使用樹狀結構處理包含價格隨機跳躍風險和隨機利率的首次通過模型評價

Abstract

近年來信用風險議題廣泛受到各界重視。Black and Cox (1976)提出一重要的信用風 險模型：首次通過模型(first passage model)，並由Kim et al. (1993), Longstaff and Schwartz (1995), and Zhou (2001)加以改進。該模型假定公司的資產價值服從 特定的隨機過程，並假定當公司資產價值低於違約門檻時則公司發生違約事件。本研 究計畫提出一樹狀結構模擬首次通過模型，並討論隨機跳躍擴散模型(the jump-diffusion process)以及利率波動對信用風險的影響。公司資產價值的隨機跳躍 可用來模擬突發事件例如法律訴訟以及金融市場騷動。加入隨機利率模型則可用來模 擬公司資產和利率變化的關連性。使用上述如此複雜的模型評價各式金融商品的信用 風險是一個困難的問題，因為封閉公式解不易求出，而且使用樹狀結構評價時，也會 因為有非線性誤差(nonlinearity error)的問題而造成評價結果巨幅震盪(見 Figlewski and Gao (1999))。 本研究計畫預計在第一年建構一個樹狀結構，該樹狀結構可隨公司資產隨機跳躍以及 債務結構改變適時調整其結構，所以可以鉅幅壓低非線性誤差。所以該樹狀結構不僅 可達成Zhou(2001)提到的要求：評價短期公司債時產生較大信用價差(credit spread)，並產生不同形狀的信用價差曲線。它也可以處理具有提前贖回、轉換特性 (American-style features)的商品，例如可賣回債券(putable bonds)。並且可用來 分析公司債務結構改變對信用價差的影響。為了討論利率變化對信用風險的影響，第 二年的計畫將提出二維樹狀結構(two dimension tree)，以同時模擬公司資產價格及 短利(short term interest rate)的隨機過程及相關性。所以該樹狀結構可描述資產 價格深受利率變動影響的公司的信用風險，例如銀行及保險公司。為了驗證樹狀結構 的正確性及一般性，第三年的計畫將做實證研究。我首先會收集股票市場和債券市場 的交易資料，用來推估公司資產價格和短利隨機過程模型的參數，將參數代入樹狀結 構後，再比較其評價結果和市場真正交易資料的差異，並用該資訊來估計首次通過模 型中一些重要，但無法直接從市場上觀察到的參數，例如違約門檻以及破產後債權人 取回資金的比例(recovery rate)。

Credit risk problems have attracted much attention recently. The first passage model (FPM hereafter), pioneered by Black and Cox (1976), Kim et al. (1993), Longstaff and Schwartz (1995), and Zhou (2001) is a credit risk model that simulates the dynamics of the firm value process. It assumes the default occurs once the firm value is below the default boundary. This project develops a tree that models FPM with the jump-diffusion process, and the stochastic interest rate. The jumps (in the jump-diffusion process) can be used to model the surprising events like lawsuits and financial turmoil. The stochastic interest rate settings can be used to model the correlations between the firm value process and the change of the interest rate. Estimating the credit risks of various financial securities under such a complex credit risk model mentioned above is indeed a hard problem since analytical formulas are hard to be derived. Evaluating this model with the tree might result in oscillated results due to nonlinearity error (see Figlewski and Gao (1999)). The first year plan of this research project will develop a tree model that is flexible enough to fit the random jumps of the firm value and the change of liability structure to suppress the nonlinearity error. Thus my tree can generate large credit spreads for short term bonds and rich dynamics of credit spread curves as mentioned in Zhou (2001). It can also deal with American-style derivatives, like putable bonds and analyze how the change of liability structure influence the credit spreads. To measure the credit risk due to the change of interest rates, the second year plan of this research project will develop a two dimension tree that model the evolutions and the correlations of the firm value process and the short term interest rate process. Thus my tree can more exactly model the credit risk of the firm whose value is sensitive to the change of the interest rate, like banks and insurance companies. To verify the robustness and correctness of the aforementioned trees, the third year plan will do empirical studies. I plan to collect the trading data from the stock markets and the bond markets to calibrate the parameters used in modeling the firm value process and the short term interest rate process. Then I can compare the pricing results of my tree model with the real world market data and use this information to estimate some parameters which is important in FPM but can not be directly observed from the market, like exogenous default boundary, recovery rate, and so on.