统计与数据科学学院知识库

期权是金融衍生品中极为重要的一种，自从它诞生之日起，学者们对于期权的研究从未停止。在所有的研究中，Black-Scholes(B-S)期权定价模型具有里程碑意义，获得了前所未有的成功。但是，B-S模型基于资产价格连续变动、交易过程无交易费用、资产价格服从对数正态分布等一系列假设之上，在一些情形下，它对标的资产价格变动过程的假设与金融市场的实际不符。因此，为了描述标的资产价格在变动过程中出现的一些特殊情形，应该采用另外的随机过程来描述。本文研究混合高斯过程下的亚式期权定价。

考虑到经典的B-S期权定价模型不能描述资产价格的长相依性和跳跃现象，运用混合高斯跳模型来描述标的资产价格的变动过程。首先，得到了几何平均亚式幂期权价格所满足的偏微分方程。其次，分别获得了几何平均亚式看涨和看跌幂期权的定价公式。最后，讨论了参数对期权价格的敏感性并进行了实例分析，结果表明，混合高斯跳模型在描述具有“跳”特征的标的资产价格时优于几何布朗运动。

由于B-S期权定价模型不能描述金融资产价格常值周期性、长相依性的特征，采用时变混合分数布朗运动描述金融资产价格的变动，并且对亚式期权定价时也考虑了标的资产的交易费用。首先，运用自融资 对冲策略得出了在离散情形下几何平均亚式期权价格所满足的偏微分方程，其次，得到了几何平均亚式看涨、看跌期权的定价公式，并分析了定价模型中的参数对期权价格的影响。最后，选取万科股票的日收盘价对所建立的定价模型进行了实例分析，验证了定价模型的有效性。

Option is a very important kind of financial derivatives. Since its birth, scholars have never stopped studying options. In all the studies, the Black Scholes (B-S) option pricing model is a milestone and has achieved unprecedented success. However, the B-S model is based on a series of assumptions. In some cases, these assumptions about the price change process of the underlying asset are inconsistent with the reality of the financial market. Therefore, in order to describe some special situations in the price change process of the underlying asset, another stochastic process should be used.This thesis studies the pricing of Asian options under mixed Gaussian process.

Considering that the classical Black Scholes (B-S) option pricing model can't describe the long-term dependence and jump phenomenon of asset prices, this thesis uses Gaussian mixture jump model to describe the change process of underlying asset prices. Firstly, the partial differential equations of geometric mean Asian power option prices are obtained. Secondly, the pricing formulas of geometric mean Asian call and put power options are obtained respectively, this thesis discusses the sensitivity of parameters to option price and makes an empirical analysis. The empirical results show that the Gaussian mixture jump model is better than geometric Brownian motion in describing the underlying asset price with "jump" characteristics.

The B-S Option pricing model can't describe the characteristics of constant value periodicity and long-term dependence of financial asset prices. The time-varying mixed fractional Brownian motion is used to describe the changes of financial asset prices, by using the self-financing  hedging strategy, the partial differential equation of the geometric average Asian option price and the pricing formulas of the geometric average Asian call and put options are obtained, and the influence of the parameters in the pricing model on the option price is analyzed, this thesis selects the daily closing price of Vanke stock to make an empirical analysis of the pricing model, and verifies the effectiveness of the pricing model.