結構式模型下評價可轉換公司債

Abstract

近年來討論可轉換公司債的論文，都使用樹狀結構來模擬股價和短利的變化，像是Hung & Wang(2002)和Chamber & Lu(2007)，這些在計算信用風險都是利用Jarrow and Tumbull(1995)提出的縮減式模型(Reduced model)，利用在外流通債券去計算信用價差來求出公司的違約機率。然而這並沒有考慮到股價高低對公司違約機率的影響。本研究以首次通過模型(First passage time model)與結構式信用風險模型(Structural Model)求出公司價值和債務水準，來計算公司違約的機率。最後研究顯示是當股價越高，公司資產相對越高破產的機率低，反之亦然。這些結果是上述論文所無法呈現出來的結果。本文的三維度模型是同時計算股價和短期利率(根據Vasicek model)，把CRR tree和Hull-White tree結合在一起。在每一個節點上的股東權益都可以視為以公司資產為標的物的向下失效買權，在隨機利率下的向下失效買權價的計算，是利用Collin-Dufresne and Goldstein (2001)所提出的.債券價格是透過倒推法(Backward Induction)，求出可轉債期初價格。

Many recent papers price convertible bonds with a tree that simulate the evolutions of stock prices and interest rates, like Hung & Wang (2002) and Chamber & Lu (2007). These works follow the reduced-form model of Jarrow and Tumbull (1995) that calibrates a corporation’s default probability with the credit spreads of its outstanding bonds. However, the relationship between the corporation’s stock price and the corporation’s default probability is not considered. This thesis addresses this problem by incorporating the first-passage time model, a popular type of structural credit risk model that models the default event with the evolution of firm value and the debt level. The resulting model suggests that a higher stock price implies a relatively higher corporate value as well as a lower default probability, and vice versa— a good property that is not possessed in aforementioned models.My three-dimensional tree models the evolution of the stock price and the short-term interest rates (following the Vasicek model) with the combination of the CRR tree and the Hull-White tree. The equity value at each tree node can be treated as a down-and-out call option on the corporation value and thus the latter value and hence the default probability can be evaluated by the stochastic-interest-rate option pricing method proposed by Collin-Dufresne and Goldstein (2001). The backward induction is then used to price convertible bonds.