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Higher quality 6" x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. Bell & Howell Information and Leaming 300 North Zeeb Road, Ann Arbor, MI 48106-1346 USA 800-521-0600 UMI ® ESSAYS ON PRICING FIXED INCOME DERIVATIVES AND RISK MANAGEMENT A Dissertation Presented by JUN ZHANG Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirement for the degree of DOCTOR OF PHILOSOPHY September 2000 Isenberg School of Management UMI Number: 9988858 Copyright 2000 by Zhang, Jun All rights reserved. UMI ® UMI Microform9988858 Copyright 2001 by Beil & Howell Information and Learning Company.
All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P. Box 1346 Ann Arbor, Mi 48106-1346 © Copyright by Jun Zhang 2000 All Rights Reserved ESSAYS ON PRICING FIXED INCOME DERIVATIVES AND RISK MANAGEMENT A Dissertation Presented by JUN ZHANG Approved as to style and content by: — ẩ Saw Ue Sanjay K.
Nawalkha, Chair Z⁄/, ir: Hossein B. Kazemi, Merhiber (_=s—_ Thomas R. Schneeweis, Member y Hossein B. Kazemi, Dire¢tor Ph.
Program Isenberg School of Management To Kai and Wesley ACKNOWLEDGEMENTS First of all, I would like to thank my parents, my husband and my son for their unconditional love, their sacrifice, and their belief in me. I would like to thank my committee member, Professor Hossein Kazemi. for his acceptance and encouragement through all the years I have been in the Finance Ph. It is Professor Kazemi who taught me the basics of finance theory and showed me the tools of finance research.
Without his guidance, [ would have had a much more difficult time finishing this dissertation. I would also like to thank my other committee member, Professor Thomas Schneeweis, for his insightful comments on my dissertation and for providing an excellent research environment. [ appreciate the participation of my outside committee member. Professor Jin Feng from the Statistics Department.
for his steadfast help and for his timely feedback on some issues in my dissertation. Further, I would like to thank Professor Ben Branch for introducing me to the practical world of finance and for being so kind to me and the other doctoral students. Thanks also to Professor Nelson Lacey and Professor Nikunj Kapadia for their constant help and support. I am grateful to all my fellow Ph.
students for being my friends and cheering me up through the ups and downs of the Ph. Finally, my most heartfelt gratitude goes to my chair, Professor Sanjay Nawalkha, who inspired me and helped me every step along the way through the whole process of writing this dissertation. Without his ideas, his knowledge, and his intensive time input. this dissertation would not have been possible.
[ am truly thankful to him. ABSTRACT ESSAYS ON PRICING FIXED INCOME DERIVATIVES AND RISK MANAGEMENT SEPTEMBER 2000 JUN ZHANG, B., HUAZHONG UNIVERSITY OF SCIENCE AND TECHNOLOGY M., UNIVERSITY OF MASSACHUSETTS AMHERST Ph. UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Sanjay Nawalkha This dissertation consists of four essays on pricing fixed income derivatives and risk management. The first essay presents pricing and duration formulas for floating rate honds and interest rate swaps with embedded options.
It combines Briys et al.’s approximation with the extended Vasicek term structure model to value caps and floors. Using this approach, it computes the durations of caps, floors. collars, floating rate bonds with collars and interest rate swaps with collars, and provides comparative statics analyses of the these durations with respect to the underlying variables such as the cap rate, the floor rate, the interest rate volatility, and the level of interest rates. The second essay explores a class of polynomial Taylor series expansions for approximating the bond return function, and examines its implication for managing interest rate risk.
The generalized duration vector models derived from alternative Taylor series expansion extend Fong and Fabozzi’s M-square model and Nawalkha and Chambers’ M-vector model, and the empirical tests show that immunization results can be improved for models g(t) =t* with @ less than | when higher order generalized duration vectors are used. The third essay develops a methodology to build recombining trees for pricing American options on bonds under deterministic volatility HTM models. Without imposing the HJM drift restriction, our approach uses the Nelson-Ramaswamy transformation to generate recombining forward rate trees. We show that the option prices obtained from our recombining trees satisfy Merton’s bond option PDE when step size approaches zero.
Numerical simulations provide evidence that this approach is efficient in pricing both European and American contingent claims. The fourth essay obtains computationally efficient trees for pricing European options under two types of proportional volatility HJM models. We construct a numeraire economy in which European options are priced using a maturity-specific equivalent martingale measure. We then show that for the two types of proportional volatility models, European option prices are independent of the forward rate drift under this maturity-specific equivalent martingale measure.
Our method is particularly beneficial when used to price long-dated caps, floors and collars because these instruments involve a large number of long-dated puts and calls. TABLE OF CONTENTS ACKNOWLEDGEMENTISS. cece cence renrccccerccceerccenrnsnrccanenaenencccesereeseW pee ene ee bene heen Eee ENE E OER UN OER R UTERO Ene DEE eben NEE nee be Een unas venabernncereneerevenveeWh LIST OF TABLES 7.0 ce cee nnn cece nce cc nccccvcccnnsvvcesnncvccccsvennccesnuvenncvenswcesancusesend I FI. INTRODUCTION eee ROHR RR EEE PRE EHH EE HE SEES HH HER HER EES HR ROE EEE HD neeeuveavencececnevencseevercecel 2.
