有信用風險的Heath-Jarrow-Morton利率模型對利率衍生性金融商品定價

Abstract

本文以Trolle and Schwartz (2008)的模型為基礎，從擴散項(diffusion term)機率分配相同的角度設定無信用風險的期限結構模型，同時在模型加入信用風險，此模型具有N個期限結構因子， N個非期限結構因子，同時讓遠期利率、隨機波動度、遠期信用利差三條隨機過程都具有Square Root的特性，亦即保證即期利率、隨機波動度、即期信用利差三者恆大於等於零。此模型具有有限維度馬可夫性質，在有限狀態變數的條件下，分別推導出風險中立下的無、有信用風險的債券價格，此模型亦符合Duffie, Pan and Singleton (2000)提出的Affine Jump Diffusion（AJD）條件，能解出歐式債券選擇權的解析解。

We provide a Heath–Jarrow–Morton model with Unspanned Stochastic Volatility (USV) and credit risk. From the diffusion term distribution point of view, we extend the Trolle and Schwartz (2008) USV HJM model and add credit risk. This model has risk free forward rate, stochastic volatility and forward credit spread with square root form. The model also has finite dimension Markovian property and has the affine jump diffusion property (AJD) as Duffie, Pan and Singleton (2000). Consequently, we can obtain defaultable bond option prices with analytic form.