On a derivative pricing theory with jumps and stochastic volatilities

首先，我们采用比例风险模型对估值问题进行了分析处理。比例风险模型被认为对违约风险和提前还款风险等财务事件风险的分析和估计具有统计意义。其次，基于Gram-Charlier展开式，推导了标的资产价格受时变Levy过程驱动时平均期权定价的近似公式。随时间变化的Levy过程对于潜在价格的驱动因素很有吸引力，因为该过程提供了一个灵活的框架来产生跳跃，捕捉随机时间变化时的随机波动，并引入杠杆效应。第三，我们提出了一种时变Levy过程下离散监测路径依赖期权的定价方法。该方法的关键是推导出过程间关节分布的多变量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.