不同利率模型之利率上限契約的評價分析

Abstract

本文應用不同利率模型評價利率上限契約，所用的利率期間結構模型包括HJM、BGM與GK模型。我們有系統地分析比較各種不同的上限利率、到期期限、與波動結構下，利率上限契約的價值差異。本文應用波動函數為時間相關的HJM模型，分別推導以短期利率與LIBOR利率為標的之利率上限契約價格。分別在BGM與GK模型下，回顧利率上限契約價格公式的推導過程。接著，本文提出2個單因子的HJM模型，稱為波動模型I與波動模型II，同時在設定的模型下，驗證遠期利率的唯一性。此外，在實務上，本文提供的數值例子與模擬的結果，可以做為未來發行新奇利率選擇權，或選擇適當利率評價模型的參考。 首先，本文分別在HJM與BGM模型下，假設與時間相關的利率波動函數，分析利率上限契約的價值。結果顯示，標準利率上限契約的價值隨著利率重設次數增加而升高，但隨著上限利率增加而降低。為了方便討論，其它各種波動型態對水平型的利率上限契約價值比率，在本文中稱為“價值比”。接著比較各種波動型態的價值比，結果顯示，隨著上限利率增加，上升型與駝峰型的價值比越高，但下降型與指數遞減型的價值比越低；同時，隨著到期期限增加，上升型與駝峰型的價值比越高，但下降型與指數遞減型的價值比越低。再者，隨著到期期限增加時，短期的單一利率上限契約價值有上升的趨勢，而長期契約價值有下降的趨勢。 其次，本文在波動模型I與波動模型II之下，檢驗利率上限契約的價值。以數值方法模擬利率上限契約價值，在不同波動結構型態、不同上限利率與到期期限下，檢驗利率上限契約價值的差異性。同時探討模型係數對利率上限契約價值的敏感度。接著，我們在BGM與GK模型下，探討利率不連續變動對利率上限契約價值的影響，得到以下的結果。隨著上限利率增加，利率上限契約價值下降的速度越小；同時，隨著到期期限增加，利率上限契約價值上升的速度越高。再者，以GK模型評價的利率上限契約比BGM模型評價的相同契約更有價值。此外，我們也討論利率波動結構對利率上限契約價值的影響。

This study evaluates the interest rate caps with various term structure models, including HJM, BGM, and GK models. We systematically analyze and compare cap values with different cap rates, maturities, and volatility structures. Applying HJM model with time-dependent volatility functions, we derive cap prices with short rates and LIBOR rates as underlyings, respectively. Under BGM and GK models, we review the derivations for the pricing formulas of interest rate caps. Next, we provide two one-factor HJM models, namely volatility model I and II, and prove the uniqueness of the forward rate under these specific models. Furthermore, this study provides some numerical examples and simulation results for the practice to issuing exotic interest rate options and choosing an adequate pricing term structure model. Firstly, this study analyzes the cap values with time-dependent volatility functions under HJM and BGM models. The result presents that the standard interest rate cap becomes more valuable as the reset dates of interest rates increase, but less valuable as the cap rate increases. For convenience, the cap values in various types relative to the flat one are called “value ratios” throughout this study. Next, as the cap rate increases, value ratios for the increase and hump types increase, but decrease for the decrease and exponential decrease types. Additionally, as the time to maturity increases, value ratios for the increase and hump types increase, but decrease for the decrease and exponential decrease types. Moreover, as time to maturity increases, short-term caplets become more valuable, but long-term caplet values decrease. Secondly, this study investigates the cap values under volatility models I and II. Using a numerical simulation, we examine the cap value differences between various volatility structures, cap rates, and maturities. Moreover, sensitivity analysis for model coefficients on cap values is also investigated. Next, this study investigates the jump effects of interest rates on cap values under BGM and GK models. The decreasing speeds of cap values would decrease as the cap rate increases. Additionally, the increasing speeds of cap values increase as time to maturity increases. Furthermore, we observe that the cap values priced by GK model are more valuable than those priced by BGM model. Finally, the volatility structure effects on cap values are also examined.