A CDO Pricing Model with Bivariate Tree Approach: Considering the Correlation between Stochastic Interest Rate and Stock Price

本文章建構結合股價與利率的可違約立體樹狀模型，用以模擬CDS指數分券的價格，並且與市場價格比較驗證。 模型考慮隨機股價以及隨機利率，並採用均值追蹤演算法結合立體樹上分支機率限制 [0,1]，求得立體樹上由期初至到期日的路徑。相較於過往的立體樹模型，本文將股價的隨機過程與利率隨機過程的相關性、公司期間所配置的股利納入考量，立體樹上的各節點數據都可以透過對數轉換成同一時間點相對應的股票價格; 違約強度公式配置也有所更動，本論文將Nelson和Siegel於1987年發表的簡化殖利率曲線模型加入違約強度公式，以反映市場殖利率環境對公司違約機率的影響。 文章後半部，應用蒙地卡羅模擬iTraxx歐洲指數中114家公司的CDS價差分券。首先從期初開始，在每一期的計算過程中，同時產生累積機率與違約機率相比較，判斷該時間點立體樹狀模型違約與否以及立體樹上X值的路徑，直到到期日模擬數據告一段落，將數值轉換成股價，再將各公司發行之CDS價差按比例建立CDO分券。接著從到期日開始，將立體樹上之各節點之分券價值，回溯折現加總得到所有公司分券的價格的現值，並與實際市場價格相比較。A defaultable bivariate tree approach is introduced to simulate the value of CDX and compare with real market price. The stochastic processes of both stock prices and interest rates are considered in my bivariate model. The model also took the dividends and the correlation between two stochastic variables into account. I also adopted Dai’s mean tracking algorithm to improve the stability of probabilities and improved the default intensity formula by Nelson, Siegel’s (1987) parsimonious yield curve model. I used the corresponding data of iTaxx index and applied Monte Carlo simulation to simulate the spread of the iTraxx tranches. On the first stage, the model is calibrated to the current CDS market. Then I output the cumulated probability and compare with default probability. The default event is identified and the route on the bivariate tree is decided during the process. One the second stage, the tranche structure is created and the spread value is evaluated. I adopted the backward induction during the process for calculating the present discounted value. The above procedures were repeated twice under two assumptions: (1) The stochastic processes of interest rates and stock prices are independent. (2) The stochastic processes of interest rates and stock prices are dependent by considering the correlation coefficients between interest rates and each company. Finally, I compared the absolute errors between independent case and dependent case. The result shows that the independent case was slightly closer to the dependent case