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PDF Ebook One-Factor Interest Rate Models: Analytic Solutions and Approximations

Submitted by antoq on Sat, 08/15/2009 - 06:30

The uncertainty attached to the future movements of interest rates is an important part of the theory of financial decision making. Most investors are risk averse and risk is linked in particular to interest rates. It is thus important to understand the factors that drive interest rates and the models associated with it. The issue of pricing interest rate derivatives has been addressed by the financial literature in a number of different ways.
One of the oldest approaches is based on modelling the evaluation of the instantaneous short interest rate. This is still quite popular for pricing interest rate derivatives and for risk management purposes, and represents the most commonly used type of dynamical stochastic model for interest rates. More over, in literature there are many models on the instantaneous short interest rate, but despite bewildering number of models, little is known about how these models compare in terms of their ability to capture the actual behavior of the short rate. Therefore, one of the aim of this study is to give information about these models and allow readers to compare them. We can separate these models into two category, which are Equilibrium models and No-arbitrage models. Vasicek (1977), Dothan (1978), Courtadon (1982), CIR (1985), Ho-Lee (1986), Black-Karasinski (1991), Hull-White (Extended Vasicek) (1993), Hull-White (Extended CIR)(1993), Ait-Sahalia (1996), Mercurio- Moraleda (2000), etc., is a list of one-factor short rate models investigated in the thesis.

Many books contain analytic solutions to some of these models, though no specific book contains all solutions. In the work, we united various solutions and, in addition we derived the analytic expression of the Courtadon Model. We solve these models analytically by using stochastic differential methods. For further reading about mathematical methods, Kloedan-Platen (1992) is recommended. Then, we deal with parameter estimation of some of these models using Turkey’s treasury zero coupon bond data. For estimating parameters we use different kinds of discretization methods and compare the results. Moreover, by this work we cope with the missing data problem. Indeed, we can observe the limited number of price of bonds which have different time to maturity since the Turkey’s finance market is not deep. But, in parameter estimation we need the time series for the one day interest rate and so we develop a methodology for forming the time series data.

In the preliminaries, we give the fundamental definitions and theorems in the mathematical and financial viewpoint of interest rate modelling. In the following chapter, we present these various types of instantaneous spot interest rate models as we mentioned above and their analytic solutions. In the third chapter, we introduce the discretization methods of some of the models mentioned in the previous chapter and parameter estimation of some of these models is done by regressing the interest rate on its first lag. Also after estimating parameters we predict the next day’s interest rate by using Monte Carlo Method. In the last chapter, we mention about interest rate trees and we form interest rate tree of the Zero Coupon Bond of Turkey by using Hull-White (Extended Vasicek) Model.

Contents
Abstract
¨Oz
Acknowledgments
Table of Contents
List of Figures
List of Tables
CHAPTER
1 INTRODUCTION
2 PRELIMINARIES
3 ONE-FACTOR SHORT RATE MODELS

    3.1 One-Factor Equilibrium Models
      3.1.1 Vasicek Model
      3.1.2 Dothan Model
      3.1.3 Rendleman-Bartter Model
      3.1.4 Courtadon Model
      3.1.5 Constant Elasticity of Variance Model
      3.1.6 Marsh-Rosenfeld Model
      3.1.7 Cox-Ingersoll-Ross Model
      3.1.8 Exponential-Vasicek Model
      3.2 No-Arbitrage Models
      3.2.1 Ho-Lee Model
      3.2.2 Hull-White Model (Extended Vasicek)
      3.2.3 Hull-White Model (Extended CIR)
      3.2.4 Black-Derman-Toy Model
      3.2.5 Black-Karasinski Model
      3.3 Some Other Extended Models
      3.3.1 CIR ++ Model
      3.3.2 Mercurio-Moraleda Model
      3.3.3 Extended Exponential Vasicek Model

4 DISCRETIZATION

    4.1 The Euler Scheme
      4.1.1 Simulation

    4.2 Milshtein Scheme

5 INTEREST RATE TREES

    5.1 A General Tree-Building Procedure
    5.2 Implementation

6 CONCLUSION
References

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