Crank-Nicolson Finite Difference Method with Sobolev Space Energy Estimate Theorem for Capital Market Prices
Keywords:
Stock prices, Crank-Nicolson, Option pricing, BS PDE and Put Option
Abstract
In this study, we have Black-Scholes analytic formula and Crank- Nicolson (CN) finite difference method for valuation of European put option which has earned the interests of researchers for determining both analytic and approximate solutions to Partial Differential Equations (PDEs) with Sobolev space energy estimate theorem. The simulations of analytical and numerical were effectively carried out. The results showed as follows: increase in volatility increases the values of option for both BS and CN prices, there are significant difference between BS and CN due to the changes of stock volatility, a little increase in the initial stock prices significantly increases the value of put option, when the strike price is greater than the initial stock price it increases put option values, Sobolev space energy estimates were used as asset value function to estimate asset prices at different maturity periods. Finally, the graphical solutions and comparisons of other parameters were discussed all in this paper which is informative to investors for the proper investment plans.
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References
Amadi, I. U., Azor, P. A. and Chims, B. E. (2020). Crank-Nicolson Analysis of Black-
Scholes Partial Differential Equation for Stock Market Prices. Academia Arena.12(1):1-
22.
Amadi, I.U.,Osu. B.O. and Davies, I.(2022). A Solution to Linear Black-Scholes Second order Parabolic Eqaution in Sobolev Spaces. International journal of Mathematics and Computer Research,10, issue 10,2938-2946.
Amadi, I.U., Davies, I. Osu , B. O. and Essi, I.D. (2022).Weak Estimation of Asset Value Function of Boundary Value Problem Arising in Financial Market, Asian journal of Pure and Applied Mathematics, 4(3),600-615.
Ankudinova, J. and M. Ehrhardt, (2008). On the numerical solution of non linear Black-Scholes equations. Comput Math. Applic:, 56:799-812. DO1: 10. 1016/j.camwa. 2008.02.005. arising in the pricing of contingent claims. Journal of financial and quantitative analysis.
Black F, Scholes M. (1973). The Pricing of Options and Corporate Liabilities. The Journal
of Political Economy. 81:637-654.
Brennan, M. and Schwartz, E. (1978). Finite Difference Methods and Jump Processes
arising in the pricing of contingent claims. Journal of financial and quantitative analysis.
5(4):461-474.
Cerna, D., V. Pasheva, N. Popivanou and G. Venlcov, (2016). Numerical solution of the Black- Scholes equation. Using cubic spline wavelets proceedings of the AIP conference, (IPC’ 16), AIP Publishing, USA DO1: 10.1063/1. 4968447.
Company, R., A. Gonzalez and L. Jodar, (2008). Numerical solution of modified Black-Scholes equation pricing stock options with discrete dividend. Math Comput. Modell., 44: 1058-1068. DO1: 10/1016/j.mcm. 2006.03.009.
Cortes, J., L, Jodar, R. Sala and P. Sevilla-Peris, (2005). Exact and numerical solution of Black–Scholes matrix equation. Applied Math. Comput., 160: 607-613. DO1: 10. 1111/j. 1475-6803. 1996. tb00592.
Diperna ,R.J and Lions, P.L. (1989) Ordinary Differential Equations, transport theory and Sobolev Spaces”. Invention maths. 98 ,511.
Dremkova, E., and Ehrhardt, M. (2011). A high-order compact method for nonlinear Black–Scholes option pricing equations of American options. International Journal of Computer Mathematics, 88(13), 2782-2797.
Fadugba,S.E and Nwozo, C.R (2013).Crank Nicolson Finite Difference Method for the valuation of options. The pacific journal of science and technology, 14,2 ,136-146.
Fadugba, S. E. and Ayegbusi, F. D. (2020). Bilaterial Risky Partial Differential Equation
Model for European Style Option. Journal of Applied Sciences. 20(3): 104-108.
Fadugba, S. E., Chukwuma, N. and Teniola, B. (2012). The Comparative Study of Finite Difference Method and monte Carlo Method for Pricing European Option.
Mathematical theory and Modelling. 2(2), 53-67.
George ,K.K. and Kenneth ,K.L.(2019). Pricing a European Put Option by numerical methods. International journal of scientific research publications, 9,issue 11,2250- 3153.
Heston.S.I. (1993). A closed form solution for options with stochastic volatility with application to bond and currency option,Rev. Financial Studies, 6 ,327.
Hull, J. C. (2012). Option. Futures and other Derivatives. Pearson Education Inc ;7.
Hull, J. C. (2013). Options. Futures and other Derivatives. (5.Ed) Edition: London Prentice Hall International.
