Option Pricing with Multivariate GARCH Models

本研究计划的最终目标是1)开发一个多元统计建模框架，该框架足够灵活，可以描述动态并考虑到金融资产之间的相关性，这可以用于风险管理，一般来说，特别是在高维上的期权定价；2)提供监管机构和监督者急需的工具，以提高金融稳定性，金融机构从业人员需要快速评估其风险敞口，改善市场流动性，并允许金融市场更有效地定价和承担风险。利用所提出的模型，多种资产的衍生品可以以理论上一致的方式定价，研究计划为该框架下的衍生品定价开发了创新的封闭形式易于计算的表达式或公式。目前，在这种灵活的框架下，唯一可能的期权定价方法依赖于耗时的蒙特卡罗模拟等数值技术，而及时为证券定价需要牺牲模型真实性的捷径。另一方面，封闭形式公式允许对当今金融市场中存在的复杂衍生品进行快速准确的定价。提出的框架还允许推导期权价格敏感性的封闭形式表达式。这些措施用于评估这些衍生产品的风险，对于有效管理与此类资产相关的风险是必要的。金融机构和政府监管机构经常进行这样的计算，以管理拥有数千种资产的投资组合的风险。因此，封闭形式的解决方案非常重要，因为它们允许在不牺牲精度的情况下进行可靠的实时风险管理。本研究推导出的公式使得在模型估计中纳入历史期权价格成为可能，从而更有效地估计模型参数。将历史期权价格纳入这个多变量框架还允许计算金融市场感知风险的创新措施，考虑到这些市场的复杂性和相互依赖性。这些创新措施将导致对复杂产品所涉及的风险进行比目前可用的更好和更一致的评估。该项目开发的灵活框架将引起学术界、金融业从业人员和监管机构的兴趣。所提出的模型对金融衍生品如何估值、如何管理其风险以及如何制定、实施和评估金融政策都有影响。拟议的研究计划将大大有利于加拿大经济提供工具，以提高市场流动性，允许金融市场有效定价和承担风险，增加金融稳定性和减少未来金融危机的可能性。

The ultimate goal of this research program is to 1) develop a multivariate statistical modelling framework, flexible enough to describe the dynamics and to take into account the correlations among financial assets, which can be used for risk management, in general, and option pricing, in particular, in high dimensions, and 2) provide the tools that regulators and supervisors crucially need to increase financial stability and that practitioners in financial institutions need to rapidly evaluate their risk exposures, improving market liquidity, and allow the financial markets to price and bear risk more efficiently. With the proposed model, derivatives on multiple assets can be priced in a theoretically consistent way and the research program develops innovative closed form readily computed expressions or formulas for derivatives pricing in this framework. Currently, the only possible methods for option pricing in such a flexible framework rely on time consuming numerical techniques like Monte Carlo simulation, and pricing securities in a timely manner requires shortcuts which sacrifice realism in the models. Closed form formulas, on the other hand, allow for fast and accurate pricing of the complex derivatives that exist in today's financial markets. The proposed framework also allows derivation of closed form expression for option price sensitivities. These measures are used to assess the risks of these derivative products and necessary for efficient management of the risks associated with such assets. Financial institutions and government regulators frequently perform such calculations to manage the risk of portfolios with thousands of assets. Therefore, closed form solutions are of immense importance because they allow reliable risk management in real time without sacrificing precision. The formulas derived in this research make it possible to incorporate historical option prices in the model estimation, leading to more efficient estimates of model parameters. Incorporating historical option prices in this multivariate framework also allows for calculating innovative measures of the perceived risk in financial markets that take into consideration the complex nature and interdependence of these markets. These innovative measures will lead to better and more consistent assessment of the risks involved in complex products than what is currently available. The flexible framework developed in this project will be of interest to academics, financial industry practitioners, and regulators. The proposed models have implications for how financial derivatives are valued, how their risks are managed, and therefore on how financial policy should be created, implemented, and evaluated. The proposed research program will greatly benefit the Canadian economy by providing tools to improve market liquidity and allow financial markets to efficiently price and bear risk, increasing financial stability and decreasing the likelihood of future financial crises.