敲定时间为随机变量的情况下奇异期权定价问题研究

Valuing Exotic Options under the Strike-Time of the Random Variable

受保险精算中定价最小死亡保证金的启发,当死亡发生时,会收到一定数额的财富作为补偿,而这笔财富当作是一种支付,它不仅依赖于原生资产的当前价格,还依赖于之前的价格信息.可以把这个支付函数看做是一种特殊期权的收益函数.又由于随机变量Tx(表示年龄为x的顾客从购买合约到死亡的时间段)的分布可以被近似地看做是几个指数分布的线性组合.假设股票价格变化服从双指数跳扩散过程.利用Lévy过程的指数停时的有关结果,给出敲定时间为随机变量的情况下累计期权的价格公式的显式解.这些定价方法可以用于与死亡相关的未定权益的定价,如各种养老金保险等.

At the time of death, a benefit payment is due, and the benefit payment is regarded as a payment function, which depends not only on the price of a stock at that time, but also on prior prices.The payment turns out to be the payoff of an option, because the distribution of the time of death Tx （that is the time-until-death random variable for a life age x from buying the contracts） can be approximated by a linear combination of exponential distributions.For simplicity,the analysis is made for the case where the time until death is exponentially distributed, which is independent with the stock price process. In this paper, stock price was assumed to follow a double exponential jump diffusion process.The results for exponential stopping of a Levy process were used to derive a closed-form formula for a Knock Out Discount Accumulator, under the strike time is the random variable.The pricing method of this paper can be used for value equity-linked death benefits such as various annuities and so on.