Short rates and bond prices in one-factor models - DiVA portal

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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