Macbeth, J. and Merville, L. (1979). An Empirical Examination of the Black-Scholes Call
Option Pricing Model. Journal of Finance. 34(5):1173-1186.
Nwobi, F. N, Annorizie, M. N. and Amadi, I. U. (2019). Crank- Nicolson Finite Difference
Method in Valuation of Options. Communications in Mathematical Finance. 8(1):93- 122.
B.O Osu. (2010).A stochastic model of the variation of the capital market price.International Journal of trade, Economics and Finance, 1, 297.
B.O. Osu and C. Olunkwa, (2014).Weak solution of Black Scholes Equation option pricing with transaction costs. International journal of applied mathematics.1 , 43.
Osu, B.O and Amadi, I U.(2022).Existence of Weak Solution With Some Stochastic Hyperbolic Partial Differential Equation. International journal of Mathematical Analysis and Modeling,5,issue 2,13-23.
Osu, B.O , Amadi, I.U. and Davies, I.(2022).An Approximation of Linear Evolution Equation with Stochastic Partial Derivative in Sobolev Space. Asian journal of Pure and Applied Mathematics, International journal of Mathematical Analysis and Modeling, 5,issue 3,13-23
Osu, B.O , Isaac, D.E and Amadi, I. U. (2022).Solutions to Stochastic Boundary Value Problems in Sobolev Spaces with its Implications in Time-Varying Investments. Asian journal of Pure and Applied Mathematics, 4(3): 581-599.
Song, L. and W. Wang, (2013). Solution of the fractional difference Black – Scholes option pricing model by finite difference method. Abs Applied Anal., 2013: 1-11. 001:10. 155/2013/194286.
Uddin, M. K. S.,Siddiki,M.N.A.A and M. A. Hossain, (2015). Numerical solution of a linear Black – Scholes models: A comparative overview. Journal of Engineering.
Razali H., (2006) The pricing efficiency of equity warrants: A Malaysian case. ICFAI journalof derivatives markets.3(3):6-22.
Rinalini, K. P. (2006). Effectiveness of the Black-Scholes Model Pricing Options in Indian
Option Market. The ICFAI Journal of Derivatives Markets. 6-19.
Wokoma, D. S.A., Amadi, I. U. and Azor, P. A. (2020). Estimation of Stock Prices using
Black Scholes Partial Differential Equation for Put Options. African Journal of
Mathematics and Statistics Studies. 3(2): 80-89.
Yueng .L.T.(2012).Crank-Nicolson scheme for Asian option. Msc thesis Department of Mathematical and actuarial Sciences ,faculty of Engineering and Science. University of Tunku Abdul Rahman. 1-97.
Zhang. H., F. Liu, I. Turner and S. Chen, (2016). The numerical simulation of the tempered fractional Black -Scholes equation for European double barrier option. Applied Math. Model; 40: 5819-5834.
Scholes Partial Differential Equation for Stock Market Prices. Academia Arena.12(1):1-
22.
Amadi, I.U.,Osu. B.O. and Davies, I.(2022). A Solution to Linear Black-Scholes Second order Parabolic Eqaution in Sobolev Spaces. International journal of Mathematics and Computer Research,10, issue 10,2938-2946.
Amadi, I.U., Davies, I. Osu , B. O. and Essi, I.D. (2022).Weak Estimation of Asset Value Function of Boundary Value Problem Arising in Financial Market, Asian journal of Pure and Applied Mathematics, 4(3),600-615.
Ankudinova, J. and M. Ehrhardt, (2008). On the numerical solution of non linear Black-Scholes equations. Comput Math. Applic:, 56:799-812. DO1: 10. 1016/j.camwa. 2008.02.005. arising in the pricing of contingent claims. Journal of financial and quantitative analysis.
Black F, Scholes M. (1973). The Pricing of Options and Corporate Liabilities. The Journal
of Political Economy. 81:637-654.
Brennan, M. and Schwartz, E. (1978). Finite Difference Methods and Jump Processes
arising in the pricing of contingent claims. Journal of financial and quantitative analysis.
5(4):461-474.
Cerna, D., V. Pasheva, N. Popivanou and G. Venlcov, (2016). Numerical solution of the Black- Scholes equation. Using cubic spline wavelets proceedings of the AIP conference, (IPC’ 16), AIP Publishing, USA DO1: 10.1063/1. 4968447.
Company, R., A. Gonzalez and L. Jodar, (2008). Numerical solution of modified Black-Scholes equation pricing stock options with discrete dividend. Math Comput. Modell., 44: 1058-1068. DO1: 10/1016/j.mcm. 2006.03.009.
Cortes, J., L, Jodar, R. Sala and P. Sevilla-Peris, (2005). Exact and numerical solution of Black–Scholes matrix equation. Applied Math. Comput., 160: 607-613. DO1: 10. 1111/j. 1475-6803. 1996. tb00592.
