Document Type : Research Article


1 University of Isfahan, Isfahan, Iran

2 Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, FIN-65101 Vaasa, Finland.


This article proposes a new numerical technique for pricing asset-or-nothing options using the Black-Scholes partial differential equation (PDE). We first use the θ−weighted method to discretize the time domain, and then use Haar wavelets to approximate the functions and derivatives with respect to the asset price variable. By using some vector and matrix calculations, we reduce the PDE to a system of linear equations that can be solved at each time step for different asset prices. We perform an error analysis to show the convergence of our technique. We also provide some numerical examples to compare our technique with some existing methods and to demonstrate its efficiency and accuracy.


[1] Muhammad Ahsan, Martin Bohner, Aizaz Ullah, Amir Ali Khan, and Sheraz Ahmad, A haar
wavelet multi-resolution collocation method for singularly perturbed differential equations
with integral boundary conditions, Mathematics and Computers in Simulation 204 (2023),
[2] Muhammad Ahsan, Weidong Lei, Amir Ali Khan, Masood Ahmed, Maher Alwuthaynani, and
Ayesha Amjad, A higher-order collocation technique based on haar wavelets for fourth-order
nonlinear differential equations having nonlocal integral boundary conditions, Alexandria
Engineering Journal 86 (2024), 230–242.
[3] Muhammad Ahsan, Siraj ul Islam, and Iltaf Hussain, Haar wavelets multi-resolution collocation analysis of unsteady inverse heat problems, Inverse Problems in Science and Engineering 27 (2019), no. 11, 1498–1520.
[4] Imran Aziz, AS Al-Fhaid, et al., An improved method based on haar wavelets for numerical
solution of nonlinear integral and integro-differential equations of first and higher orders,
Journal of Computational and Applied Mathematics 260 (2014), 449–469.
[5] Fischer Black and Myron Scholes, The pricing of options and corporate liabilities, Journal
of political economy 81 (1973), no. 3, 637–654.
[6] G Hariharan and K Kannan, A comparative study of haar wavelet method and homotopy
perturbation method for solving one-dimensional reaction-diffusion equations, International
Journal of Applied Mathematics and Computation 3 (2011), no. 1, 21–34.
[7] Desmond J Higham, An introduction to financial option valuation: mathematics, stochastics
and computation, vol. 13, Cambridge University Press, 2004.
[8] Ram Jiwari, A haar wavelet quasilinearization approach for numerical simulation of burgers
equation, Computer Physics Communications 183 (2012), no. 11, 2413–2423.
[9] Devendra Kumar and Komal Deswal, Two-dimensional haar wavelet based approximation
technique to study the sensitivities of the price of an option, Numerical Methods for Partial
Differential Equations 38 (2022), no. 5, 1195–1214.
[10] Ulo Lepik and Helle Hein, ¨ Solving pdes with the aid of two-dimensional haar wavelets, pp. 97–
105, Springer International Publishing, Cham, 2014.
[11] Mohammed S. Mechee, Zahir M. Hussain, and Zahrah Ismael Salman, Wavelet theory: Applications of the wavelet, Wavelet Theory (Somayeh Mohammady, ed.), IntechOpen, Rijeka,
[12] Liu Meng, Meng Kexin, Xing Ruyi, Shuli Mei, and Carlo Cattani, Haar wavelet transform
and variational iteration method for fractional option pricing models, Mathematical Methods
in the Applied Sciences 46 (2023), no. 7, 8408–8417.
[13] RC Mittal, Harpreet Kaur, and Vinod Mishra, Haar wavelet-based numerical investigation of
coupled viscous burgers’ equation, International Journal of Computer Mathematics 92 (2015),
no. 8, 1643–1659.
[14] M Ratas, SK Jena, and S Chakraverty, Application of haar wavelet based methods for solving
wave propagation problems, AIP Conference Proceedings, vol. 2293, AIP Publishing, 2020.
[15] Umer Saeed and Mujeeb ur Rehman, Haar wavelet–quasilinearization technique for fractional
nonlinear differential equations, Applied Mathematics and Computation 220 (2013), 630–648.
[16] Burma Saparova, Roza Mamytova, Nurjamal Kurbanbaeva, and Anvarjon Akhatjonovich
Ahmedov, A haar wavelet series solution of heat equation with involution, Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 86 (2021), no. 2, 50–55.
[17] Steven E Shreve et al., Stochastic calculus for finance ii: Continuous-time models, vol. 11,
Springer, 2004.
[18] Saeed Vahdati, Mohammad Reza Ahmadi Darani, and Mohammad Reza Ghanei, Haar
wavelet-based valuation method for pricing european options, Computational Methods for
Differential Equations 11 (2023), no. 2, 281–290.
[19] Amit K Verma and Diksha Tiwari, Higher resolution methods based on quasilinearization and
haar wavelets on lane–emden equations, International Journal of Wavelets, Multiresolution
and Information Processing 17 (2019), no. 03, 1950005.
[20] Amit Kumar Verma, Mukesh Kumar Rawani, and Carlo Cattani, A numerical scheme for
a class of generalized burgers’ equation based on haar wavelet nonstandard finite difference
method, Applied Numerical Mathematics 168 (2021), 41–54.
[21] Ruyi Xing, Meng Liu, Kexin Meng, and Shuli Mei, Coupling technique of haar wavelet transform and variational iteration method for a nonlinear option pricing model, Mathematics 9
(2021), no. 14, 1642.