统计与数据科学学院知识库

       亚式期权作为一种典型的路径依赖型奇异期权，其定价方法的研究一直是金 融学领域研究的热点问题之一。目前，已有文献很少同时考虑到金融市场的随机 性、模糊性以及分形特征。但实际中，金融市场不仅具有随机性和分形特征，也 具有模糊性。因此，本文考虑在模糊状态下用混合次分数跳过程研究几何亚式期 权定价问题，从而更加全面地刻画金融资产的价格变动规律。本文主要研究内容 如下：

       第一，考虑到金融资产价格的长记忆性及跳跃现象，基于混合次分数布朗运 动，结合泊松过程构建了几何亚式期权定价模型。首先，根据混合次分数Itô公式， 得到混合次分数跳过程Itô公式及其股价所满足随机微分方程的解析解；其次，运 用风险中性原理给出几何亚式看涨期权的定价公式；最后，通过数值实验讨论不 同 Hurst 指数 H 和泊松强度??下，股价、无风险利率及波动率等参数与期权价格 的关系。

       第二，进行 Monte Carlo 模拟和实证分析。本部分主要给出了混合次分数跳 过程下，标的资产波动率参数估计表达式的推导过程。然后，根据标的资产的实 际数据得到各参数估计值，进而进行 Monte Carlo 模拟及拟合效果分析，验证了 该模型的合理性和实用性。

       第三，进一步考虑金融市场模糊性，引入模糊集理论构建几何亚式模糊期权 定价模型。首先，基于已建立的几何亚式期权定价模型，通过三角模糊数和多元 扩张原理，得到几何亚式看涨模糊期权价格的区间端点，给出模糊定价公式；其 次，数值模拟分析了置信度和 Hurst 指数对模糊价格的影响；最后，将本文所建 立模型与经典 BS 模型进行对比。 结果表明，混合次分数跳过程模型下的模拟路径更接近真实路径，且在此模 型下，结合模糊集理论得到的模糊价格区间也更为合理，更有助于金融投资者的 决策。

       As a typical path-dependent singular option, the pricing method of Asian option has always been one of the hot issues in the field of finance. At present, few literatures have considered randomness, fractal characteristics and fuzziness of financial markets at the same time. However, in practice, the financial market has not only randomness and fractal characteristics, but also fuzziness. Therefore, in this thesis, we consider using mixed sub-fractional process with jump to study the geometric Asian option pricing problem in fuzzy state, so as to describe the price change law of financial assets more comprehensively. The main research contents of this thesis are as follows:

       Firstly, considering the long memory and jump phenomenon of financial asset prices, a geometric Asian option pricing model is constructed based on mixed sub-fractional Brownian motion and Poisson process. Firstly, by using the mixed sub-fractional Itô formula, the analytical solution of the stochastic differential equation satisfying the mixed sub-fractional jump process Itô formula and its stock price is obtained. Secondly, the pricing formula of geometric Asian call option is given based on the risk-neutral principle. Finally, the relationship between stock price, risk-free interest rate and volatility and option price is discussed by numerical experiments with different Hurst index H and Poisson intensity λ.

         Secondly, Monte Carlo simulation and empirical analysis are carried out. This part mainly gives the derivation of expression for estimating the volatility parameter of the underlying asset. under the mixed fractional jump process. Then, the estimated values of each parameter are obtained according to actual data of the underlying asset., and the Monte Carlo simulation and fitting effect analysis are carried out to verify the rationality and practicability of the model.

         Thirdly, considering the fuzziness of financial market further, the geometric Asian fuzzy option pricing model is constructed by introducing fuzzy theory. Firstly, based on the established geometric Asian option pricing model, the range end points of the geometric Asian call fuzzy option price are obtained through the triangular fuzzy number and the multiple expansion principle, and the fuzzy pricing formula is given. Secondly, the influence of confidence and Hurst index on fuzzy price is analyzed by numerical simulation. Finally, the model established in this thesis is compared with the classical BS model. The results show that the simulated path under the mixed fractional jump process model is closer to the real path. And under this model, the fuzzy price interval obtained by combining the fuzzy theory is more reasonable and more helpful to the decision of financial investors.