臺灣博碩士論文加值系統

本文旨在探討Heath-Jarrow-Morton (HJM)架構下之Ritchken-Chuang (RC)利率模型，其二元再重合樹（binomial recombining tree）評價美式利率選擇權的評價效能。一般化的HJM利率模型裡，無法找到有限個具馬可夫性質(Markovian)的狀態變數來描述利率期間結構(term structure)之動態行為。因此，HJM模型的狀態變數具有路徑相依的性質，運用於評價長天期選擇權契約時，二元樹會面臨節點數目隨切割數的增加，呈現指數遞增之評價不效率問題。RC利率模型採用對遠期利率波動結構為時間之定態函數(deterministic function)的作法，在HJM架構下建構出具下述特色的評價模型：（一）遠期利率波動結構的設定比一般模型有彈性，包括可描述駝峰型波動結構；（二）解決HJM架構下二元樹無法評價長天期選擇權的問題。
本文使用RC模型推得的歐式利率選擇權封閉解，來探討不同的波動結構設定對利率選擇權評價結果的影響；並且藉已知的歐式選擇權封閉解，來檢視RC二元再重合樹狀圖評價美式選擇權時數值方法的收斂速度與數值解的正確性。探討的問題依序如下：（一）相同的歐式利率選擇權在指數遞減型波動結構假設下所算出的價格與駝峰型波動結構假設下所求得的價格，兩者間存在差異；（二）檢測RC模型之二元再重合樹狀在不同的內差法搭配下，所造成的收斂速度差異；（三）測試RC模型之二元再重合樹狀圖在不同的資訊矩陣設定下，對於數值解精確度與收斂速度的影響。
研究結果顯示：（一）若實證結果為駝峰型波動結構的市場條件下，一般模型將波動結構簡化假設為指數遞減函數的作法，會造成模型評價的錯誤，顯見RC模型之假設對於市場資訊描述的正確性；（二）二次內插法是增進RC模型數值解收斂速度的適當選擇；（三）同樣的切割數下，較大的資訊矩陣，其算出的數值解有較高的正確度。

Without identifying special conditions on volatility structures, the evolution of the term structure can not be made Markovian with respect to a finite dimensional Markovian system under the generalized Heath-Jarrow-Morton (HJM) paradigm. For the flaw in implementing the non-recombining binomial lattice procedure, efficient HJM algorithm is not available for accurately pricing most types of long-term American contracts. Specifying the forward rate volatility structure as a special deterministic function of time only, Ritchken & Chuang (RC) model in the HJM paradigm develops a feasible approach to avoid the exploding-tree problem faced by HJM. In addition to incorporating full information on the term structure, the most significant advantage of RC model is that this model can price interest rate derivatives under various patterns of volatility structures, including hump volatility structure.
To find out the practical pricing ability of the RC model, the numerical analysis of this article investigates two issues. One is that the appropriate setting of volatility structures is necessary to heighten the accuracy of the pricing result. The other is the convergence speed of the RC algorithm in using different interpolation methods and the different size of the information matrix.
The result of this study shows as the following : (1) If the empirical test verifies the hump volatility structure is true, then the additional consideration of the volatility structure in the RC model will not be redundant. (2) The quadratic interpolation method is the best way to accelerate the convergence speed of the solution of the RC algorithm. The bigger size of the information matrix, the better accuracy of the numerical solution.

第一章 總論………………………………………………………… 1
第一節 研究動機與目的…………………………………………… 1
第二節 研究架構…………………………………………………… 3
第二章 利率模型概述……………………………………………… 5
第一節 一般利率均衡模型概述…………………………………… 5
第二節 無套利機會利率模型概述………………………………… 7
第三節 波動結構的設定對模型馬可夫性質的影響……………… 21
第三章 RC利率模型……………………………………………… 26
第一節 前言………………………………………………………… 26
第二節 連續時間下之RC模型…………………………………… 27
第三節 間斷時間下之RC模型…………………………………… 31
第四章 RC模型評價效能之分析………………………………… 37
第一節 前言………………………………………………………… 37
第二節 波動結構之設定對於選擇權評價影響之探討…………… 37
第三節 不同的內插法對於RC模型數值解其正確度影響之探討 39
第四節 資訊矩陣大小對RC模型數值解其正確度影響之探討… 42
第五章 結論與建議………………………………………………… 45
附錄 ……………………………………………………………… 47
參考文獻 ……………………………………………………………… 52

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