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研究生:林育民
研究生(外文):Yu-Min Lin
論文名稱:應用蒙地卡羅法對HJM模型下的利率衍生性商品定價
論文名稱(外文):Pricing Interest Rate Derivatives in HJM Model by Monte Carlo Method
指導教授:岳夢蘭岳夢蘭引用關係
指導教授(外文):Meng-Lan Yueh
學位類別:碩士
校院名稱:國立中央大學
系所名稱:財務金融研究所
學門:商業及管理學門
學類:財務金融學類
論文種類:學術論文
論文出版年:2004
畢業學年度:92
語文別:英文
論文頁數:53
中文關鍵詞:一般化HJM模型準隨機序列蒙地卡羅模擬高斯HJM模型利率衍生性商品
外文關鍵詞:Monte Carlo simulationQuasi-random sequencesInterest rate derivativesGeneral HJM modelGaussian HJM model
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HJM 模型是一個非常一般化的利率模型,它只需要外生給定期初的利率期間結構和
債券報酬率的波動性期間結構。本文提供一個利率上限評價模型,其波動性結構可
以非常一般化。當我們檢視一因子HJM模型下的利率衍生性商品評價時,我們考慮
兩種不同的波動性結構,一個為指數下降型,另一個為駝峰型。我們利用蒙地卡羅
模擬結合有效的債券報酬隨機過程和準隨機序列來評價一些利率衍生性商品,包
括,純折價債券選擇權、利率上限、利率交換選擇權。本文的結論顯示我們可以利
用準隨機序列較準確地評價這些利率衍生性商品。另外,我們也提供利用兩因子高
斯HJM模型對利率交換選擇權評價時的一些特性。
Heath, Jarrow and Morton (hereafter HJM) model is a very general interest rate model,
their only required inputs are the initial yield curve and the volatility structure for pure
discount bond (PDB) price return. Here we provide the interest rate caps pricing model
with very general volatility structure. When we test the valuation of interest rate
derivatives in one-factor HJM model, we consider two different volatility structures as (i)
exponentially decaying (ii) humped. We use Monte Carlo simulation combined with
efficient bond return process and quasi-random sequences to price several interest rate
derivatives included PDB option, caps and swaptions. The result of this thesis is that we
can price these interest rate derivatives accurately by Monte Carlo simulation combined
with quasi-random sequences. We also show some characteristics of two-factor Gaussian
HJM model when pricing interest rate swaptions.
1. Introduction……………………………………………………………………..1
2. The Models………………………………………………………………………3
2.1 PDB Option under a One-Factor HJM Model…………………………..4
2.1.1 Interest Rate Model……………………………………….………...4
2.1.2 Pricing Model under General HJM Model………………..……....5
2.1.3 Monte Carlo Simulation……………………………………………5
2.2 Caps under a One-Factor HJM Model…………………………...………6
2.2.1 Interest Rate Model…………………………………………………6
2.2.2 Pricing Model under General HJM Model…………………..……6
2.2.3 Monte Carlo Simulation……………………………………………8
2.3 Swaptions under a Two-Factor HJM Model…………………..…………9
2.3.1 Interest Rate Model…………………………………………………9
2.3.2 Pricing Model under General HJM Model………………………10
2.3.3 Monte Carlo Simulation………………………………………..…10
2.4 Forward Rate Correlation Coefficient Matrix…………………….……11
3. Variance Reduction……………………………………………………………13
3.1 Antithetic Variable Technique………………………………………...…13
3.2 Quasi-Random Sequences…………………………………………….…13
3.2.1 Faure Sequences…………………………………………...………14
3.2.2 Sobol Sequences……………………………………………………15
3.3 Characteristic of Quasi-Random Sequences……………………………16
3.3.1 Low Time Interval…………………………………………………17
3.3.2 High Time Interval…………………………………………...……18
4. Numerical Results…………………………………………………………..…21
4.1 PDB Option………………………………………………………….……21
4.1.1 The One-Factor HJM Model………………………………...…….21
4.1.2 Vasicek Model Used to Calculate The Initial Term Structure.…..22
4.1.3 Closed-Form Solution for a European Call Option on PDB…….22
4.1.4 Contract…………………………………………………………….24
4.2 Interest Rate Caps………………………………………………………..30
4.2.1 The One-Factor HJM Model………………………………………30
4.2.2 Vasicek Model Used to Calculate The Initial Term Structure...…30
4.2.3 Closed-Form Solution for a Interest Rate Cap…………………...30
4.2.4 Contract………………………………………………………….….33
4.3 Interest Rate Swaptions………………………………………………….38
iv
4.3.1 The Two-Factor HJM Model……………………………………....38
4.3.2 Vasicek Model Used to Calculate The Initial Term Structure…...39
4.3.3 Contract……………………………………………………………..39
4.3.4 Table and Figures Results………………………………………….40
5. Conclusions…………………………………………………………………….51
References…………………………………………………………………………...52
Brigo, D., and F. Mercurio, 2001, “Interest Rate Models Theory and Practice”, Berlin,
New York.
Buhler, W., M. Uhrig-Homburg, U. Walter, and T. Weber, 1999, “An Empirical
Comparison of Forward-Rate and Spot-Rate Models for Valuing Interest-Rate
Options”, The Journal of Finance, pp. 269-305.
Carverhill A., and K. Pang, 1998, “Efficient and Flexible Bond Option Valuation in
The Heath, Jarrow and Morton Framework”, Risk Books, pp. 183-189.
Chuang I-Tuan., 2000, “Implementing Two-Factor Interest Rate Model with
Path-Dependent State Variables”, Journal of Financial Studies, Vol.8, No. 2, pp. 1-25.
Clewlow L., and C. Strickland, 1998, “Pricing Interest Rate Exotics in Multi-Factor
Gaussian Interest Rate Models”, PaperChrisandLes, pp. 1-35.
Driessen J., P. Klaassen, B. Melenberg, 2002, “The Performance of Multi-Factor Term
Structure Models for Pricing and Hedging Caps and Swaptions”, pp. 1-36.
Galanti S. and A. Jung, 1997, “Low-Discrepancy Sequences : Monte Carlo Simulation
of Option Prices”, The Journal of Derivatives, pp. 63-83.
Hull, J.C., 2003, “Options, futures, & Other Derivatives”, 5th ed., United States of
America: Prentice-Hall, Inc.
Jarrow, R.A., 1996, “Modeling Fixed Income Securities and Interest Rate Options”,
McGraw-Hill Companies, Inc.
Jarrow, R.A., 2000, “Derivative Securities”, 2th ed., South-Western College.
Joy C., P.P. Boyle, and Ken Seng Tan, 1998, “Quasi-Monte Carlo Method in
Numerical Finance”, Risk Books, pp. 269-280.
Lund J., 1998, “Review of Continuous-Time Term-Structure Models”, Department of
Finance, The Aarhus School of Business, pp. 1-11.
53
Moraleda J.M. and T.C.F. Vorst, 1997, “Pricing American interest rate claims with
humped volatility models”, Journal of Banking & Finance 21, pp. 1131-1157.
Radhakrishnan A.R., 1998, “Does Correlation Matter in Pricing Caps and Swaptions ”,
Department of Finance, Stern School of Business, New York University, pp. 1-24.
Rebonato R., 1999, “On the simultaneous calibration of multifactor lognormal interest
rate models to Black volatilities and to the correlation matrix”, Journal of
computational Finance, Vol. 2, No. 4, pp. 5-27.
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