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Valuation of a Basket CDS & CDO

Valuation of a Basket CDS & CDO. John C. Hull < Option, Futures, and Other Derivatives > 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.

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Valuation of a Basket CDS & CDO

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  1. Valuation of a Basket CDS & CDO John C. Hull < Option, Futures, and Other Derivatives >6E, Chap.21.8

  2. 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。

  3. 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。

  4. 各別債券之違約條件機率

  5. 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。

  6. Pricing kth-to-Default Swapfor Large N

  7. 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。

  8. Exercise 21.11 • 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?

  9. Convertible Bonds John C. Hull < Option, Futures, and Other Derivatives >6E, Chap.21.9

  10. 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。

  11. 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。

  12. 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

  13. Pricing Model-parameter

  14. Example 21.1

  15. u = 1.1519 • d = 0.8681 • a = 1.0126 • p_u = 0.5167 • p_d = 0.4808

  16. Exercise 21.28 • 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.)

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