مدل بلک شولز تعمیم یافته تحت نوسانات گارچ با محاسبه ارزش در معرض خطرشرطی در قیمت گذاری مشتقه

نوع مقاله : مقاله پژوهشی

نویسندگان

1 گروه ریاضی مالی، دانشکده علوم پایه، دانشگاه آیت ا... بروجردی(ره)، بروجرد، ایران.

2 استادیار گروه ریاضی مالی، دانشکده علوم پایه، دانشگاه آیت ا... بروجردی(ره)، بروجرد، ایران.

3 گروه ریاضی، دانشکده علوم پایه، دانشگاه آیت ا... بروجردی (ره)، بروجرد، ایران.

10.22051/jfm.2023.43857.2828

چکیده

بازارهای مالی نقش اساسی در توسعه اقتصادی هر کشوری دارد، لذا بررسی دقیق این بازارها از جنبه‌های مختلف ضروری به نظر می‌رسد. حضور در این بازارها همواره با ریسک بالایی همراه است و به ‌منظور کاهش ریسک ابزارهای مختلفی پدید آمده است. اختیار معامله، متداول­ترین ابزار معامله‌ای است که به بازارهای مالی معرفی ‌شده است. مدل بلک شولز برای قیمت‌گذاری طیف وسیعی از قراردادهای اختیار معامله استفاده می‌شود. فرض اساسی در این مدل ثابت بودن نوسان بازده‌ها است که فرض مناسبی در دنیای واقعی نیست. هدف این پژوهش توسیع مدل بلک شولز تحت نوسانات تصادفی است. ابعاد نوآوری پژوهش شامل تحلیلی از عملکرد قیمت‌گذاری مدل‌ها در طول دوره‌های کوتاه‌مدت، میان‌مدت و بلندمدت با ارزش در معرض خطر شرطی برای هر یک از قیمتها است. برای این منظور، از داده‌های ایران‌خودرو در بازه 1/9/ 1399 تا 23/9/ 1401 مورد استفاده قرار گرفت و نوسانات گارچ استاندارد و گارچ آستانه محاسبه و در مدل توسعه‌یافته بلک شولز به­کار گرفته شد. در ادامه قیمت اختیار خرید تحت مدل بلک شولز  با نوسانات تاریخی، مدل بلک شولز توسعه‌یافته با گارچ استاندارد، گارچ آستانه و گارچ نمایی محاسبه شد. نتایج پژوهش حاکی از آن است که قیمت اختیار با نوسان تاریخی به قیمت واقعی بازار نزدیکتر است و همچنین، با توجه به ارزش­در معرض خطر شرطی بیشتر، ریسک واقعی‌تر نشان می‌دهد. در نهایت، برای آزمون نتایج بدست آمده قیمت اختیار خرید در هر چهار نوسان مفروض با روش مونت کارلو محاسبه و نتایج بدست آمده برای قیمت­ها تایید شد.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Generalized Black-Scholes Model under Garch Volatility with Conditional value-at-risk Calculation in Derivative Pricing

نویسندگان [English]

  • Hossein Nasrollahi 1
  • Mohammad Reza Haddadi 2
  • Manizheh Goudarzi 3
1 Financial Mathematics Department, Ayatollah Borujerdi University, Borujerd, Iran.
2 Assistant Professor at Financial Mathematics, Faculty of Basic Sciences, Ayatollah Borujerdi University, Borujerd, Iran.
3 Department, of Mathematics and Statistics, Ayatollah Borujerdi University, Borujerd, Iran.
چکیده [English]

Financial markets play an essential role in the economic development of any country, therefore, a detailed examination of these markets from various aspects is necessary. Being in these markets is always associated with a high risk, and various tools have emerged in order to reduce the risk. Option is the most common trading tool that has been introduced to the financial markets. The Black-Scholes model is used to price a wide range of options contracts. The basic assumption in this model is the constant volatility of returns, which is not a suitable assumption in the real world. The aim of this research is to extend the Black-Scholes model under stachistic volatilities. The innovation of this research includes an analysis of the pricing performance of the models during the short, medium and long term periods with conditional value at risk for each of the prices. For this purpose, the data of Iran Khodro was used between 21/11/2020 to 14/12/2022 and the standard GARCH and threshold GARCH volatilities were calculated and used in the developed Black-Scholes model. In the following, the call option price was calculated under the Black-Scholes model with historical volatilities, the developed Black-Scholes model with standard GARCH, TGARCH and exponential GARCH. The results of the research indicate that the option price with historical volatility is closer to the real market price and also shows a more real risk due to the conditional value at risk. Finally, to test the obtained results, the price of the call option in all four assumed volatilities was calculated with Monte Carlo method and the obtained results were confirmed for the prices.

کلیدواژه‌ها [English]

  • GARCH Volatility
  • Option Pricing
  • Black-Scholes Model
  • TGARCH
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