含違約風險之可轉換債券評價

Valuation of Convertible Bonds with Default Risk

王心吟(Hsin-Yin Wang)

指導教授 ： 彭栢堅

國立臺灣大學/理學院/數學研究所/碩士(2009年)

https://doi.org/10.6342/NTU.2009.10535

摘要

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。

關鍵字

可轉換債券 ； 二項式模型 ； BDT 模型 ； Ho-Lee 模型 ； 違約風險 ； Chambers-Lu 模型 ； Hung-Wang 模型

並列摘要

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→∞.

並列關鍵字

Convertible Bond ； Binomial (CRR) Model ； Black-Derman-Toy (BDT) Model ； Ho-Lee Model ； Default Risk ； Chambers-Lu Model ； Hung-Wang Model

參考文獻

[2] Black, F., E. Derman, and W. Toy. “A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options.” Financial Analysts Journal, Vol. 46, Issue 1. (1990), 33-39.

[4] Chambers, D. R. and Q. Lu. “A Tree Model for Pricing Convertible Bonds with Equity, Interest Rate, and Default Risk.” The Journal of Derivatives, Vol. 14, Issue 4. (2007), 25-46.

[5] Cox, J. C., S. A. Ross, and M. Rubinstein. “Option Pricing: A Simplified Approach.” Journal of Financial Economics, Vol. 7, Issue 3. (1979), 229-263.

[6] Ho, T. and S. B. Lee. “Term Structure Movements and Pricing Interest Rate Contingent Claims.” Journal of Finance, Vol. 41, No.5. (1986), 1011-1029.

[7] Hull, J. C. “Options, Futures, and Other Derivatives.” Prentice Hall, New Jersey, fifth Edition, 2003.