含違約風險之可轉換債券評價

Abstract

本文探討利用股價、短期利率和違約風險三因子模型評價可轉換債券。其中，利用二項式 (CRR) 模型生成股價二元樹；使用Ho-Lee模型與Black-Derman-Toy (BDT)模型生成短期利率二元樹；違約風險使用Chambers-Lu模型與Hung-Wang模型分別推導各期間的風險中立違約機率通式，比較兩模型違約機率通式，可發現在到期日越長的無風險債券現值越小假設下，Chambers-Lu模型的違約機率大於Hung-Wang模型的違約機率。 利用股價、短期利率和違約風險生成六元樹找出風險中立機率，可知此模型不具有完備性，因此可轉換債券價格並非唯一的。可利用風險中立機率計算可轉換債券價格區間，並發現Hung和Wang文中所使用之機率並非風險中立機率。另外，比較和討論當分割期數趨近無限大時，可轉換債券價格區間收斂性。

This paper presents the valuation of convertible bonds by using stock price, short rate, and default risk. We used binomial (CRR) Model to construct binomial stock price tree; while short rate was transformed into two binomial trees by using Ho-Lee Model and Black-Derman-Toy (BDT) Model. Two different equations were obtained while analyzing default risk by Chambers-Lu Model and by Hung-Wang Model. By investigate these two equations, we demonstrated if the longer the duration, the smaller the price of default-free bonds. Then, default probabilities obtained by Chambers-Lu Model tend to be greater than the probabilities obtained by Hung-Wang Model. After using stock price, short rate, and default risk to construct hexnomial tree and calculate risk-neutral probabilities, we concluded that the model is incomplete since these prices were not unique. Convertible bond price intervals were calculated using by risk-neutral probabilities. We observed that the probabilities used in the paper introduced by Hung and Wang were not risk-neutral probabilities. Moreover, we compared and discussed the convergence convertible bond price intervals as the number of periods→∞.