Diperna ,R.J and Lions, P.L. (1989) Ordinary Differential Equations, transport theory and Sobolev Spaces”. Invention maths. 98 ,511.
Dremkova, E., and Ehrhardt, M. (2011). A high-order compact method for nonlinear Black–Scholes option pricing equations of American options. International Journal of Computer Mathematics, 88(13), 2782-2797.
Fadugba,S.E and Nwozo, C.R (2013).Crank Nicolson Finite Difference Method for the valuation of options. The pacific journal of science and technology, 14,2 ,136-146.
Fadugba, S. E. and Ayegbusi, F. D. (2020). Bilaterial Risky Partial Differential Equation
Model for European Style Option. Journal of Applied Sciences. 20(3): 104-108.
Fadugba, S. E., Chukwuma, N. and Teniola, B. (2012). The Comparative Study of Finite Difference Method and monte Carlo Method for Pricing European Option.
Mathematical theory and Modelling. 2(2), 53-67.
George ,K.K. and Kenneth ,K.L.(2019). Pricing a European Put Option by numerical methods. International journal of scientific research publications, 9,issue 11,2250- 3153.
Heston.S.I. (1993). A closed form solution for options with stochastic volatility with application to bond and currency option,Rev. Financial Studies, 6 ,327.
Hull, J. C. (2012). Option. Futures and other Derivatives. Pearson Education Inc ;7.
Hull, J. C. (2013). Options. Futures and other Derivatives. (5.Ed) Edition: London Prentice Hall International.
Macbeth, J. and Merville, L. (1979). An Empirical Examination of the Black-Scholes Call
Option Pricing Model. Journal of Finance. 34(5):1173-1186.
Nwobi, F. N, Annorizie, M. N. and Amadi, I. U. (2019). Crank- Nicolson Finite Difference
Method in Valuation of Options. Communications in Mathematical Finance. 8(1):93- 122.
B.O Osu. (2010).A stochastic model of the variation of the capital market price.International Journal of trade, Economics and Finance, 1, 297.
B.O. Osu and C. Olunkwa, (2014).Weak solution of Black Scholes Equation option pricing with transaction costs. International journal of applied mathematics.1 , 43.
Osu, B.O and Amadi, I U.(2022).Existence of Weak Solution With Some Stochastic Hyperbolic Partial Differential Equation. International journal of Mathematical Analysis and Modeling,5,issue 2,13-23.
Osu, B.O , Amadi, I.U. and Davies, I.(2022).An Approximation of Linear Evolution Equation with Stochastic Partial Derivative in Sobolev Space. Asian journal of Pure and Applied Mathematics, International journal of Mathematical Analysis and Modeling, 5,issue 3,13-23
Osu, B.O , Isaac, D.E and Amadi, I. U. (2022).Solutions to Stochastic Boundary Value Problems in Sobolev Spaces with its Implications in Time-Varying Investments. Asian journal of Pure and Applied Mathematics, 4(3): 581-599.
Song, L. and W. Wang, (2013). Solution of the fractional difference Black – Scholes option pricing model by finite difference method. Abs Applied Anal., 2013: 1-11. 001:10. 155/2013/194286.
Uddin, M. K. S.,Siddiki,M.N.A.A and M. A. Hossain, (2015). Numerical solution of a linear Black – Scholes models: A comparative overview. Journal of Engineering.
Razali H., (2006) The pricing efficiency of equity warrants: A Malaysian case. ICFAI journalof derivatives markets.3(3):6-22.
Rinalini, K. P. (2006). Effectiveness of the Black-Scholes Model Pricing Options in Indian
Option Market. The ICFAI Journal of Derivatives Markets. 6-19.
Wokoma, D. S.A., Amadi, I. U. and Azor, P. A. (2020). Estimation of Stock Prices using
Black Scholes Partial Differential Equation for Put Options. African Journal of
Mathematics and Statistics Studies. 3(2): 80-89.
Yueng .L.T.(2012).Crank-Nicolson scheme for Asian option. Msc thesis Department of Mathematical and actuarial Sciences ,faculty of Engineering and Science. University of Tunku Abdul Rahman. 1-97.
Zhang. H., F. Liu, I. Turner and S. Chen, (2016). The numerical simulation of the tempered fractional Black -Scholes equation for European double barrier option. Applied Math. Model; 40: 5819-5834.
Published
2024-05-08
How to Cite
I.U, A., P., O., & P.A., A. (2024). Crank-Nicolson Finite Difference Method with Sobolev Space Energy Estimate Theorem for Capital Market Prices. GPH - International Journal of Mathematics, 7(03), 122-136. https://doi.org/10.5281/zenodo.11151560
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