统计与数据科学学院知识库

      近年来, 随着金融市场及金融全球化的加速发展, 金融市场突发事件的发生愈加频繁, 使得投资者面临的金融市场不确定性及投资风险随之增加. 在此背景下, 如何合理规避金融市场不确定性及投资风险显得尤为重要. 期权和投资组合作为现代金融学理论研究的核心内容, 可以较好地规避金融市场不确定性及投资风险. 对于期权定价和投资组合而言, 最为关键的是构建能够精确描述金融市场数据动态变化特征的模型. 但现有关于描述金融基础资产价格趋势的模型存在对其动态变化特征考虑不全面的问题. 此外, 随着金融市场一体化的迅猛发展, 利率不再是一个常数, 并且研究发现利率的随机波动特征对期权定价及最优投资决策具有显著影响. 因此, 本文以金融基础资产价格的动态变动特征及市场利率随机性为切入点, 构建符合金融市场特征的模型对期权定价及投资组合问题的研究具有重要的理论意义和应用价值.

      本文的研究重点是基于金融市场基础资产价格的随机波动、尖峰、厚尾、跳跃扩散及市场利率的随机波动特征, 构建了具有不同特征的4/2随机混合欧式期权定价模型, 并基于50ETF期权数据检验4/2随机混合模型在欧式期权定价中的定价表现. 此外, 基于4/2随机混合模型在期权定价中的定价表现, 构建了4/2-CIR跳扩散投资组合模型, 获得了该模型下的最优投资策略及效用损失, 并探讨了模型参数对最优风险敞口及效用损失的影响. 具体内容及主要结论如下:

       (1) 基于现有随机波动率及随机利率混合模型的不足, 构建4/2-CIR随机混合模型对欧式期权定价问题展开研究. 首先, 综合考虑金融资产随机波动及市场利率随机性特征对期权定价结果的影响, 构建了4/2-CIR随机混合模型. 其次, 获得了4/2-CIR随机混合模型下标的资产对数价格的特征函数及期权定价公式. 最后, 通过数值算例分析随机利率因子对模型定价结果的影响, 并根据实际数据对模型参数进行校准, 进而探讨模型的定价性能及精度. 研究表明, 利率随机性对期权定价结果具有显著影响, 并且在4/2随机波动率模型中引入随机利率因子有助于提高已有4/2随机波动率模型的定价精度.

      (2) 考虑到金融资产价格的动态变化特征对期权定价结果具有显著影响, 提出了能够全面描述基础资产随机波动、尖峰、厚尾及跳跃等特征的4/2跳跃扩散随机混合模型. 首先, 在基础资产价格具有随机波动特征的前提下, 将双指数扩散过程引入到描述基础资产价格动态变化特征的模型中, 构建了4/2跳扩散随机混合模型. 其次, 获得了4/2 跳扩散随机混合模型下相应的特征函数及欧式期权定价公式. 最后, 利用粒子群优化算法对模型参数进行估计, 并基于4/2跳扩散模型下欧式期权定价公式及参数估计值检验模型的定价性能和精度. 结果表明, 相较于已有4/2随机波动率模型、3/2随机波动率模型及Heston随机波动率模型, 本文所构建4/2跳扩散随机混合模型在欧式期权定价问题中的定价表现最优. 此外, 4/2跳扩散随机混合模型对实值期权及虚值期权的定价精度优于平值期权.

      (3) 为全面描述基础资产动态变化特征及市场利率随机性特征, 提出了4/2-CIR跳扩散随机混合模型. 首先, 在等价鞅概率测度下, 基于基础资产价格的随机波动、尖峰、厚尾、跳跃及市场利率的随机性特征, 构建了4/2-CIR跳扩散随机混合模型. 其次, 获得了4/2-CIR跳扩散随机混合模型下标的对数价格的特征函数及欧式期权定价公式. 最后, 利用粒子群优化算法对模型参数进行估计, 并通过对比分析法检验4/2-CIR跳扩散随机混合模型的定价性能及优势. 研究表明, 4/2-CIR跳扩散随机混合模型不仅能够全面描述基础资产的动态变化特征和市场利率的随机波动特征, 而且能够反应基础资产与波动率之间的杠杆效应. 此外, 与4/2-CIR随机混合模型及4/2跳扩散随机混合模型相比, 4/2-CIR跳扩散模型在欧式期权定价中的定价性能最优.

      (4) 在效用函数最大化准则下, 构建了股票指数满足4/2-CIR跳扩散过程的投资组合模型. 首先, 在幂效用函数下, 获得了基于4/2-CIR跳扩散投资组合模型所得HJB 方程的最优解、最优投资策略及最优风险敞口. 其次, 证明了4/2-CIR跳扩散投资组合模型下HJB方程解及投资策略的最优性. 最后, 通过数值例子分析了模型参数对最优风险敞口及效用损失的影响, 并通过采用不同模型下投资策略所产生的损失大小, 分析基于4/2-CIR跳扩散投资组合模型所得投资策略的优势. 结果表明, 风险厌恶系数、投资期限及风险溢价因子对最优风险敞口具有显著影响. 此外, 风险溢价因子对短视损失具有显著影响. 并且与现有4/2随机波动率模型及4/2-CIR随机混合模型下的投资策略相比, 4/2-CIR跳扩散混合模型下的投资策略最优.

