带跳的多因素随机波动率模型的统计分析及金融衍生品定价

Financial derivatives valuation has been a central issue in the theory of asset pricing. The key to the derivatives valuation is to model the underlying asset price dynamics, in which the volatility is the most important variable. The existing asset pricing models do not perform well empirically in practice because of the impacts of many very short-lived but erratic extreme events usually found in financial market and these events are not fully considered in the existing asset pricing models. This project considers the underlying asset price dynamics following a continuous-time multifactor stochastic volatility model with jumps, in which the stochastic volatility is driven by the covariance matrix specified by the asymmetric affine jump-diffusion processes. At first, by using the data analysis method, statistical calculation and simulation technique, the following issues such as the statistical properties of big jump risks, model selection and parameter estimations, stochastic correlation, and the integrated volatility and volatility term structure, are discussed based on the real market data. Then, numerical methods on some derivatives valuation are modelled based on the above proposed model by applying numerical approximation and stochastic analysis approaches. With this research project, we will obtain the behaviors of the asset price dynamics, and the efficient numerical schemes for both the statistical inference of high dimensional model and the derivatives valuation, which will be able to provide some new theories and methods for asset pricing and risk management, as well as the basis for the scientific decision-making in practice. These will be also beneficial for improving the development of such disciplines as Mathematics, Statistics and Finance in Guangxi, and the talent cultivation and regional economy.

金融衍生品定价是资产定价理论研究领域的一个中心课题，衍生品定价的关键是估计其标的资产价格的变动过程，波动率是刻画这种变动过程的最重要变量。由于金融市场常受许多极其短暂且不稳定极端事件的冲击，现有的资产定价模型没有完全考虑这些冲击，已被证实存在不足。本项目考虑标的资产满足一类连续时间带跳的多因素随机波动率模型，其中波动率是非对称的仿射跳扩散过程描述的协方差阵过程。首先，应用数据分析和统计计算与模拟技术，借助市场数据研究大跳跃风险的统计特性、模型选择与参数估计、随机相关性和波动率的协整与期限结构；然后，基于此模型，应用数值近似与随机分析探讨几类衍生品定价方法。通过研究获取资产价格变动行为和高维模型统计推断与衍生品定价的有效算法。研究结果将为资产定价与风险管理提供新的理论和方法，也将为实际应用的科学决策提供依据。这有利于推动广西的数学、统计学与金融学等学科的发展，并为人才培养和地方经济建设服务。

金融衍生品定价是资产定价理论研究领域的一个中心课题，衍生品定价的关键是估计其标的资产价格的变动过程，波动率是刻画这种变动过程的最重要变量。由于金融市场常受许多极其短暂且不稳定极端事件的冲击，现有的资产定价模型没有完全考虑这些冲击，已被证实存在不足。本项目考虑标的资产满足一类连续时间带跳的多因素随机波动率模型，其中波动率是非对称的仿射跳扩散过程描述的协方差阵过程。本课题研究了三个方面的问题：1.带跳跃风险的多因素随机波动率模型下金融衍生品定价的几个深入问题。a. 美式连续分期付款期权、美式期权、回望期权、障碍期权、亚式期权，以及创新设计了一类外向型重置期权，对这些金融衍生品建立了相应的定价模型，给出了数值计算实例，分析了模型参数的敏感性，结果很有科学价值和实际应用意义；b. 构建了一类瞬时利率运动模型，并给出了基于债券的美式期权定价算法。2. 应用数据分析和统计计算与模拟技术，借助市场数据实证研究了大跳跃风险的统计特性，随机相关性和波动率的协整与期限结构，并应用于保险精算建模和Markov决策过程；3.对多维随机延迟积分-微分方程研究了数值算法，给出了稳定性理论结果。这些研究结果将为资产定价与风险管理提供新的理论和方法，也将为实际应用的科学决策提供依据，并为人才培养和地方经济建设服务。