Implementing Two-Factor Interest Rate Model with Path-Dependent State Variables

路徑相關的兩因子利率模型

10.6545/JoFS.2000.8(2).1

莊益源(I-Yuan Chuang)

利率模型 ； HJM模型 ； 路徑相闕 ； Vasicek及Ho-Lee模型 ； Interest rate model ； HJM mode! ； path dependent ； Vasicek and Ho-Lee model

中國財務學刊

8卷2期（2000 / 08 / 31）

1 - 24

英文

在普遍架構之下的HJM模型，債券價格與點利率都是非馬可夫性的。因此，此類型的模式有路徑相關的特性而需要保持追蹤歷史的資訊才能做衍生性金融產品的訂價。過去的演算法需要先以前進的方式建立樹狀架構來求得每一點狀態變數的極大與極小值，再倒推以求得衍生性金融產品的價格。本文旨在提供二因子模型有效率之演算法，我們找出狀態變數的極大與極小值的方式，因而僅需倒推運算，此節省了電腦記憶體空間也改進了運算速度。最後，我們並提供一些歐式選擇權的封閉模型。

In the general HJM paradigm, the dynamics of bond prices and spot interest rate are intrinsically non-Markovian. Thus, the model has the path dependent property which requires the information starting from date O. Previous algorithms require the forward scan to keep the information of the maximum and minimum of the state variables at each node. The purpose of this paper is to provide an efficient lattice in the case of two correlated sources of uncertainty We identify the pattern for the maximum and minimum of the state variable and our algorithms require only backward recursion. This procedure saves the computer memory and improve the execution speed. Finally, examples of exact solutions for European claim s will also be provided.

1. Baxter, M.,A. Rennie(1996).Financial Calculus.Cambridge University Press.
2. Bhar, R.,C. Chiarella(1996).Working paper, School of Finance and Economics, University of Technology, Sydney.
3. Brenner, R.,R. Jarrow(1993).A Simple Formula for Options on Discount Bonds.Advances in Futures and Options Research,6,45-51.
4. Geman, H.(1989).Working paper, ESSEC.
5. Heath, D.,R. Jarrow,,A. Morton(1992).Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation.Econometrica,60,77-105.
6. Ho, T.,S.B. Lee(1986).Term Structure Movements and Pricing Interest Rate Contingent Claims.Journal of Finance,41,1011-1029.
7. Hull, J.(1999).Options, Futures, and Other Derivative Securities.Englewood Cliffs, NJ:Prentice-Hall.
8. Hull, J.,A. White(1992).In the Common Interest.
9. Hull, J.,A. White(1990).Pricing Interest Rate Derivative Securities.Review of Financial Studies,3,573-592.
10. Jamshidian, F.(1990).Bond and Option Evaluation in the Gaussian Interest Rate Model.Working paper, Merryll Lynch Capital Markets:New York.
11. Jamshidian, F.(1989).Journal of Finance.
12. Li, A.,P. Ritchken,L. Sankarasubramanian(1995).Lattice Models for Pricing American Interest Rate Claims.Journal of Finance,50,719-737.
13. Moraleda, J.,T. Vorst(1997).Pricing American Interest Rate Claims with Humped Volatility Models.Journal of Banking and Finance,21,1131-1137.
14. Musiela, M.,M. Rutkowski(1997).Martingale Methods in Financial Modelling.New York:Springer, Berlin, Heidelberg.
15. Oksendal, B.(1998).Stochastic Differential Equation.
16. Ritchken, P.,I. Y. Chuang(2000).Interest Rate Option Pricing with Volatility Humps.Review of Derivatives Research,3,237-262.
17. Ritchken, P.,L. Sankarasubramanian(1995).Volatility Structures of Forward Rates and the Dynamics of the Term Structure.Mathematical Finance,5,55-72.
18. Vasicek, O.(1977).An Equilibrium Characterization of the Term Structure.Journal of Financial Economics,5,177-188.
19. Wilmott, P.,J. Dewynne,S.D. Howison(1993).Option Pricing: Mathematical Models and Computation.Oxford Financial Press.

1. (2008)。美國次級房貸危機：證券化風險之啟示。永豐金融季刊，41，55-111。