評價可轉換公司債之信用風險模型研究

Abstract

可轉換公司債同時具備有債券和權益的特性，它允許債券的持有人，在特定的時間點得 依事先約定好的轉換比例轉換成股票，所以可轉債的價格深受股價風險、利率風險以及 發行者的信用風險影響，而且提前轉換的特性也讓可轉債沒有封閉評價公式解，而需使 用數值方法求解。Brennan and Schwartz (1977,1980)模擬公司價值和利率的隨機過程， 並使用Merton(1974)的結構式模型處理信用風險。然而市場無法直接觀察公司價值，而 且Merton 模型無法處理到期日前公司違約的問題。Tsiveriotis and Fernandes (1998) 模擬股價的隨機過程，並將可轉債拆成權益和債券分別評價。另一方面，Hung and Wang (2002)和Chambers and Lu (2007)則建構了股價和利率的四元樹，並採用Jarrow and Turnbull (1985)的縮減式模型，利用信用風險價差的期限結構來推論違約機率。在上 述評價模型中，公司的股價只用來判斷債券是否要轉換成股票，卻和公司違約機率大小 無關。然而股價的高低不僅反應公司的財務狀況，也間接反映公司未來違約的可能性。 此外，上述論文的四元樹模型會出現機率為負的情形，而且他們假定回收率(recovery rate)為外生給定的常數，也與實證結果不符。 本研究計畫會從修改現有四元樹評價模型的瑕疵出發，我會將Dai(2009)所提出的三 元樹結構加入到原有的四元樹模型中，處理因短利過高而造成機率為負的問題。此外也 會根據實證研究發現的回收率和違約機率的關係(見Hamilton, (2007))，搭配縮減式模型 利用風險價差內生建構回收率。接下來我會將結構式模型引進到可轉債的四元樹評價模 型，修改Merton (1974)的方法，用股價倒推公司資產價值，以及公司違約的機率。最 後我會將結構式模型和縮減式模型結合，同時利用風險價差和股票價格來建構公司信用 風險模型，再將該信用風險模型融入四元樹模型來評價可轉債。

Convertible bonds have hybrid attributes of both fixed-income securities and equities. The bond holder can convert the bonds into stocks at certain time points with a predetermined conversion ratio. Thus the price of the convertible bond is deeply influenced by the stock price, the interest rate, and the credibility of the bond issuer. Due to the existence of early conversion, the convertible bond cannot be priced analytically and must be priced by numerical methods. Brennan and Schwartz (1977, 1980) model the stochastic processes of the firm value and the short term interest rate. They use the structural model proposed by Merton (1974) to model the credit risk. However, the firm value can not be directly observed from the markets and the default prior to bond’s maturity date can not be modeled under Merton’s approach. Tsiveriotis and Fernandes (1998) model the stochastic process of the stock price. They price the convertible bond by separate the bond’s value into equity and debt components. On the other hand, Hung and Wang (2002) and Chambers and Lu (2007) construct quadtrees for simulating the stock price and the short rate. They adopt the reduced model proposed by Jarrow and Turnbull (1985) and infer the default probabilities by calibrating the term structure of credit spreads. In their models, the endogenously modeled stock price process is only used to determine the optimal conversion strategy and the stock prices are irrelevant to the default probabilities. However, the stock price level indeed reflects the financial status of the firm and the likelihood of default. In addition, the branching probabilities of their quadtrees can be negative. They assume the recovery rate is an exogenously given constant, which is also inconsistent with empirical studies. The first major goal of my project is to fix the aforementioned pitfalls of the quadtree models. To deal with negative branching probabilities problem due to high short rates, the trinomial branches proposed in Dai (2009) will be employed to modify the structure of the quadtrees. Next, I will substitute the relationship between the recovery rate and the default probabilities proposed in an empirical study document (see Hamilton (2007)) into the reduced model, and model endogenous recovery rates by calibrating the credit spreads. Next, I will incorporate the structural model into quadtrees for pricing convertible bonds. The endogenously defined stock prices can be used to infer the firm values and the default probabilities by modifying the method suggested in Merton (1974). Finally, I will integrate both the reduced model and the structural model to model the credit risk of the firm. Both the credit spreads and the stock prices will be fully utilized to infer the default probabilities. This information can be integrated into quadtrees for pricing convertible bonds.