مقایسۀ کارایی مدل های قیمت گذاری اختیار تحت پرش، چولگی و کشیدگی غیر نرمال (مقاله علمی وزارت علوم)
درجه علمی: نشریه علمی (وزارت علوم)
آرشیو
چکیده
مدل بلک شولز فرض می کند که بازده سهام از توزیع نرمال با نوسان ثابت پیروی می کند. درصورتی که شواهد تجربی در بازارهای مالی نشان می دهند که بازده سهام چولگی و کشیدگی غیرنرمال دارد. به منظور انعکاس بهتر ویژگی های سری بازده دارایی ها، مدل هایی برای تعمیم مدل بلک شولز برای قیمت گذاری دقیق تر اختیار معامله معرفی شده است. مدل مرتون، مدل کو و گرام چارلیه توسعه یافته هایی از مدل بلک-شولز هستند که با افزودن یک فرایند پرش مرکب پواسون، اثر چولگی و کشیدگی در چگالی قیمت دارایی ها را مدل سازی می کنند. این پژوهش دقت مدل های قیمت گذاری اختیار خرید معامله بلک شولز، مرتون، مدل کو و گرام چارلیه را بررسی و مقایسه می کند و تأثیر چولگی و کشیدگی غیرنرمال را بر قیمت گذاری تحلیل می کند. نتایج نشان می دهد که مدل گرام چارلیه در شرایط چولگی منفی و کشیدگی غیرنرمال خطای کمتری از مدل های پرش انتشار مرتون و مدل کو دارد. درمقابل، در شرایط چولگی و کشیدگی پایین، مدل کو عملکرد بهتری از خود نشان می دهد.A Comparative Analysis of Option Pricing Models Under Jump Dynamics, Skewness, and Non-Normal Kurtosis
The Black-Scholes model assumes log-normal stock returns with constant volatility, yet empirical evidence reveals significant deviations, including skewness and excess kurtosis in financial markets. To better capture these characteristics, extended models incorporating jump processes and non-normal distributions have been developed. This study evaluates the pricing accuracy of four option pricing models—the Black-Scholes model, the Merton jump-diffusion model, the Kou double-exponential jump model, and the Gram-Charlier expansion model—with a focus on their performance under varying degrees of skewness and kurtosis. Our findings indicate that the Gram-Charlier model outperforms the Merton and Kou models in scenarios with negative skewness and leptokurtic distributions. Conversely, the Kou model demonstrates superior accuracy under conditions of low skewness and kurtosis. These results highlight the importance of selecting appropriate pricing models based on the underlying return distribution characteristics. Keywords: Options, Merton's Diffusion Jump Model, Gram-Charlier Model, Skewness, Kurtosis JEL Classification : G11, G12 Introduction . The Black-Scholes model assumes that stock returns follow a normal distribution with constant volatility. However, empirical evidence in financial markets shows that stock returns exhibit significant non-normal skewness and kurtosis. To better capture these characteristics of asset return series, several models have been developed to generalize the Black-Scholes framework for more accurate option pricing. The Merton jump-diffusion model and the Kou model extend the Black-Scholes approach by incorporating a compound Poisson jump process, which allows these models to account for skewness and kurtosis in asset price distributions. An alternative methodology, the Gram-Charlier expansion, addresses skewness and kurtosis effects through a different approach - it uses Hermite polynomials to approximate the probability distribution of asset prices. This study systematically examines and compares the pricing accuracy of four key models: the standard Black-Scholes model, the Merton jump-diffusion model, the Kou model, and the Gram-Charlier expansion. Our analysis specifically focuses on how these models perform under varying conditions of skewness and excess kurtosis, providing insights into their relative strengths and limitations. Materials & Methods This study analyzes historical closing prices for three major Iranian financial instruments: (1) Iran Khodro Company (ticker: Khodro) from May 16, 2020 to August 28, 2024; (2) Social Security Investment Company (ticker: Shasta) from March 2, 2022 to August 28, 2024; and (3) Charisma Equity Fund (ticker: Aharm) from December 20, 2021 to August 28, 2024. Volatility estimates for all three securities were computed using historical volatility methods based on the price series. We then calculated European call option prices using four distinct pricing models: the standard Black-Scholes model, Gram-Charlier expansion, Merton jump-diffusion model, and Kou double-exponential jump model. Finally, we conducted a comparative analysis between the model-derived prices and actual market prices to evaluate model performance. Findings This study incorporates the effects of skewness and excess kurtosis into various option pricing models. Our comparative analysis reveals that the Gram-Charlier model demonstrates superior pricing accuracy, exhibiting lower errors compared to both the Merton jump-diffusion model and Kou model under conditions of negative skewness and leptokurtic distributions. Conversely, the Kou model outperforms alternative approaches in markets characterized by low skewness and kurtosis. These findings make two significant contributions to the option pricing literature. First, they provide empirical evidence for model selection criteria based on distributional characteristics of underlying assets. Second, they demonstrate that optimal model choice depends critically on specific market conditions. Our results suggest that practitioners should carefully consider the statistical properties of asset returns when selecting pricing models, rather than relying on a single universal approach. Discussion and Conclusion The results of this study indicate that for the Shasta symbol, option prices calculated using the Gram-Charlier model significantly outperform those derived from the Kou and Merton models. This superior performance stems from the presence of negative skewness and excessive kurtosis in the data. We conclude that the Gram-Charlier model produces better results than the Merton and Kou jump models when applied to datasets exhibiting negative skewness and high kurtosis. For the Khodro symbol, the Merton model demonstrates greater pricing accuracy compared to both the Gram-Charlier and Kou models, a result attributable to the high kurtosis present in this dataset. Regarding the Leverage symbol, the Kou model provides more accurate option pricing than the Gram-Charlier model, owing to the relatively normal skewness and kurtosis characteristics of the data. These findings lead to two key inferences: First, the Kou model proves particularly suitable for datasets displaying normal characteristics or for cases where normality has been rejected by statistical tests but without significant abnormal skewness and kurtosis. Second, the Merton model serves as an appropriate option pricing model for non-normal datasets characterized primarily by high kurtosis without accompanying skewness.
تبلیغات
