Option Pricing Using Modification of Black Scholes Merton Model with GJR-GARCH

Abstract

Option pricing is an important topic in modern finance as it plays a role in investment strategy and risk management. The Black-Scholes-Merton (BSM) model introduced in 1973has become the standard in option pricing, but the assumption of constant volatility and symmetry makes this model often less suitable for volatile and asymmetric market conditions. This study aims to modify the BSM by incorporating volatility estimated through the Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) model, and compare its performance with the Fractional Black-Scholes-Merton (FBSM) model. The data used is the daily closing price of Apple Inc. shares for the period January 7, 2022 to March 7, 2025 obtained from Yahoo Finance. The research procedure includes stationarity test using Augmented Dickey-Fuller (ADF), ARIMA identification, heteroscedasticity testing, and volatility estimation with GJR-GARCH (1,1) model. European call option prices are estimated by BSM and FBSM using both historical volatility and GJR-GARCH volatility. The results show that the FBSM model with GJR-GARCH (1,1) volatility provides the most accurate estimation with a Mean Absolute Percentage Error (MAPE) of 4.08%. In contrast, the BSM with historical volatility yields a MAPE of 17.25%. These findings confirm that the integration of FBSM with GJR-GARCH volatility is more realistic and reliable in option pricing under dynamic and asymmetric market conditions.