Valuation of a Basket CDS & CDO - John C. Hull - 6E, Chap.21.8

Pricing kth-to-Default Swap 技術問題 如何管理各種信用風險間的相關性？ Example 如果各債券間相關性很低，要全部都不 違約是非常不可能的，所以 First-to- Default Swap 的權利金必然較高。

Gaussian Copula Model 假設各債券之違約時間轉換為常態分布後，會 服從多元常態分布，且相關係數為常數 ρ 。

各別債券之違約條件機率

Pricing kth-to-Default Swap 給定相關係數，我們可以由上式求算 kth-to- Defualt Swap 的理論價格。

Pricing kth-to-Default Swap for Large N

Implied Copula Correlation 把前述的程序反向操作過來，透過 kth- to-Default Swap 的市場價格，我們可以 反求隱含相關係數。 注意到，實證上有「 Correlation Smile 」 的現象。

Exercise How does a 5-year nth-to-default CDS work? Consider a basket of 100 reference entities where each reference entity has a probability of defaulting in each year of 1%. As the default correlation between the reference entities increases what would you expect to happen to the value of the swap when (a) n = 1 and (b) b = 25?

Convertible Bonds - John C. Hull - 6E, Chap.21.9

Convertible Bond 相當於普通債附帶兩種選擇權  Conversion ：債權人可以在特定時點之 後，自由地將債券轉換成股票。  Call ：債務人可以在特定條件下，以特定 價格提前償還債務。但債權人可以在此 時選擇轉換。 優點 解決代理人問題、製造訊息效果。

Pricing Convertible Bond 可轉換債的價值隨著股價漲跌改變。我 們可以建構二元樹模擬股價路徑，進而 評價可轉換債。 此處，我們將信用風險納入考慮：公司 有可能突然倒閉，股價降至零，債券價 值為面額乘上回復率。

Pricing Model -structure  股價（ GBM ）： dS = (r-q)Sdt + σSdW  倒閉機率（ Poisson ）： λdt 二元樹 在每個時間間隔 Δt ：  有 p_u 的機率，股價上漲 u  有 p_d 的機率，股價下跌 d  有 1-exp(λΔt) 的機率，公司倒閉

Pricing Model -parameter

市場資料 股票現價 50% 波動率 30% 股息率 0% 無風險利率 5% 倒閉率 1% 回復率 40% Example 21.1 債券資料 面額 $100 利息率 0% 到期九個月 轉換條款 任何時間兩股 贖回條款 任何時間 $113

u = d = a = p_u = p_d =

Exercise A 3-year convertible bond with a face value of $100 has been issued by company ABC. It pays a coupon of $5 at the end of each year. At the end of the first year, it can be exchanged for 3.6 shares immediately after the coupon date. At the end of the second year, it can be exchanged for 3.5 shares immediately after the coupon date. The current stock price is $25 and the stock volatility is 25%. No dividends are paid on the stock. The risk-free interest rate is 5% with continuous compounding. The yield on bonds issued by ABC is 7% with continuous compounding and the recovery rate is 30%. (a) Use a three-step tree to calculate the value of the bond. (b) How much is the conversion option worth? (c) What difference does it make to the value of the bond and the value of the conversion option if the bond is callable any time within the first 2 years for $115? (d) Explain how your analysis would change if there were a dividend payment of $1 on the equity at the 6-month, 18-month, and 30-month points. (Hint: Use equation (20.2) to estimate the default intensity.)