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研究生： 陳昱廷
Chen, Yu-Ting.
論文名稱： 結合Hull-White模型與Nelson-Siegel模型之應用
Application on The Combination of Nelson-Siegel and Hull-White Models
指導教授： 鍾經樊
Chung, Ching-Fan
口試委員: 張焯然
Chang, Jow-Ran
蔡子晧
Tsai, Tzu-Hao
學位類別： 碩士
Master
系所名稱： 科技管理學院 - 計量財務金融學系
Department of Quantitative Finance
論文出版年： 2020
畢業學年度： 108
語文別： 英文
論文頁數： 35
中文關鍵詞： 利率期限結構 、Hull-White模型 、Nelson-Siegel模型 、無風險利率 、信用利差 、估值
外文關鍵詞： term structure of interest rate, Hull-White model, Nelson-Siegel model, risk-free interest rate, credit spread, pricing
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    • 本文結合Nelson-Siegel靜態模型與Hull-White單因子動態模型，稱之為NS-HW模型，並以其研究無風險利率的期限結構。在該模型中，利率期限結構將由Nelson-Siegel模型描述其主體，其動態行為則以Hull-White模型勾勒。模型整合後的期限結構僅由六個參數、期限和當前時點$t$所影響。藉此模型，未來任何時點、任何期限的利率分配將可以直接推導得出，從而使未來利率的模擬更加直接且容易。NS-HW模型除了適用於利率模擬外，也能模擬如零息債券，利率上/下限和利率交換選擇權等利率相關之商品及路徑獨立之利率衍生品之價格。

    除了無風險利率外，假設零息債券的信用風險和利率風險互相獨立且風險中立測度下的違約損失率為1，具風險之零息債券價格可以拆解為無風險折現因子和存活機率的乘積。其中存活機率之於出險比，正如同折現因子之於利率，存在相同的函數形式，且出險比也具有期限結構，因此可以套用NS-HW模型進行估計，並以估計參數進行出險比的預測。再加總預測的無風險利率與出險比，即可以以此方法來預測具風險的利率及利率商品。

    本文收集了美國國庫債券和高盛集團發行之公司債券之報價資料，通過最小化債券實際價格與模型計算之債券價格之平方和，可以估計出NS-HW模型中的六個參數。這些參數可用於預測未來利率和模擬利率衍生品價格，我們將模擬的利率上限價格與Black模型計算的價格進行比較，以呈現NS-HW模型的配適能力。另外，本文也以NS-HW模型模擬利率上、下限、利率交換選擇權作為模型的應用。此外，本文分別模擬了以無風險和具風險利率計算持有不同債券的5天及1個月之99％風險值，以了解基於NS-HW模型在不同樣本與商品下所建議的資本計提。

    This thesis combines Nelson-Siegel static model with Hull-White one factor dynamic model to have a new model called as NS-HW model to study the term structure of interest rate. In this model, the term structure is mainly described by Nelson-Siegel model and its dynamic behavior is formulated with Hull-White model. As the result, yield curve turns out to be only driven by six parameters, time to maturity, and the current period under investigation. Therefore, the distribution of future interest rate can be derived at any time with any maturity, which makes the simulation of future interest rate much easier. In addition, NS-HW model is appropriate for simulating prices of financial instruments that are functions of interest rates, like zero-coupon bonds, interest rate caps/floors and swaptions.

    In addition, by assuming firm’s credit risk and interest rate risk are independent, and the loss given default is 1 under the risk neutral measure, the price of a zero-coupon bond issued by corporate can be discomposed into the product of risk-free discount factor and survival probability. The survival probability is to the odds ratio as discount factor is to interest rate. They both follow a same function form. Considering that odds ratio also follows a term structure, hence it can be estimated with the NS-HW model and then be predicted. Then the risky interest rates can be predicted by taking a sum of predicted risk-free interest rates and predicted odds ratios.

    Finally, this thesis collects quote prices of U.S. Treasury bonds and quote prices of corporate bonds issued by Goldman Sachs. By minimizing the sum of square of the difference in bond prices calculated by NS-HW model and actual bond prices, the six parameters can be estimated. Those parameters can then be used to predict future interest rates and to simulate interest rate derivatives. We compare the simulated caplet price with price calculated by Black model (1976) to show the capability of NS-HW model. caplet, floorlet, and swaption are simulated and presented as applications of NS-HW model. In addition, we also simulate 5-day and 1-month 99% value at risk of bonds calculated with risk-free and risky interest rates to see how much the capital charge will be suggested by NS-HW model under different situation.

    Introduction----------1 Literature Review----------2 Traditional Theories----------2 Static Models----------3 Dynamic Models----------3 Functional Data Analysis----------4 Value at Risk (VaR) ----------4 Methodology----------5 The Nelson-Siegel Interest Rate Model----------5 The Hull-White Model----------7 Estimation of Parameters----------11 Forecast Interest Rate and Bond Price----------13 Interest Rate With Credit Risk----------15 Estimate Parameters of O(t,T) and Forecast ̃R(t,T) ----------16 Loss of zero coupon bond----------18 Loss of coupon bond----------19 Calculate VaR----------20 Empirical Results----------20 The Data----------20 Estimate Parameters----------21 Estimateβ1,β2,β3,λ,c,σ----------21 Estimateβ’1,β’2,β’3,λ’,c’,σ’ ----------22 Forecast Interest Rate----------24 Pricing interest rate derivatives----------24 European Interest rate caplet and floorlet----------24 European Interest rate swaption----------25 Simulate Value at Risk of bonds----------26 VaR of Zero-coupon Bonds----------26 VaR of Coupon Bonds----------27 Conclusion----------28 References----------30 Appendix A----------31 Appendix B----------33 Appendix C----------33

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