统计与数据科学学院知识库

期权作为金融市场的衍生品发展越来越快,被众多投资者和风险管理者运用, 尤其是波动率衍生品的不断壮大,对金融管理提出了更高要求.B-S 模型是应用 最广泛的研究期权定价的模型之一,但通过历史数据和实证研究发现:其对数收 益不遵循标准正态分布的假设条件,而且不能够很好地刻画资产自相似、长相依 等特性.因此,为了更准确描述标的资产价格变化情况,学者们更多选择用随机波 动模型来刻画金融资产标的价格. 本文研究了混合高斯 Heston 随机波动模型下的欧式期权定价及统计模拟分 析.第一部分主要得到混合高斯 Heston 随机波动模型下的欧式期权定价.首先得 到混合高斯Heston随机波动模型满足的偏微分方程,然后得到模型中波动率方程 解的存在唯一性,接着讨论解的 p 阶矩的性质定理,最后结合偏微分方程满足的 边界条件,得到混合高斯 Heston 随机波动模型的解析解. 第二部分是关于双混合分数Heston模型解的存在唯一性,主要适用于解决短 期期权的拟合问题.得到双混合分数Heston随机波动模型资产价格方程解的存在 唯一性,由于其解的复杂性,对模型中波动率和股票价格随机微分方程进行欧拉 离散化. 第三部分选取上证 50ETF期权进行统计模拟分析.对选取的数据进行描述性 统计分析,验证了金融市场存在波动集聚、尖峰厚尾和非对称等特征,并对未知参 数进行估计和敏感度分析,最后用 Monte Carlo 模拟法对混合高斯 Heston 随机波 动模型进行有效性分析.研究表明,采用混合高斯 Heston 随机波动模型比 Heston 模型更加接近其真实值.研究结果可为期权定价的理论和发展提供更多新的依 据.

With the development of financial markets, options have been used by many investors and risk managers as derivatives. In particular, the continuous growth of volatility derivatives has put forward higher requirements for financial management. The B-S model is one of the most widely used models to study option pricing. But through historical data and empirical research show: firstly, its logarithmic returns do not follow the assumption of a standard normal distribution; secondly, it cannot well describe the characteristics of self-similarity and long-term dependence of assets. Therefore, in order to more accurately describe the changes in the price of the underlying asset, scholars choose to use the stochastic fluctuation model to characterize the underlying price of financial assets. The European option pricing and statistical simulation analysis are studied under the mixed Gaussian Heston stochastic volatility model. In the first part, mainly gets European option pricing under the mixed Gaussian Heston stochastic volatility model. First, the partial differential equation are obtains which satisfied the mixed Gaussian Heston stochastic stochastic model, then gets the existence and uniqueness of the solution of the volatility equation in the model, next discusses the theorem about the nature of the p-order moment of the solution, and the analytical solution of the mixed Gaussian Heston stochastic volatility model is finally obtained by combining the boundary conditions satisfied by the partial differential equation. In the second part, it is about the existence and uniqueness of the solution of the double mixed fractional Heston model, which is mainly applicable to the problem of fitting short-term options. The existence and uniqueness of the solution of the asset price equation of the double mixed-fraction Heston stochastic wave model are obtained, due to the complexity of its solution, Euler discretization is performed on the volatility and stochastic differential equations of stock prices in the model. In the third part, the 50ETF options are selected for statistical simulation analysis. Perform descriptive statistical analysis on the selected data to verify characteristics such as the existence of agglomerations, thick tails, and asymmetry in the financial market, and estimate and sensitivity analysis of unknown parameters. Monte Carlo simulation method is used to analyze the effectiveness of the mixed Gaussian Heston stochastic volatility model. It shows that the mixed Gaussian Heston stochastic wave model is closer to its true value than the Heston model. The research results can provide more new basis for the theory and development of option pricing.