Using the LIBOR Market Model to Price the Interest Rate Derivatives: A Recombining Binomial Tree Methodology

以LMM利率模型評價利率衍生性商品：結合節點二項樹方法

LMM(LIBOR Market Model)利率期限模型是一個很複雜的利率模型，要在此模型下推導評價封閉公式解或是數值評價方法都很困難。由於LMM模型具備非馬可夫的特性，建構利率樹時會發生節點無法重合的問題，而導致利率樹的節點個數成爆炸性成長，使得電腦無法有效地使用該利率樹進行評價。本篇論文利用Ho, Stapleton, and Subrahmanyam(HSS) 介紹的造樹法來建構節點重合的LMM利率樹。我們首先改寫Poon與Stepleton(2005)對LMM建構的離散時間數學模型，接下來推導達期利率的條件期望值和條件變異數──這些資料對利率樹的建構十分重要。最後我們利用HSS提供的建樹法建構利率樹，並用該利率樹評價利率衍生性金融商品。第五節的數值資料驗證我們的數值模型可提供精確的評價結果。

LIBOR market model (LMM) is a complicated interest rate model and it is hard to be evaluated both analytically and numerically. Because of the non-Markov property of the LMM, a naively implemented tree model will not recombine. Thus the size of this naive tree model will grow explosively and the tree cannot be efficiently evaluated by computers. This paper proposes a recombining LMM tree model by taking advantages of tree construction methodology proposed by Ho, Stapleton, and Subrahmanyam (HSS). We first rewrite the discrete mathematical model for LMM suggested by Poon and Stapleton (2005). Then we derive the conditional means and the variances of the discrete forward rates which are important for the tree construction. Finally, our recombining trees for pricing interest rate derivatives are built by taking advantages of the tree construction methodology proposed by HSS. Numerical results illustrated in Section 5 suggest that our method can produce convergent and accurate pricing results for interest rate derivatives.