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U.U.D.M. Project Report 2009:13

Short rates and bond prices in one-factor models

Muhammad Naveed Nazir

Examensarbete i matematik, 30 hp Handledare och examinator: Johan Tysk Juni 2009

Department of Mathematics Uppsala University

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Acknowledgement

All prays to Almighty Allah who induced the man with intelligence, knowledge and wisdom. It is He who gave me ability, perseverance and determination to complete this thesis. Teachers are lighthouses spreading the light of knowledge and wisdom everywhere and guiding the new generation so that they can cruise safely towards their destination. They are performing the job, which Allah himself acknowledge as the noblest to all jobs; the job of teaching. They will get its reward not only from Allah but also in the form of immense respect that every student carries for them in the core of his heart. I offer my sincerest thanks and deepest gratitude to my research supervisor Prof. Johan Tysk for his inspiring and valuable guidance, encouraging attitude and enlightening discussions enabling me to pursue my work with dedication. I would like to say a big thanks to all the teachers who taught me during the entire program. They did not only teach me how to learn, they also taught me how to teach, and their excellence has always inspired me. I would like to take this opportunity to thank my colleagues, friends who were always there for evocative discussions, invaluable advice and support. Many thanks and my deepest gratitude to my parents who have kept me motivated through the extreme hard time and encourage me during the good times.

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Abstract

Interest rate modeling has gained special attention during the last few decades which has resulted in reliable models. These models are in common use for future evolution of interest rate; one way to accomplish this task is by describing the future evolution of the short rate. The goal of the thesis is to provide a detail analysis of bond pricing using one factor short rate model. The thesis explores and provides an insight into the practicality and the intervariability between different models.

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Table of Contents

Table of Contents ........................................................................................................................ 1 1 Introduction ......................................................................................................................... 2 2 Basics Definition.................................................................................................................. 3

2.1 Bond............................................................................................................................. 3 2.2 Bond Characteristics .................................................................................................... 3 2.3 Bond Types .................................................................................................................. 4 2.4 Bond price .................................................................................................................... 6 2.5 Bond Yield ................................................................................................................... 6 2.6 Relation between Yield and Bond Price ...................................................................... 6 2.7 Bond Convexity and Duration..................................................................................... 9 3 One Factor Short Rate Models ......................................................................................... 10 3.1 Short Rate Model ...................................................................................................... 10 3.2 Vasicek Model ........................................................................................................... 10 3.3 The Hull White Model (Extended Vasicek Model) ................................................. 16 3.4 Cox Ingersoll Ross Model ......................................................................................... 19 3.5 The Hull White Model (Extended CIR Model) ...................................................... 23 3.6 Dothan Model ........................................................................................................... 24 3.7 Black-Derman-Toy Model........................................................................................ 26 4 Model's Evaluation ............................................................................................................ 28 4.1 Vasicek Model ........................................................................................................... 28 4.2 Extended Vasicek Model ........................................................................................... 28 4.3 Cox Ingersoll Ross Model ......................................................................................... 29 4.4 Extended Cox Ingersoll Ross Model......................................................................... 29 4.5 Black Derman Toy Model ......................................................................................... 29 4.6 Short Rate Model Versus Other Models................................................................... 30 Conclusion: ............................................................................................................................ 31 References .................................................................................................................................. 32

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1 Introduction

This thesis defines and analyzes a simple one-factor model of the term structure of interest rates. 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 modeling 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. Therefore, one of the aims of this study is to give information about these models and allow readers to compare them. In this thesis, we are discussing on-factor short rate models, Vasicek model (1977), HullWhite (extended Vasicek model) (1993), Cox Ingersoll Ross model (1985), Hull-White (extended CIR model) (1993), Dothan model (1978), Black-Derman-Toy model (1980). The thesis is organized as follows: Chapter 2 provides with the basic introduction to bonds. The Chapter to follow will discuss different short rate models for bond pricing. Finally, Chapter 4 summarises the results and discussions with concluding remarks.

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2 Basics Definition

2.1 Bond

A bond is a debt instrument which is issued for a period of more than one year for the purpose of capital rising by borrowing. A bond is a promise to repay the principal with interest (coupons) on a specific date (maturity). Some bonds do not pay interest but all bonds require a repayment principal.

2.2 Bond Characteristics

A bond depends on the following characteristics,

? Issuers Bonds are issued by public authorities, credit institutions, companies and supranational institutions in the primary markets. The most common process of issuing bonds is through underwriting. In underwriting, one or more security firms or banks, forming a syndicate, buy an entire issue of bonds from an issuer and resell them to investors.

There is little difference between bonds issued by companies and those issued by the national government. A federal government bonds have a low risk by default, where as corporate bonds are considered higher risk. International bonds are complicated by different currencies. These types of bonds are issued in foreign market to the issuer's home market and the currency type is the currency of the foreign market. Examples of some international bonds are Euro bonds, Foreign bonds, Global bonds etc.

? Priority Determined by the probability of when the issuer will pay back the money. The priority shows your place in the queue of the company for payments. If you have unsubordinated (senior) security then you will be the first in the queue to receive

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payment from bankruptcy of its assets. But if you have subordinated (junior) debt security then you will receive payment after the senior (unsubordinated) security holders have received their shares.

? Coupon Rate (Interest Rate) Interest payment received by the bondholder is called "coupon". Most bond issuer companies pay interest every sixth month, but it is also possible to pay monthly, quarterly or annually. The coupon is expressed as a percentage of its value. If the bond coupon rate is fixed (or stays fixed) and market rate rises then bond price are reduced or if market rate falls then bond price will rise.

? Redemption The return of an investor's principal in a security, such as a stock, bond, or mutual fund. Or we can say as, both investor and issuers are exposed to interest rate risks because they agree either to receive or pay a set coupon rate over a specific period of time.

2.3 Bond Types

? Callable (Redeemable) bond gives rights to bond issuer but not obligation to redeem his issue of bonds before its maturity. Issuers have to pay bond holders premium. There are two types of this bond. o American Callable bonds can be called by the issuer any time after the call protection. o European Callable bonds can be called by the issuer only on pre-specified dates.

? Convertible bonds give rights to bondholder but not obligation to convert their bonds into different securities, like shares. This only applies on corporate bonds.

? Puttable bonds give rights to bondholder but not obligation to sell their bonds back to the issuer at a predefined date and price. Investors can ask for the repayment of the bond.

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