【作者】 李飞；

【导师】 向开理；

【作者基本信息】 西南财经大学 ， 数理金融学， 2021， 硕士

【摘要】 本文在模糊环境下基于分数布朗运动研究了具有随机执行价的两类欧式看涨幂期权定价问题。由于资产收益率存在着明显的尖峰厚尾以及长程相关性特征,分数布朗运动相比标准布朗运动可以更好地描述资产价格的变化,此外金融市场存在波动性和信息不充分等特点,随机环境下期权定价模型中的参数比如无风险利率、波动率不可能为一个准确的数据,模糊理论的提出成为解决这一问题的办法之一,正是在这样的背景下,本文考虑到不确定性同时包括了随机性和模糊性两方面,并且在随机环境下还考虑到执行价格随机可以一定程度上降低风险,对两类欧式幂期权定价进行了研究,主要内容和结论包括以下三方面。第一个方面是随机环境下的两类欧式幂期权定价问题。考虑到金融资产收益率“尖峰厚尾”的特点,本文使用分数布朗运动来描述股票价格的变化过程,同时考虑到执行价格随机,使用了风险中性定价的方法推导出分数布朗运动下随机执行价的两类欧式看涨幂期权的定价公式,得到了对应的解析解。第二个方面是模糊环境下基于分数布朗运动的随机执行价的两类欧式幂期权定价问题。我们考虑到不确定性不仅包括随机性还包括模糊性,因此利用模糊集理论的知识将无风险利率、波动率以及股票价格进行了模糊处理,得到了在两种不确定环境下的两类欧式看涨幂期权的模糊价格区间,作为本部分的结束,考虑到为了便于投资者决策,本文还将得到的模糊价格区间进行了去模糊处理,得到了不确定环境下两类欧式幂期权的可能性均值。第三个方面是运用Python进行了数值模拟。主要研究了重要两个参数赫斯特指数H以及置信度α对两类看涨欧式幂期权价格的影响,同时也研究了传统Black-Schole模型中的股价、执行价格、无风险利率等参数对本文定价模型的影响,结果均符合经济学意义,此外在数值模拟的最后一部分还将本文的模型与传统模型进行了对比,主要包括执行价格随机与固定的模型对比、模糊环境下资产价格遵循分数布朗运动与标准布朗运动的模型对比以及模糊环境下的可能性均值与非模糊环境下传统模型期权价格对比。更多还原

【Abstract】 In this paper,based on the fractional Brownian motion in a fuzzy environment,two types of European call power options pricing problems with random strike prices are studied.Due to the obvious sharp peaks and thick tails and long-term correlation characteristics of asset returns,fractional Brownian motion can better describe changes in asset prices compared to standard Brownian motion.In addition,financial markets are characterized by volatility and insufficient information,and a random environment The parameters in the option pricing model such as risk-free interest rate and volatility cannot be an accurate data.The proposal of fuzzy theory has become one of the ways to solve this problem.It is against this background that this article takes into account the uncertainty At the same time,it includes both randomness and ambiguity.In a random environment,it also considers that the random execution price can reduce the risk to a certain extent.The pricing of two types of European power options is studied.The main content and conclusions include the following three aspects.The first aspect is the pricing of two types of European power options in random environments.Taking into account the "peak and thick tail" characteristics of financial asset returns,this paper uses fractional Brownian motion to describe the changing process of stock prices.At the same time,considering the random execution price,risk-neutral pricing is used to derive random execution under fractional Brownian motion.The corresponding analytical solutions are obtained for the pricing formulas of the two types of European call power options.The second aspect is the pricing problem of two types of European power options based on the random strike price of fractional Brownian motion in fuzzy environment.We consider that uncertainty includes not only randomness but also ambiguity.Therefore,we use the knowledge of fuzzy set theory to blur the risk-free interest rate,volatility and stock price,and obtain two types of European styles under two uncertain environments.The fuzzy price range of call power options is the end of this section.Considering that in order to facilitate investors’decision-making,this paper will also de-fuzzify the fuzzy price range obtained in this paper,and obtain the possibility of two types of European power options under uncertain environments.The third aspect is the use of Python for numerical simulation.Mainly studies the impact of two important parameters Hurst index H and confidence level α on the prices of two types of European power options.At the same time,it also studies the stock price,execution price,risk-free interest rate and other parameters in the traditional Black-Schole model.The results of this article’s pricing model are all in line with economic significance.In addition,in the last part of the numerical simulation,the model of this article is compared with the traditional model,which mainly includes the comparison of random and fixed execution prices,and asset prices in fuzzy environments.The model comparison of fractional Brownian motion and standard Brownian motion,and the comparison of the probability mean value in fuzzy environment and the traditional model option price in non-fuzzy environment.更多还原