## Numerical Solution on Black-Scholes Equation of Call Option and Sobolev Space Energy Estimates Theorem

Amadi I. U. *

Department of Mathematics & Statistics, Captain Elechi Amadi Polytechnic, Rumuola, Port Harcourt, Nigeria.

Onyeka P.

Department of Mathematics & Statistics, Ignatius Ajuru University of Education, Rumuolumeni, Port Harcourt, Nigeria.

Nwagor P.

Department of Mathematics & Statistics, Ignatius Ajuru University of Education, Rumuolumeni, Port Harcourt, Nigeria.

Azor P. A.

Department of Mathematics & Statistics, Federal University, Otuoke, Nigeria.

*Author to whom correspondence should be addressed.

### Abstract

This paper considered the notion of European call option which is geared towards solving analytical and numerical solutions. In particular, we examined the Black-Scholes closed form solution and Black-Scholes Partial Differential Equation (BSPDE) using Crank-Nicolson finite difference method. The explicit price of both options is found accordingly. The numerical solutions were compared to the closed form prices of Black-Scholes formula; and all results showed as follows: increase in volatility increases the values of option for both BS and CN, 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 call option, when the strike price is greater than the initial stock price it devalues the call option, increasing interest rate at difference levels increases the value of call option, Sobolev energy estimates were used as asset value function to estimate asset prices at different interest rates. To this end, the graphical solutions and comparisons of other parameters were discussed for the purpose of decision making in time varying investment plans.

Keywords: Stock prices, Crank-Nicolson, option pricing, BS PDE, call option

**How to Cite**

*Asian Journal of Pure and Applied Mathematics*,

*5*(1), 357–369. Retrieved from https://globalpresshub.com/index.php/AJPAM/article/view/1858

### Downloads

### References

Hull JC. Options. Futures and Other Derivatives. Pearson Education Inc. 2012;7.

Black F, Scholes M. The pricing of options and corporate liabilities. The journal of political Economy. 1973;81:637-654.

Macbeth J, Merville L. An Empirical Examination of the Black-Scholes call Option pricing Model, Journal of Finance. 1979;34(5):1173-1186.

Hull JC. Options, Futures and other Derivatives,( 5.ed) Edition: London prentice Hall International; 2003.

Razali H. The pricing efficiency of equity warrants: A Malaysian case, ICFAI Journal of Derivatives Markets. 2006;3(3):6-22.

Rinalini KP. Effectiveness of the Black Scholes model pricing options in Indian option market. The ICFAI Journal of Derivatives Markets. 2006;6-19.

Nwobi FN, Annorzie MN, Amadi IU. Crank-Nicolson finite difference method in valuation of options. Communications in Mathematical Finance. 2019;8(1):93-122.

Brennan M, Schwartz E. Finite Difference Methods and jump Processes Arising in the Pricing of Contingent Claims. Journal of Financial and Quantitative Analysis. 1978;5(4):461-474.

Etheride A. A course in Financial calculus, New York. Cambridge University Press; 2002.

Wilmott P, Howison S, Dewynne J. Mathematics of Financial Derivatives. Cambridge University press, New York; 2008.

Higham DJ. An introduction to financial option valuation: Mathematics, Statistics and computation. Department of Mathematics university of Strathclyde, Cambridge University Press; 2004.

Sargon D. Pricing Financial Derivatives with the Finite Difference Method. Degree project in Applied Mathematics and Industrial Economics, KTH Royal Institute of Technology School of Science of Engineering Sciences. 2017;1-142.

Bokes T. Probabilistic and Analytic methods for pricing American style Asian options. Dissertation thesis Comenius University of Bratislava, faculty of Mathematics, Physics and Informatics. 2011;1-188.

Amadi IU, Davies I, Osu BO, Essi ID. Weak Estimation of Asset Value Function of Boundary Value Problem Arising in Financial Market, Asian journal of Pure and Applied Mathematics; 2022.

Osu BO, Isaac DE, Amadi IU. Solutions to Stochastic Boundary Value Problems in Sobolev Spaces with its Implications in Time-Varying Investments. Asian Journal of Pure and Applied Mathematics. 2022;4(3):581-599.

Osu BO, Olunkwa C. Weak solution of Black Scholes Equation option pricing with transaction costs. International Journal of applied mathematics. 2014;1:43.

Osu BO, Eze EO. Obi CN. The impact of stochastic volatility process on the values of assets. Scientific African. 2020;9(2020):7.

Yueng LT. Crank-Nicolson scheme for Asian option. Msc thesis Department of Mathematical and actuarial Sciences, faculty of Engineering and Science. University of Tunku Abdul Rahman. 2012;1-97.

Amadi IU, Osu BO, Davies I. A Solution to Linear Black-Scholes Second order Parabolic Eqaution in Sobolev Spaces. International journal of Mathematics and Computer Research. 2022;10(10):2938-2946.

Osu BO, Amadi IU. Existence of Weak Solution with Some Stochastic Hyperbolic Partial Differential Equation. International journal of Mathematical Analysis and Modeling. 2022;5(2):13-23.

Osu BO, Amadi IU, Davies I. 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. 2022;5(3):13-23.

Osu BO. A stochastic model of the variation of the capital market price”. International Journal of trade, Economics and Finance. 2010;1:297.

Diperna RJ, Lions PL. Ordinary Differential Equations, transport theory and Sobolev Spaces. Invention Maths. 1989; 98:511.

Heston SI. A closed form solution for options with stochastic volatility with application to bond and currency option, Rev. Financial Studies. 1993;6:327.

Fadugba SE, Nwozo CR. Crank Nicolson Finite Difference Method for the valuation of options. The pacific Journal of Science and Technology. 2013;14(2):136-146.

Osu BO. A stochastic model of the variation of the capital market price”. International Journal of trade, Economics and Finance. 2010;1:297.

Diperna RJ, Lions PL. Ordinary Differential Equations, transport theory and Sobolev Spaces. Invention Maths. 1989;98:511.

Yueng LT. Crank-Nicolson scheme for Asian option. Msc thesis Department of Mathematical and actuarial Sciences, faculty of Engineering and Science. University of Tunku Abdul Rahman. 2012;1-97.