2002 Fiscal Year Final Research Report Summary

On the approximation to the optimal rigion

Research Project

Project/Area Number	12630028
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Single-year Grants
Section	一般
Research Field	Economic statistics
Research Institution	Hitotsubashi University
Principal Investigator	TAKAHASHI Hajime Hitotsubashi University, Graduate School of Economics, Professor, 大学院・経済学研究科, 教授 (70154838)
Project Period (FY)	2000 – 2002
Keywords	Barrier option / Square root boundary / Nonlinear renewal theorem / Forward risk adjusted model
Research Abstract	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.