兰州财经大学机构知识库

金融市场的迅速发展,使得金融衍生产品不断增加,而期权作为金融衍生产 品的一种,因为其自身具备的杠杆性和套期保值特性而引起了广泛关注.在期权 定价的研究过程中,研究人员运用布朗运动和分形布朗运动模型对期权进行了定 价研究,但实证研究表明:经典布朗运动模型和分形布朗运动模型均不能准确刻 画金融资产的自相似和长相依等特性,而多分形布朗运动模型能够更好的刻画金 融市场尖峰后尾的特征.因此,对于多分形布朗运动模型下资产定价的研究是很 有必要的. 本文基于金融随机分析和期权定价理论,采用多分形随机利率模型来描述金 融资产标的价格变化情况.主要研究内容有两部分.第一部分是对期权定价公式 的推理证明.首先,构造投资组合并运用无风险对冲原理和随机微分方程理论推 理得出期权所满足的随机微分方程.然后,通过热传导方程对偏微分方程进行求 解,并推导得出期权在多分形布朗运动模型下具有交易费用和随机利率的欧式期 权定价公式. 第二部分采用上证 50ETF 期权进行模拟分析.首先,对所研究数据进行描述 统计分析得到数据的基本特征,并对其中的参数进行估计.然后,分别用经典 B-S 和多分形随机利率模型对标的资产价格进行模拟,并与标的资产的真实值进行比 较.最后,将模拟得到的标的资产的价格带到相应期权定价公式中,计算出期权的 价格,并与期权的真实值进行对比.研究表明:与经典的 B-S 模型相比,多分形随 机利率模型下的期权定价结果更加接近期权的真实值.因此说明多分形随机利率 模型对期权定价是有效并且可行的.

The rapid development of financial markets has led to the increasing number of financial derivatives. As a financial derivative, options are widely concerned with their leverage and hedging characteristics. On the research of option pricing, researchers use respectively Brownian motion and fractional Brownian motion model to study option pricing, but empirical studies show that both classical Brownian motion model and fractional Brownian motion model cannot accurately describe the self-similarity and long-term dependence characteristics of financial assets, while multifractional Brownian motion model can better describe the peak and tail characteristics of financial markets. Therefore, it is necessary to study the asset pricing under the multifractional Brownian motion model. In this paper, based on the financial stochastic analysis and option pricing theory, uses the multifractional stochastic interest rate model to describe the price change of financial assets.There are two parts in this paper. The first part is the proof of reasoning of the option pricing formula. Firstly, The stochastic equations are verifed by used constructing a portfolio and applying a risk-free hedging principle and stochastic differential equation theory. Then, the partial differential equations are solved by the heat conduction equation, and it is concluded that the 兰州财经大学硕士毕业论文 多分形随机利率模型下的欧式期权定价研究 III European option pricing formula with the transaction cost and random interest rate under the mutifractional Brownian motion model. In the second part, using the 50 ETF option is used for simulation analysis. Firstly, the basic characteristics of the data are obtained by descriptive statistical analysis of the obtained data, and the values of the parameters to be estimated are obtained. Then, the classic B-S model and the extended model are used to simulate the price of the asset and comparing with the real price change path. Finally, the price of the simulated underlying asset is brought to the corresponding option pricing formula, and the price of the option is calculated and compared with the true value of the option. Compared with the classic B-S model, the option pricing under the multifractional stochastic interest rate model is closer to the true value of the option. Therefore, the model of multifractional stochastic is effective and feasible for option pricing.