几类典型双斜过程的性质及其在金融衍生品定价中的应用研究

In this project, we mainly investigate some properties of several typical doubly skewed processes—doubly skewed OU (Ornstein-Uhlenbeck), doubly skewed CIR (Cox-Ingersoll-Ross) and doubly skewed geometric Brownian motion. The transition density function, the distribution of first hitting time and the occupation time are considered in the beginning. Based on the theoretical research, we study the pricing of European call/put option, barrier option, power exchange option under doubly skewed model in riskless and risk market. At last, we compute the estimations of drift coefficient, diffusion coefficient, skew parameter and skew point of doubly skewed OU, doubly skewed CIR and doubly skewed geometric Brownian motion using MCMC (Markov Chain Monte Carlo). This enriches the theoretical studies of doubly skewed process. And this is the first time to price financial derivatives under doubly skewed process which has important theoretical significance and practical value.

本项目旨在研究几类典型双斜过程--双斜OU (Ornstein-Uhlenbeck)、双斜CIR (Cox-Ingersoll-Ross) 和双斜几何布朗运动的理论性质，包括转移密度、首达时分布、占位时等。在此研究的基础上，我们主要利用谱展开和测度变换的方法讨论无违约风险和带有违约风险的市场中双斜模型下欧式看涨／看跌期权，障碍期权、幂交换期权等典型金融衍生品的定价问题。最后利用MCMC (Markov Chain Monte Carlo)方法给出双斜OU、双斜CIR和双斜几何布朗运动的漂移系数、扩散系数、斜参数和斜点的参数估计。这丰富了双斜过程的理论研究，并首次尝试在双斜过程模型下做金融衍生品的定价，具有重要的理论意义和应用价值。

本项目主要研究双斜模型的理论性质及这些模型下衍生品的定价问题，对我国的外汇、利率衍生品具有重要意义。双斜模型适合刻画的资产动态，在我国市场上均能被频繁观察到，比如美元兑港币汇率持续稳定在7.75和7.85之间，以及上证50指数存在障碍水平等。利用双斜模型拟合这些动态具有其他模型不可比拟的优势，具有重要的理论意义和应用价值。.主要研究内容如下：.1）计算了双斜OU、双斜CIR的理论性质，包括转移密度、首达时分布、占位时等；.2）给出了不带违约风险和带违约风险的市场中几何布朗运动、GARCH、TGARCH模型下欧式看涨/看跌、幂交换期权的价格；.3）构造了双斜CIR、双斜CEV模型的二叉树模型并给出了该模型下欧式、美式期权的价格；.4）计算了粘性双指数跳扩散过程下违约时的拉普拉斯变换，并分析了价格聚集和跳风险对违约时分布和债券信用利差的影响；.5）实证分析了上证50指数存在障碍水平，提出利用局部波动率模型来描述标的资产动态，并推导得到该模型下欧式期权的价格。.主要成果有：.1）Analytical valuation of power exchange options with default risk (一作), Finance Research Letters, 2019..2）On the probability of default in a market with price clustering and jump risk (通讯), Mathematics and Financial Economics, 2020..3）A closed-form GARCH valuation model for power exchange options with counterparty risk (通讯), Probability in the Engineering and Informational Sciences, 2020..4）On the ttransition density and first hitting time distributions of the doubly skewed CIR process (一作), Methodology and Computing in Applied Probability, 2020.