Martingales and Super-Martingales Relative to a Convex Set of Equivalent Measures

In the paper, the martingales and super-martingales relative to a convex set of equivalent measures are systematically studied. The notion of local regular super-martingale relative to a convex set of equivalent measures is introduced and the necessary and sufficient conditions of the local regularity of it in the discrete case are founded. The description of all local regular super-martingales relative to a convex set of equivalent measures is presented. The notion of the complete set of equivalent measures is introduced. We prove that every bounded in some sense super-martingale relative to the complete set of equivalent measures is local regular. A new definition of the fair price of contingent claim in an incomplete market is given and the formula for the fair price of Standard Option of European type is found. The proved Theorems are the generalization of the famous Doob decomposition for super-martingale onto the case of super-martingales relative to a convex set of equivalent measures.

非风险中性定价意义下欧式未定权益定价及其套期保值策略

在非风险中性定价意义下,研究了欧式未定权益的定价和套期保值策略.通过期权价格过程的分布,利用等价鞅测度,分别在有红利和无红利两种情况下,得到了广义的欧式期权定价公式,也给出了欧式卖权和买权之间的平价关系;利用伊藤公式,得到欧式卖权和买权的套期保值策略.这些结果包含了在风险中性意义下原始的欧式期权定价公式和套期保值策略.

有摩擦金融市场中的美式未定权益定价

本文研究了高借款利率下投资策略受限制的美式未定权益的定价问题.文章通过引入反映上述金融市场摩擦的辅助的无摩擦金融市场类给出了美式未定权益的上下套期保值价格hup(K)和hlow(K)的定价公式.进一步,在基于金融市场无套利的准则下证明了[hlow(K),hup(K)]是美式未定权益的无套利价格区间.最后在投资策略受到某些具体限制的情形下,以美式看涨期权为例,给出了上下套期保值价格的显式表达式或估计式.

基于连续红利支付和随机波动率的未定权益定价模型

研究了具有连续红利支付和随机波动率的未定权益定价问题,利用等价鞅测度的方法推导了风险中性下的欧式未定权益定价公式.

带有随机波动率的交换期权定价

通过利用计价单位的变换及等价鞅测度,得到了在随机波动率下的交换期权定价公式.

完备的随机波动率模型的统计推断

完备的随机波动率模型是David G.Hobson and Rogers L G 于1998年引入一类新的随机波动率模型,与现有波动率模型相比,它有很多优点.本文讨论了这类模型的估计问题。通过测度变换,将系数σ(·)的估计转化为—般回归函数的估计问题,给出了该系数的核估计量,并证明了所得估计量的均方收敛性。