Applied Mathematics and Computation 366 (2020) 124693 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Robust numerical algorithm to the European option with illiquid markets D. Ahmadian a,∗ , O. Farkhondeh Rouz a , K. Ivaz a , A. Safdari-Vaighani b a Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran b Department of Mathematical and Computer Sciences, Allameh Tabataba’i University, Tehran, Iran a r t i c l e i n f o Article history: Received 25 November 2018 Revised 12 July 2019 Accepted 26 August 2019 MSC: 65M12 35Q99 91G60 91G20 Keywords: European call option Illiquid markets Newton’s method Kantorovich theorem Positivity a b s t r a c t In this paper, we consider illiquid European call option which is arisen in nonlinear Black– Scholes equation. In this respect, we apply the Newton’s method to linearize it. Based on the obtained linear equation, we obtain the approximate solutions recursively in two steps. Finally, based on the conditions of Kantorovich theorem, we investigate the convergence analysis of the Newton’s method on the proposed problem. Finally the positivity of the so- lution is discussed. © 2019 Elsevier Inc. All rights reserved. 1. Introduction Markets liquidity is an issue of very high concern in financial risk management. In a perfect liquid market the option pricing model becomes the well-known linear Black–Scholes problem. The celebrated Black–Scholes model is based on sev- eral restrictive assumptions such as liquid, frictionless and complete markets. Taking into account one or some of these parameters to have more reliable option price causes a nonlinear Black–Scholes equation with a nonlinear volatility func- tion (see [10,14,37] and [38]). In recent years nonlinear Black–Scholes models have been used to build transaction costs, market liquidity or volatility uncertainty into the celebrated Black–Scholes concept, and if in these models the volatility of the underlying asset is assumed constant we will have the classical linear Black–Scholes model (see [2,23,25] and [31]). Nonlinear Black–Scholes equations have been considered with nonlinear volatility that depends also on time t, underlying asset price S and the second derivative of option price V(S, t) with respect to S. For instance, the Leland model, assumed the nonzero transaction cost to be a fixed fraction of the volume of transactions [36]. Moreover, the Risk Adjusted Pricing Methodology (RAPM) model was introduced by Kratka in 1998 [33] as an attempt to include effects of transaction cost as well as risk from volatile portfolios into the Black–Scholes model. ∗ Corresponding author. E-mail addresses: d.ahmadian@tabrizu.ac.ir (D. Ahmadian), omid_farkhonde_7088@yahoo.com (O. Farkhondeh Rouz), ivaz2003@yahoo.com (K. Ivaz), asafdari@atu.ac.ir (A. Safdari-Vaighani). https://doi.org/10.1016/j.amc.2019.124693 0096-3003/© 2019 Elsevier Inc. All rights reserved.