一个与永久可转换债券有关的变分不等式

发布时间：2018-05-02 12:30

  本文选题：可转换债券 + 变分不等式　； 参考：《苏州大学》2012年硕士论文

【摘要】：本文研究下述变分不等式的边值问题：其中 问题(1)-(2)与一个永久可转换债券的定价问题有关。χ表示公司的总资产,f(χ)表示公司可转换债券的总价值,χ-f(χ)是公司股票的价格。δ为股票的分红率,σ为公司总资产的波动率,r为无风险利率,rδ。c是债券的分红率,γ是可转债全部转换为股票后在公司总资产中所占的比率,0γ1,α是公司总资产的上限。(参见[1]) 问题(1)-(2)虽然是一维问题,但因为算子N是二阶非线性的且在χ=0是退化的,该问题没有显示解,且理论研究具有相当的难度。文[1]的作者运用随机分析的方法,证明了问题(1)-(2)解的存在性,但他们未能证明解的唯一性,对自由边界的位置也没能作出估计。 本文利用自由边界问题理论中的惩罚方法,通过适当的逼近论证和精细的估计,证明了问题(1)-(2)在C([0,α])∩W2,∞((η,α))(η为(0,α)中任一数)中解的存在性和唯一性,并且得到了自由边界点的上下界估计。本文所用的方法,还可以推广来研究相应的具有有限到期时间的(与时间有关的)可转换债券的定价问题。
[Abstract]:In this paper, we study the boundary value problems of the following variational inequalities: The problem is related to the pricing of a permanent convertible bond. 蠂 denotes the total assets of the company f (蠂) denotes the total value of the convertible bond, 蠂-f (蠂) is the price of the company's stock, 未 is the dividend ratio of the stock, 蟽 is the fluctuation of the total assets of the company R is the risk-free interest rate, r 未. C is the dividend ratio of bonds, 纬 is the ratio of convertible bonds to stocks, and a is the upper limit of the total assets of the company. (see [1]) Although the problem is one-dimensional, the operator N is second-order nonlinear and degenerate at 蠂 ~ 0. The problem does not show a solution, and the theoretical study is quite difficult. The authors of paper [1] proved the existence of the solution of the problem by means of stochastic analysis, but they failed to prove the uniqueness of the solution and to estimate the position of the free boundary. In this paper, by using the penalty method in the theory of free boundary problems, we prove the existence and uniqueness of the solution of the problem (1) in C ([0, 伪]) 螕 W 2, 鈭，

本文编号：1833939