On a derivative pricing theory with jumps and stochastic volatilities

首先，采用比例风险模型，这已被公认为是统计上有意义的分析和估计的金融事件风险，如违约风险和提前偿付风险，我们提供了一个分析处理的估值问题。其次，当标的资产价格由时变Levy过程驱动时，我们基于Gram-Charlier展开给出了一个平均期权定价的近似公式。时变Levy过程是有吸引力的基础价格的驱动因素，因为该过程提供了一个灵活的框架，产生跳跃，捕捉随机波动的随机时间变化，并引入杠杆效应。第三，提出了时变Levy过程下离散监控路径依赖期权的定价方法。该方法的关键是推导过程跨期联合分布的多元特征函数的一般公式。

Firstly, adopting the proportional hazard model, which has been recognized to be statistically meaningful for analyzing and estimating financial event risks such as default risk and prepayment risk, we provided an analytical treatment for the valuation problems. Secondly, we developed an approximate formula based on the Gram-Charlier expansion for pricing average options when the underlying asset price is driven by time-changed Levy processes. The time-changed Levy processes are attractive to use for a driving factor of underlying prices because the processes provide a flexible framework for generating jumps, capturing stochastic volatility as the random time change, and introducing the leverage effect. Thirdly, we proposed a pricing method for discretely monitored path-dependent options under the time-changed Levy processes. The key to the method is to derive a general formula for the multivariate characteristic functions of the intertemporal joint distribution of the processes.