       In recent years, with the accelerated development of the financial market and financial globalization, financial market emergencies occur more and more frequently, so that investors face the financial market uncertainty and investment risk increases. In this context, it is particularly important to rationally hedge against uncertainty and investment risks of financial markets. The option pricing and investment portfolio, as the core content of modern finance theory, can better avoid the uncertainty and investment risks of financial markets. For option pricing and investment portfolio, the most critical thing is to construct models that can accurately characterize the dynamics of financial market data. However, the existing models describing the price trends of the underlying financial assets suffer from incomplete consideration of their dynamic characteristics. In addition, with the rapid development of financial market integration, the interest rate is no longer a constant, and it has been found that the stochastic fluctuation characteristic of the interest rate has a significant impact on option pricing and optimal investment decisions. Therefore, this thesis takes the dynamic change characteristics of financial underlying asset prices and the stochastic characteristics of the interest rate as the entry point to construct a model that conforms to the characteristics of the financial market, which is of great theoretical significance and application value to the research of option pricing and portfolio problems.

       The research focus of this thesis is to construct 4/2 stochastic hybrid European option pricing models with different characteristics based on the stochastic volatility, spikes, thick tails, jump diffusion of the underlying asset prices and stochastic fluctuation characteristic of the interest rate, and to test the pricing performance of the 4/2 stochastic hybrid models in European option pricing based on the 50ETF option data. In addition, based on the pricing performance of 4/2 stochastic hybrid models in option pricing, a 4/2-CIR jump diffusion portfolio model is constructed, the optimal investment strategy and utility loss are obtained under the model, and the effects of the model parameters on the optimal risk exposure and utility loss are explored. The details and main conclusions are as follows:

      (1) Based on the shortcomings of the existing stochastic volatility and stochastic interest rate hybrid models, a 4/2-CIR stochastic hybrid model is constructed to study the pricing of European options. Firstly, a 4/2-CIR stochastic hybrid model is constructed considering the influence of the stochastic characteristics of financial assets and interest rate on the option pricing results. Secondly, the characteristic function of the logarithmic price of the underlying asset and the European option pricing formula are obtained under the 4/2-CIR stochastic hybrid model. Finally, a numerical example is given to analyze the influence of interest rate factors on the pricing results of the model, and the model parameters are calibrated according to the actual data, and then the pricing performance and accuracy of the model are discussed. The results show that interest rate randomness has a significant impact on option pricing results, and the introduction of interest rate factors into the 4/2 stochastic volatility model can improve the pricing accuracy of the existing 4/2 stochastic volatility model.

      (2) Considering that the dynamic characteristics of financial asset prices have a significant impact on the option pricing results, this thesis proposes a 4/2 jump diffusion stochastic hybrid model, which comprehensively describes the stochastic volatility, spikes, thick tails, and jumps characteristics of the underlying asset. Firstly, on the premise that the price of the underlying asset has the characteristic of stochastic fluctuation, the double exponential diffusion process is introduced into the model describing the dynamic change characteristics of the underlying asset price, and the 4/2 jump diffusion stochastic hybrid model is constructed. Secondly, the corresponding characteristic function and the European option pricing formula are obtained under the 4/2 jump diffusion stochastic hybrid model. Finally, the model parameters are estimated by particle swarm optimization algorithm, and the pricing performance and accuracy are tested of the model based on the European option pricing formula and parameter estimates under the 4/2 jump diffusion stochastic hybrid model. The results show that compared with the existing 4/2 stochastic volatility model, 3/2 stochastic volatility model and Heston stochastic volatility model, the 4/2 jump diffusion stochastic hybrid model has the best pricing performance in the European option pricing problem. In addition, the 4/2 jump diffusion stochastic hybrid model has better pricing accuracy for in the money and out of the money option than at the money option.

      (3) In order to describe the dynamic characteristics of underlying assets and the stochastic characteristics of market interest rates, a 4/2-CIR jump diffusion stochastic hybrid model is proposed. Firstly, under equivalent martingale probability measure, a 4/2-CIR jump diffusion stochastic hybrid model is constructed based on the random fluctuations, spikes, thick tails, jumps of the underlying asset prices and the stochastic characteristics of interest rate. Secondly, the characteristic function of logarithmic price and the European option pricing formula are derived under the 4/2-CIR jump diffusion stochastic hybrid model. Finally, the model parameters are estimated by particle swarm optimization algorithm, and the pricing performance and advantages are tested of the 4/2-CIR jump diffusion stochastic hybrid model by comparative analysis. The results show that the 4/2-CIR jump diffusion stochastic hybrid model can not only fully describe the dynamic characteristics of the underlying asset and the stochastic fluctuation characteristic of the market interest rate, but also reflect the leverage effect between the underlying asset and volatility. In addition, compared with 4/2-CIR stochastic hybrid model and 4/2 jump diffusion stochastic hybrid model, 4/2-CIR jump diffusion stochastic hybrid model has the best pricing performance in European option pricing.

      (4) Under the utility function maximization criterion, a portfolio model is constructed in which the stock index satisfies a 4/2-CIR jump diffusion process. Firstly, the optimal solution of the HJB equation, investment strategy and risk exposure are obtained based on the 4/2-CIR jump diffusion portfolio model under the power utility function. Secondly, the optimality of HJB equation solution and investment strategy are proved under 4/2-CIR jump diffusion portfolio model. Finally, the effects of model parameters on optimal risk exposure and utility loss are analyzed by numerical examples, and the advantages of investment strategies based on 4/2-CIR jump diffusion portfolio model are analyzed by using the loss size of investment strategies under different models. The results show that risk aversion coefficient, investment duration and risk premium factors have significant effects on optimal risk exposure. In addition, the risk premium factor has a significant impact on short-sighted losses. Compared with the existing 4/2 stochastic volatility model and 4/2-CIR stochastic hybrid model, the investment strategy is the best under 4/2-CIR jump diffusion stochastic hybrid model.