Empirical and Comparative Analysis of the Efficiency of Nonlinear Option Pricing Models (Quantum-Based and Heston) and Conventional Linear Models (Black–Scholes and Binomial Tree) in the Tehran Stock Exchange

Abstract

Accurate option pricing, as one of the key tools for risk management and price discovery, requires the use of models capable of adapting to nonlinear behaviors and structured market volatilities. The present study aims to empirically evaluate the efficiency of both linear and nonlinear option pricing models in the Iranian market by examining four analytical frameworks: Black–Scholes and the Binomial Tree as linear models, and Heston and an extended quantum model based on the nonlinear Schrödinger equation as nonlinear models. The research data include 30 actively traded call option contracts listed on the Tehran Stock Exchange, each with a minimum of 70 trading days during the period from 2016 to 2022. In the numerical solution section, the quantum and Heston models were implemented using the method of lines and the fourth-order Runge–Kutta algorithm in a Python programming environment. The comparative evaluation criteria consisted of Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE). Out-of-sample results indicate that, under most conditions—particularly for at-the-money (ATM) options and in high-volatility environments—the quantum model provides the most accurate estimates. The Heston model demonstrates competitive performance at medium maturities and moderate volatility levels; the Binomial Tree model performs comparably to the benchmark over short horizons; and the Black–Scholes model, overall, exhibits the lowest accuracy. Moreover, employing nonlinear frameworks not only reduces pricing deviation but also leads to relative savings in computational costs compared to linear iterations based on implied volatility extraction. This study is also distinguished from a computational perspective, as it represents the first application in Iranian option pricing literature that integrates the method of lines and Runge–Kutta within a Python-based numerical implementation, moving beyond the traditional reliance on regression-based and curve-fitting techniques. Although the limited depth of Iran’s derivatives market, trading halts, and the absence of a comprehensive structured data repository remain research challenges, the findings suggest that employing nonlinear models—particularly those based on quantum dynamics—can provide a novel pathway toward fair pricing, the design of effective hedging strategies, and the advancement of financial engineering in the Iranian market.