Finite mixture models and their use for option pricing and risk management

这项研究计划的最终目标是考察有限混合模型在金融计量经济学中的使用，并有助于理解如何使用这些模型为衍生品定价。有限混合模型是密度的凸组合，因为它们在指定基础随机变量的分布时提供了简约的灵活性，所以很有吸引力。可以根据需要将附加分量添加到该分布，以达到所需的任何精度来近似任何条件分布。这甚至适用于与金融关系最密切的高度倾斜和尖峰条件分布。该提案的结果将对社会产生积极影响，并具有明显的经济效益，因为它提供了加快金融创新的工具，以改善市场流动性，并使金融市场能够有效定价和承担风险。如果监管得当，这将增强金融稳定，降低未来市场崩盘或金融危机的可能性。我们会考虑三项特别的申请。 首先，将利用混合框架的灵活性。例如，模型可以被限制为只有一个条件方差过程以及条件偏度和超额峰度。此外，条件方差过程可以是不同类型的；一些可能具有不对称性，而另一些可能是弱非平稳的。最后，可以用具有恒定但“大”变化的分量来扩充模型，这些分量为模型添加了跳跃特征。因此，有限混合模型为测试这些特性的重要性提供了一个统一的框架，这是使用现有替代方案很难做到的。由于混合模型可以使用简单的计量经济学技术进行估计，因此比较这些特征在不同市场、不同资产类别和不同时间的相对重要性是可行的。研究结果将为市场参与者提供有关市场特征差异的重要见解。 其次，将在该框架内推导出期权价格的封闭形式的解，允许研究人员包括用于估计的期权数据，并根据观察到的期权价格来校准模型。通过校准参数的长时间序列，可以分析市场参与者在最近的全球金融危机期间预期的变化。由于混合模型可以有纯跳跃成分和弱非平稳成分，并产生每个成分的概率，因此本研究允许分析这些元素中的哪些元素发生了变化。因此，通过使用混合模型，我们不仅可以测量“什么”发生了变化，还可以测量“为什么”这种情况发生了变化，这是现有方法难以衡量的。这些信息为监管机构和监管者提供了增强金融稳定性的关键工具。 最后，利用可处理的多变量混合模型可以被构造的事实。例如，在这一框架中，个股和市场指数的动态可以以与资本资产定价模型完全一致的方式联合建模。如此保守的结构性联系对于期权定价和风险管理至关重要。可以对组成道琼斯工业平均指数的全部30只股票进行建模，并以内部一致的方式对这30只股票和指数的所有期权进行定价。此外，通过推导出的闭合形式的期权定价公式，可以利用现有的期权数据来获得个股的隐含贝塔，该方法可以用来反驳隐含相关性和协偏性和峰度的度量。这些措施对于金融机构进行风险评估和管理至关重要。

The ultimate goal of this research proposal is to examine the use of finite mixture models in financial econometrics and to contribute to the understanding of how derivatives can be priced using these models. Finite mixture models, which are convex combinations of densities, are attractive because of the parsimonious flexibility they provide in the specification of the distribution of the underlying random variable. Additional components can be added to the distribution as needed to approximate, to any accuracy desired, any conditional distribution. This works even for the highly skewed and leptokurtic conditional distributions most relevant in finance. The results of the proposal will positively impact society and have clear economic benefits as it provides tools to speed financial innovation to improve market liquidity and allow financial markets to efficiently price and bear risk. Properly regulated, this will increase financial stability and decrease the likelihood of future market crashes or financial crises. Three particular applications will be considered. First, the flexibility of the mixture framework will be exploited. For example, the model can be restricted to have only one conditional variance process and conditional skewness and excess kurtosis. Also, the conditional variance processes could be of different types; some might have asymmetries while others might be weakly non-stationary. Finally, the model can be augmented with components with constant but “large” variances which add jump-like features to the model. Thus, the finite mixture model offers a unified framework for testing the importance of these features, something that is difficult using existing alternatives. Because mixture models can be estimated using simple econometric techniques it is feasible to compare the relative importance of these features across markets and asset classes and through time. The results will offer important insights to market participants about differences in market features. Secondly, closed form solutions for option prices will be derived within this framework allowing researchers to include option data for estimation and to calibrate the model to observed option prices. With long time series of calibrated parameters, changes in market participants’ expectations during, for instance, the recent global financial crisis can be analyzed. Since the mixture model can have pure jump components and components that are weakly nonstationary and yields the probability of each of the components this research allows to analyse which of these elements have changed. Thus, by using mixture models we can measure not only “what” has changed but also “why” this has changed, something which is difficult to gauge with existing methods. This information provides regulators and supervisors with crucial tools to increase financial stability. Finally, the fact that tractable multivariate mixture models can be constructed is exploited. For example, in this framework the dynamics of individual stocks and the market index can be jointly modeled in a manner which is fully consistent with the capital asset pricing model. The structural links so conserved are vital for option pricing and risk management. It is possible to model the entire set of 30 stocks comprising the Dow Jones Industrial Average and to price the set of all options on the 30 stocks and the index in an internally consistent way. Moreover, with the derived closed form option pricing formulas the available option data can be used to obtain implied betas of individual stocks and the methodology can be used to back out implied correlations and measures of coskewness and cokurtosis. These measures are essential for the risk assessment and management done by financial institutions.