q-分布理论及其在期权定价中的应用

发布时间：2018-05-19 10:03

本文选题：Tsallis熵 + q-分布　；参考：《武汉理工大学》2012年硕士论文

【摘要】：随着现代信息技术的快速发展和全球一体化程度的不断提高,各种新型衍生金融产品的出现已经势不可挡。期权作为金融衍生工具的重要一员,是投资者进行资本套期保值的常用理财手段。在真实金融市场中,标的资产收益率并不服从高斯分布,而是服从一种具有尖峰厚尾特征的分布。著名的标准B-S期权定价模型是目前最重要且应用最为广泛的定价模型,它是基于收益率服从高斯分布的假设下推导而来的,但这无疑会带来与实际价格的偏差,从而产生了“波动微笑”。如今已有很多学者研究了期权定价的修正模型,如随机波动模型、截断列维飞跃模型、α-稳定分布模型等,然而这些方法的使用都较为复杂且得不到闭形解。由于标准B-S模型结果存在偏差的根本原因在于其标的资产收益率服从高斯分布这个假设,而从复杂系统Tsallis熵理论中推导出的q-分布能较好地表示具有尖峰厚尾特性的多尺度分布,因此我们通过修正B-S期权定价模型的假设推导出修正的期权定价公式。由于目前关于q-分布族的理论研究并不完善,因此本文首先将进一步完善q-分布的统计性质,即推导几种常用q-分布族的统计数字特征(期望、方差和k阶矩)和参数估计方法(矩估计、极大似然估计和q-极大似然估计),以期完善其理论基础；然后介绍了标准B-S模型、基于q-高斯分布假设下的B-S期权定价模型和离散化期权定价理论；最后运用实际金融指数及标的资产价格进行实证分析,在实证中,首先估计分布的参数并根据均方误差来比较不同估计方法的优劣,然后对所选取的期权数据进行定价模型的对比。结果表明,极大似然估计方法与其他两种估计方法相比较优,因此在对标的资产收益率数据进行分布参数估计时,我们选取极大似然估计方法。针对四种期权定价模型的对比结果表明,基于q-高斯分布的B-S期权定价模型比标准B-S模型更接近实际价格,而基于高斯和q-高斯分布的两种离散化模型同样高估了期权的价格。因此,在实际期权定价中,基于q-高斯分布的B-S期权定价模型较为可行有效。
[Abstract]:With the rapid development of modern information technology and the increasing degree of global integration, the emergence of various new derivative financial products has become irresistible. As an important member of financial derivatives, option is a common financial means for investors to carry out capital hedging. In the real financial market, the return on underlying assets is not distributed from Gao Si, but from a distribution with sharp and thick tail. The famous standard B-S option pricing model is the most important and widely used pricing model, which is derived from the assumption of Gao Si distribution, but it will undoubtedly bring deviation from the actual price. This creates a "wave smile". Nowadays, many scholars have studied the modified model of option pricing, such as stochastic volatility model, truncated Levelian model, 伪 -stable distribution model, etc. The fundamental reason for the deviation of the results of the standard B-S model lies in the assumption that the return rate of the underlying asset is distributed from Gao Si. The q-distribution derived from the Tsallis entropy theory of complex systems can well represent the multi-scale distribution with the characteristic of peak and thick tail, so we derive the modified option pricing formula by modifying the hypothesis of B-S option pricing model. Since the current theoretical research on q-distribution family is not perfect, this paper will further improve the statistical properties of q-distribution, that is, to deduce the statistical numerical characteristics of several commonly used q-distributive families. Variance and k-order moments) and parameter estimation methods (moment estimation, maximum likelihood estimation and q-maximum likelihood estimation) in order to improve their theoretical basis, then the standard B-S model is introduced. B-S option pricing model and discrete option pricing theory based on q- Gao Si distribution hypothesis. First, the parameters of the distribution are estimated and the advantages and disadvantages of different estimation methods are compared according to the mean square error, and then the pricing model of the selected option data is compared. The results show that the maximum likelihood estimation method is better than the other two methods, so we choose the maximum likelihood estimation method when we estimate the distribution parameters of the return data of the underlying assets. The comparison results of four option pricing models show that the B-S option pricing model based on q- Gao Si distribution is closer to the actual price than the standard B-S model, while the two discrete models based on Gao Si and q- Gao Si distribution also overestimate the option price. Therefore, the B-S option pricing model based on q- Gao Si distribution is feasible and effective in the actual option pricing.
【学位授予单位】：武汉理工大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：F224;F830.9

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