金融高频大数据下的风险推断及其与多元标的衍生品定价和金融风险管理的交叉融合研究

This project shall focus on the following subject matters: inference for integrated volatility matrices using high-frequency high-dimensional financial data; multi-asset derivatives pricing; and derivatives pricing and financial risk management incorporating high-frequency realized measures. For the volatility matrix inference problem, we shall adopt the theory of Dambis, Dubins and Schwarz (the well known DDS theorem) and use the time change technique to resolve the heavy dependence of existing methods on the sparsity assumption made for the target volatility matrix as well as the assumption that restricts the level of leverage effect (i.e., the negative dependence between price changes and volatility changes) and hence, the conditions of using our theory and method are much more realistic than that in the literature. We shall also investigate the effect of complicated market microstructure noises including both random additive noise and rounding error on the estimation of volatility matrices, and propose new methods that are readily applicable to this more realistic situation. For the multi-asset derivatives pricing problem, we shall derive the closed-form relation between the Copulas under the physical measure and the equivalent martingale measure (EMM) using Levy Copula methods, answering the unresolved fundamental question that whether the structure of nonlinear dependence among underlying assets' price processes remains unchanged under the measure change from the physical one to the EMM. Because the existing realized-measures-based derivatives pricing and risk management methods are all built on the assumption that realized measures are estimates of the relevant risk targets with high precision, the above new theories and methods thus have direct implications in developing new empirical pricing and risk management models that incorporate high-frequency realized measures.

本申请项目关注：高频高维波动率矩阵的推断；多元标的（multi-asset）衍生品定价理论研究；及由此自然延伸的高频数据方法与多元标的衍生品定价及风险管理的交叉融合研究。面向实际情境，我们拟使用连续鞅论的Dambis, Dubins, Schwarz时间刻度变换（time change）技术解决文献里波动率矩阵估计方法对矩阵稀疏性和杠杆效应水平限制假设的依赖；解决高频数据中带有价格离散干扰的复杂市场微观结构效应对波动率矩阵估计的影响；使用Levy Copula方法解决文献里多元标的衍生品定价理论不能给出基础资产价格间非线性相关的一般相依性在物理测度和风险中性测度下的联系及其金融经济学解释的问题；由于文献中结合高频数据的衍生品定价和风险管理方法多建立在精确的已实现风险度量（realized measures）上，上述新的理论方法将启示我们发现更加贴近实际的高频实证定价及风险管理模型和方法。

在金融市场存储数据越发膨胀的今天，发展挖掘利用高频甚至高维高频金融数据的计量理论方法，对金融风险管理及金融衍生品定价具有重要实际意义。本项目在以下应用基础研究方面取得重要结果：（一）作为研究高频金融数据资产价格波动率或波动率矩阵性质的基础，我们深入研究波动率过程的特征，即波动率过程的积分波动率，我们在具有市场微观结构噪声和价格跳跃的情形下，首次给出了波动率的波动率的非参数估计及其渐近性质，并给出了一个判断波动率过程是否具有扩散项的假设检验方法。（二）在一般化Roll(1984)模型下，对随机加性微观结构噪声做参数模型限制，同时考虑价格离散因素对波动率估计的影响。我们发现用非线性最小二乘估计及粒子滤波相结合的估计方法，能给出非常准确并实际可行的波动率估计，模拟及实证结果都显示了我们方法相较传统方法的优越性。（三）在同时有市场微观结构噪声和日内周期效应下，给出了基于Khmaladze鞅转换方法的波动率函数形式检验方法，首次在复杂的市场微结构下给出均衡价格波动率函数形式的高频检验方法，为我们更深刻的认识波动率在高频下的性质做了基础理论工作。（四）在因子模型下大维波动率矩阵的推断问题中，我们基于股票市场公共因子的波动和市场波动高度相关这一特性，利用时间变更采样方法，解决高频时间序列数据的弱相依性和非平稳性问题，放松了大维随机矩阵理论在应用中对于时间序列相依性的限制。（五）在使用多元无限可分分布及Levy Copula理论研究多元标的衍生品定价问题方面，我们在一类流行的Levy过程即CGMY过程的时变布朗运动表示形式基础上，进一步发现了关于时变从属过程的一种新的分解理论，并在此基础上提出了两种关于CGMY 过程路径的序贯模拟方法，克服了已有文献中的模拟方法模拟偏差难以界定，模拟误差参数设定盲目的缺点。