统计与数据科学学院知识库

期权作为当代金融衍生品市场的新兴产品,越来越成为金融界不可或缺的组成部分.而期权定价问题也一直是学者和投资者热议的话题之一,1973年B-S(Black-Scholes)公式的出现为期权、期货等金融衍生品定价提供了强有力的理论支持,引起了学术界的广泛关注.但是经典B-S模型的收益率与实际收益率呈现出的非正态性、非线性、非独立性等经验性特征不一致,因此学者们主要从两方面着手对其进行改进.一方面为了描述金融产品的长相依特性提出了分形布朗运动模型.另一方面为了描述标的资产的“尖峰厚尾”、不连续性、“波动率微笑”等特征提出了跳跃扩散模型、随机利率与随机波动率模型.虽然这些模型能较好的拟合真实金融数据,但是与真实价格相比还是存在一定的差异.

为了更好地描述期权价格变化,本文在Heston模型的基础上对其进行改进提出了带跳的Heston-CIR混合模型和混合分形Heston-CIR模型.研究内容主要从以下三个部分展开.第一部分主要求解带跳的Heston-CIR混合模型下的欧式看涨期权价格.在Heston模型之上又考虑了随机利率和突发事件对衍生品价格的影响,建立带跳的随机波动率与随机利率(Heston-CIR)混合模型.先将标的资产所满足的随机微分方程经测度变换,变换到远期测度下.然后,利用快速傅里叶变换法解得该模型下的期权价格.

第二部分为混合分形Heston-CIR模型下的美式看跌期权定价研究.首先,为了刻画标的资产呈现出的波动率微笑和长相依等特性,基于分形市场理论用分数布朗运动和标准布朗运动的线性组合代替布朗运动,即构建混合分形Heston-CIR模型描述标的资产价格.然后,分别研究了标的资产价格和利率所满足随机微分方程的解的唯一性和存在性,并研究了利率方程的Euler格式离散化的强收敛性.

第三部分为模拟分析结果.首先,选取标的资产历史数据做描述性统计分析.其次,将真实数据与不同模型下的标的资产价格路径作比较.最后,采用最小二乘Monte Carlo算法得到不同到期日下美式看跌期权的价格,并运用数值模拟证明了提出模型的合理性.

As an emerging product of the contemporary financial derivatives market,option is becoming an indispensable part of the financial market. Option pricing is one of the most popular topics among scholars and investors.In 1973,the emergence of B-S(Black-Scholes) formula provided strong theoretical support for the pricing of financial derivatives such as options and futures,which attracted extensive attention from academic circles.However,the return rate of the classical B-S model is inconsistent with the empirical characteristics of the real return,such as non-normality, non-linearity and non-independence,so scholars mainly improve it from two aspects.On the one hand,fractional Brownian motion model is proposed to describe the long-term dependence of financial assets.On the other hand,in order to describe the characteristics of the underlying assets,such as "peak thick tail",discontinuity and "volatility smile",jump diffusion model,random interest rate and random volatility model  model are proposed.Although these models can fit the real financial data well,there are still some differences compared with the real prices.

In order to describe the variation of option price better,based on the Heston model,mixed Heston-CIR model with jump and mixed fractional Heston-CIR model are proposed.There are three parts as follows in this article.In the first part,European call option pricing under Heston-CIR mixed model with jump is solved.Based on the classical Heston model,a hybrid stochastic volatility and stochastic interest rate (Heston-CIR) model with jump is constructed by considering the impact of random interest rate and emergency on the price of financial products.Firstly,the stochastic differential equation satisfied by the underlying asset is transformed to the forward measure by measure transformation.Then,the European option price under the model is solved by the fast Fourier transform method.

In the second part,American put option pricing under mixed fractional Heston-CIR model is studied.First of all,in order to reflect the volatility smile and the long dependence of the underlying asset,a mixed fractional Heston-CIR model is constructed to describe the underlying asset price based on the fractional market theory.The linear combination of the standard Brownian motion and the fractional Brownian motion is used to replace the Brownian motion.Secondly,the uniqueness and existence of the solutions to the stochastic differential equations satisfied by the underlying asset price and interest rate are proved respectively.The strong convergence of the Euler scheme discretization of the interest rate equation is also proved.

In the third part,the results of simulation analysis are given.Firstly, the historical data of the underlying assets are selected for descriptive statistical analysis.Secondly,the real data are compared with the underlying asset price paths under different models.Finally,the least squares Monte Carlo algorithm is used to obtain the prices of American put options with different expiration dates,and the rationality of the proposed model is proved by numerical simulation.