International Journal of Nonlinear Analysis and Applications

Volatility in the Black-Scholes equation

Document Type : Research Paper

Author

Department of Computer Science, University of Garmsar, Garmsar, Iran

Abstract

Generally, the Black-Scholes model and its analyses can be presented in several different ways, ranging from the highly theoretical to the very applied approach. In this article, we will show in detail the applied methodology, the calculations and which effects the applied stress will have on the Black-Scholes option pricing model. One of the aims of nonlinear analysis is to investigate related topics to the analysis of partial differential equations and their applications. To provide for the further development of the Black-Scholes model and the Black-Scholes partial differential equation, we study some related problems. For example, we conclude that the number of call options and volatility increases at the same time.

Keywords

[1] A. Awasthi and T. Riyasudheen, An accurate solution for the generalized Black-Scholes equations governing option pricing, AIMS Math. 5 (2020), no. 3, 2226–2243.
[2] A. Baxter, A. Martin and A. Rennie, Financial Calculus: An introduction to derivative Pricing, Cambridge University Press, Cambridge, England, 1996.
[3] F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, J. Politic. Econ. 81 (1973), no. 3, 637–654.
[4] Z. Brzezniak and T. Zastawniak, Basic Stochastic Processes, Springer-Verlag, Heidelberg, 1998.
[5] G. Caginalp and D. Balenovich, Asset flow and momentum: Deterministic and stochastic equations, Phil. Trans. Royal Soc. London. Ser. A: Math. Phys. Engin. Sci. 357 (1999), 2119-2133.
[6] G. Caginalp and M. Desantis, Multi-group asset flow equations and stability, Discrete Cont. Dyn-B 16 (2011), 109–150.
[7] G. Caginalp, B. Ermentrout, A kinetic thermodynamics approach to the psychology of fluctuations in financial markets, Appl. Math. Lett. 3 (1990), 17–19.
[8] O. Calin, Deterministic and stochastic topics in computational finance, World Scientific, Singapore, 2017.
[9] N. Champagnat, M. Deaconu, A. Lejay, N. Navet and S. Boukherouaa, An empirical analysis of heavy-tails behavior of financial data: The case for power laws, HAL archives-ouvertes, 2013.
[10] J. Cohen, George Church and company on genomic sequencing, blockchain, and better drugs, Science (2018).
[11] R.D. Cohen, The relationship between the equity risk premium, duration and dividend yield, Wilmott Magazine 2002 (2002), 84–97.
[12] M. Desantis and D. Swigon, Slow-fast analysis of a multi-group asset flow model with implications for the dynamics of wealth, PLoS ONE 13 (2018).
[13] I. Glauche, A classical Approach to the Black and Scholes Formula and its Critiques, Discretization of the model, Working paper at Duke University, April, 2001.
[14] J. Huang and Z.D. Cen, Cubic spline method for a generalized Black-Scholes equation, Math. Prob. Eng. 2014 (2014).
[15] J. Hull, Options, Futures and Other Derivatives, Pearson Education India, 2003.
[16] M.K. Kadalbajoo, L.P. Tripathi and P. Arora, A robust nonuniform B-spline collocation method for solving the generalized Black–Scholes equation, IMA J. Numer. Anal. 34 (2014), 252–278.
[17] M.K. Kadalbajoo, L.P. Tripathi and A. Kumar, A cubic B-spline collocation method for a numerical solution of the generalized Black–Scholes equation, Mathe. Comput. Modell. 55 (2012), 1483–1505.
[18] F. Klebaner, Fima C, Introduction to stochastic calculus with Applications, Imperial College Press, London, 1999.
[19] J.R. Liang, J. Wang, W.J. Zhang, W.Y. Qiu and F.Y. Ren, Option pricing of a bi-fractional Black–Merton–Scholes model with the Hurst exponent H in [12, 1], Appl. Math. Lett. 23 (2010), 859–863.
[20] R. Mohammadi, Quintic B-spline collocation approach for solving generalized Black–Scholes equation governing option pricing, Comput. Math. Appl. 69 (2015), 777–797.
[21] C.S. Rao, Numerical solution of generalized Black–Scholes model, Appl. Math. Comput. 321 (2018), 401–421.
[22] M.K. Salahuddin, M. Ahmed and S.K. Bhowmilk, A note on numerical solution of a linear Black-Scholes model, GANIT: J. Bangladesh Math. Soc. 33 (2013), 103–115.
[23] S. Ross, S An Introduction to Mathematical Finance, Cambridge, England: Cambridge University Press, 1999.
[24] R.H. De Staelen and A.S. Hendy, Numerically pricing double barrier options in a time-fractional Black–Scholes model, Comput. Math. Appl. 74 (2017), 1166–1175.
[25] R. Valkov, Fitted finite volume method for a generalized Black–Scholes equation transformed on finite interval, Numer. Algorithms 65 (2014), 195–220.
[26] P. Wilmott, S. Howison and J. Dewynne, The Mathematics of Financial Derivatives, Cambridge University Press, Cambridge, 1997.
[27] M. L. Zheng, F. W. Liu, I. Turner and V. Anh, A novel high order space-time spectral method for the time fractional Fokker–Planck equation, SIAM J. Sci. Comput. 37 (2015), 701–724.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 9
September 2023
Pages 357-365
  • Receive Date: 22 September 2022
  • Revise Date: 07 December 2022
  • Accept Date: 13 December 2022