On the approximation to the optimal rigion

我们考虑了新类型的障碍类型选项。与普通障碍期权不同的是，我们的障碍与时间的平方根成正比(指数平方根障碍敲出期权)，普通障碍期权根据潜在的布朗运动具有直线或恒定障碍。由于布朗运动的起伏与时间的平方根成正比，所以我们的势垒比其他势垒更自然。我们用Siegmund(1985)和Morimoto(1999)的结果来计算必要的概率。此外，我们还考虑了指数平方根障碍击倒期权的离散时间版本。我们利用Takahashi-Woodroofe(1981,1982)给出的非线性更新定理的渐近展开式来计算必要条件概率。将我们的结果与Aitsahelia-Lai(2000)的结果相结合，可以得到美式期权的范围，但最终的结果还没有出来。

We considered new type of barrier type options. Unlike the ordinary barrier options, which have either straight line or constant barrier in terms of the underlying Brownian motion, our barrier is proportional to the square root of the time (Exponential Square root Barrier Knockout Option). Since the fluctuation of Brownian motion is proportional to the square root of the time, our barrier is more natural than the others. We have used the results of Siegmund (1985) and Morimoto (1999) for calculating the necessary probabilities. In addition to the above, we have also considered the discrete time version of the Exponential Square Root Barrier Knockout Options. We have used the asymptotic expantion for the non-linear renewal theorem given by Takahashi-Woodroofe (1981, 1982) for calculating necessary conditional probabilities. The extentons to American type option may be obtained by combining our result with Aitsahilia-Lai (2000), the final result is yet to come though.