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

本研究计划的最终目标是检验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，因为它们在底层随机变量的分布规范中提供了简约的灵活性而具有吸引力。可以根据需要向分布中添加额外的分量，以达到所需的任何精度，近似任何条件分布。这甚至适用于与金融最相关的高度倾斜和细峰度条件分布。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，因为它提供了加速金融创新的工具，以提高市场流动性，并使金融市场能够有效地定价和承担风险。如果监管得当，这将提高金融稳定性，降低未来市场崩溃或金融危机的可能性。将考虑三个特定的申请。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和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；有些可能是不对称的，而另一些可能是弱非静止的。最后，可以用具有恒定但“大”方差的组件对模型进行扩充，从而为模型添加跳变特征。因此，有限混合模型为测试这些特征的重要性提供了一个统一的框架，而使用现有的替代方法很难做到这一点。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，将在此框架内推导出期权价格的封闭形式解，允许研究人员将期权数据纳入估计并校准模型以观察到的期权价格。利用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的预期变化。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，因此这项研究允许分析这些元素中的哪些发生了变化。因此，通过使用混合模型，我们不仅可以衡量“什么”发生了变化，还可以衡量“为什么”发生了变化，这是现有方法难以衡量的。这些信息为监管者和监督者提供了增强金融稳定的关键工具。最后，利用了可处理的多元混合模型的可构造性。例如，在这个框架中，个股和市场指数的动态可以以一种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。对构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只股票和该指数的所有期权进行定价，这是可能的。此外，通过推导出的封闭式期权定价公式，可以利用现有的期权数据获得个股的隐含贝塔，并利用该方法推导出隐含相关性以及余偏度和余峰度的度量。这些措施对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.