, M. 2019. “A New Approach for the Black-Scholes Model with Linear and Nonlinear Volatilities”, Mathematics, 7, 760.
- Heo, Y., Han, H., Jang, H., Choi, Y., Kim, J. 2019. “Finite Difference Method for the TwoDimensional Black-Scholes Equation with a Hybrid Boundary Condition”, Journal of the Korea Society for Industrial and Applied Mathematics, 23(1), 19-30.
- 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. , Kim, J. 2018. “Finite Difference Method for the Black-Scholes Equation Without Boundary Conditions”, Computational Economics, 51, 961-972.
- Kim, S., Jeong, D., Lee, C., Kim, J.,2020. “Finite Difference Method for the Multi-Asset Black-Scholes Equation”, Mathematics, 8(3), 391.
- Koleva, M., Mudzimbabwe, W., 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.
- Leentvaar, C.C. 2003. “Numerical Solution of the Black–Scholes Equation with a Small Number of Grid Points” (Master’s Thesis), Delft University of Technology, Delft, The Netherlands, 1-88.
- Leland, H.E., 1985. “Option Pricing and Replication with Transaction Costs”, Journal of Finance, 40, 1283-1301.
- Lesmana, D.C., 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., 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., Fugger, J. 2015. “Finite Difference Schemes for a Nonlinear BlackScholes 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.
- Rao, S.C.S. 2018. “Numerical Solution of Generalized Black-Scholes Model”, Applied Mathematics and Computation, 321, 401-421.
- Raol, P., Prasad Goura, V.M.K. 2020. “A New High-Order Compact Finite Difference Method for Genralized Black-Scholes Partial Differential Equation: European Call Option”, Journal of Computational and Applied Mathematics, 363, 464-484.
- Rigal, A. 1994. “High Order Compact Finite Difference Schemes for Unsteady OneDimensional 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., 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., Randall, C. (2000). Pricing Financial Instruments- The Finite Difference Method”, John Wiley & Sons, Inc.,New York.
- Wilmott, P., Howison, S., Dewynne, J. (1995). “The Mathematics of Financial Derivatives”, Cambridge University Press, New York.
- Yan, R., Yang, X., Sun, S., 2020. “A class of explicit-implicit alternating parallel difference methods for the two-dimensional Black-Scholes equation”, International Journal of Computer Mathematics, https://doi.org/10.1080/00207160.2020.1804 554
- Zhao, J. Davison, M., Corless, R. M. 2007. “Compact Finite Difference Method for American Option Pricing”. Journal of Computational and Applied Mathematics, 206, 306-321.
| APA | Gulen S (2021). A Numerical Discussion for the European Put Option Model. , 132 - 140. 10.18185/erzifbed.758426 |
| Chicago | Gulen Seda A Numerical Discussion for the European Put Option Model. (2021): 132 - 140. 10.18185/erzifbed.758426 |
| MLA | Gulen Seda A Numerical Discussion for the European Put Option Model. , 2021, ss.132 - 140. 10.18185/erzifbed.758426 |
| AMA | Gulen S A Numerical Discussion for the European Put Option Model. . 2021; 132 - 140. 10.18185/erzifbed.758426 |
| Vancouver | Gulen S A Numerical Discussion for the European Put Option Model. . 2021; 132 - 140. 10.18185/erzifbed.758426 |
| IEEE | Gulen S "A Numerical Discussion for the European Put Option Model." , ss.132 - 140, 2021. 10.18185/erzifbed.758426 |
| ISNAD | Gulen, Seda. "A Numerical Discussion for the European Put Option Model". (2021), 132-140. https://doi.org/10.18185/erzifbed.758426 |
| APA | Gulen S (2021). A Numerical Discussion for the European Put Option Model. Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 14(1), 132 - 140. 10.18185/erzifbed.758426 |
| Chicago | Gulen Seda A Numerical Discussion for the European Put Option Model. Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi 14, no.1 (2021): 132 - 140. 10.18185/erzifbed.758426 |
| MLA | Gulen Seda A Numerical Discussion for the European Put Option Model. Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol.14, no.1, 2021, ss.132 - 140. 10.18185/erzifbed.758426 |
| AMA | Gulen S A Numerical Discussion for the European Put Option Model. Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi. 2021; 14(1): 132 - 140. 10.18185/erzifbed.758426 |
| Vancouver | Gulen S A Numerical Discussion for the European Put Option Model. Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi. 2021; 14(1): 132 - 140. 10.18185/erzifbed.758426 |
| IEEE | Gulen S "A Numerical Discussion for the European Put Option Model." Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 14, ss.132 - 140, 2021. 10.18185/erzifbed.758426 |
| ISNAD | Gulen, Seda. "A Numerical Discussion for the European Put Option Model". Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi 14/1 (2021), 132-140. https://doi.org/10.18185/erzifbed.758426 |
