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

点击翻译按钮获取中文摘要

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.