PRICING AND DURATION OF FLOATERS AND INTEREST RATE SWAPS WITH EMBEDDED OPTIONS. cece ecw cence weet ecnneccccenewecnneunseercasncentteecesnsseesses)5 22 ae ng oe FattDerivation HSER KEKE EKER HEE HESEFESEERAR KERR HH HHAH RRR HEHE RE ceuuuacucsneccccanveccneseusvened 2. Price and Duration Relationships. Pricing and Durations of Caps, Floors, Collars, Floaters, and SŠWAapS.cccces ec cccne cence cece cnneercanvcsseucccsnnnne L® 2 2.
GENERALIZED DURATION VECTOR MODELS FOR MANAGING 3. Introduction SOR RSH HERE HE EHH HEHE KEKE HEHE 0h do đc HSH RYH EEE HET EER YY ORR HERRERO HY Oe we ce-----v. The Generalized Duraton Vector Models. Review of M-Square Model and M-Vector ModelL.
The Generalized Duration Vector Models. Empirical pric T ests wes nh. SEER R ERROR OEE EH EE REO EE ED REE HERR R EEE PEE RENEE HY EEE RE DORE ME DERE EME 10 2. RECOMBINING FORWARD RATE TREES — A NEW APPROACH TO PRICING INTEREST RATE CONTINGENT CLAIMS UNDER HIM MODEL WTTH DETEERRMTNISTIC VOLATTLTTTES.
c eee ce cee nen ce ee ne ne cee nveveennewetreerrcesreee ke eseree 49 42.4 ; Review | of HJM Forward Rate 33 Model. che The MiodelÌ.-~ tr nh vn nhe — *» ~~ đc.--~ con em ke rời 64 —_ sec sex. 65 Extension to Multi-Factor ModeÌs. ke eeeeee ¬ ¬— OS 5.
PRICING LONG-DATED CAPS, FLOORS, AND COLLARS UNDER PROPORTIONAL VOLATILITY HJM FORWARD RATE MODELS. -- ni nh nàn KÝ Km nh Hư vi vi vi 74 5. Maturity-Specific Equivalent Martingale Measure in the Numeraire Economy. Proportional Volatlity HIM Models.- cà cà ceee enero ee nies 81 5.
The Simple Proportional Volatility HJM Model. Hybrid Proportional Volaulty HIM Miodel. Recombining Trees for Pricing European Calls and Puts Based on Proportional Volatility HJM Models.-- cà eeằe«eee 85 5. Recombining Tress for Pricing Caps and Floors Based on Proportional Volatility HIM ModelÌs._ nành ens conene 90 5.
kh ve ekvvveeDˆ BIBLIOGRAPHY —. LIST OF TABLES Table 3. Deviation of Actual Values from Target Values for D' Strategy. Deviation of Actual Values from Target Values for D* Strategy .45 Deviation of Actual Values from Target Values for DỶ Strategy.
Deviation of Actual Values from Target Values for D* Strategy .47 Deviation of Actual Values from Target Values for D® Strategy. European Put Option Prices for Deterministic Volatility HJM Model. American Put Option Prices for Deterministic Volatility HJM Model. European Call Option Prices for Proportional Volatility HJM Model.
European Put Option Prices for Proportional Volatility HIM Model. Recombining Trees to Pricing Caps and Floors for Proportional Volatility HIM Model. | LIST OF FIGURES Figure 2. Duration of Cap vs.
Cap Rate and Duration of Floor vs. Duration of Collar vs. Cap Rate and Floor Rate. Durations of Cap, Floor and Collar vs.
Interest Rate Volatility. Durations of Cap, Floor and Collar vs. Instantaneous Interest Rate. Duration of Floater vs.
Cap Rate, Floor Rate, Interest Rate Volatility and [nstantaneous Ínterest Rate. Duration of Swap vs. Interest Rate Volaninty: and Instantaneous Interest Rate. 29 CHAPTER I Introduction The dramatic increase in interest rate volatility over the past twenty-five years has led to an increased use of interest rate derivatives and other risk management tools by financial institutions.
Correctly pricing interest rate derivatives and understanding their complex interest risk characteristics, refining existing methodology and developing new approaches for managing interest rate risk have become important issues in financial research and practice. This dissertation focuses on four different but related issues in pricing fixed income derivatives and risk management. The essays are self-contained. and can be read independently of each other.
The risk measure “duration” has been traditionally used to measure and hedge the interest rate risk of fixed income securities. Since duration was discovered about four decades before the development of the continuous-time term structure models, the research on duration models has evolved somewhat independently of the research on term structure models. In the past few years. some research effort has been made to integrate duration models with the term structure models.
The first essay of this dissertation provides an application of using continuous time term structure model in characterizing the interest rate risk of variable rate instruments with embedded options within the traditional duration framework. This essay combines Briys et al.’s [1991] approximation with the extended Vasicek term structure model (see Hull and White [1990b]) to value caps and floors. and computes the durations of caps. floors, collars, floating rate bonds with collars and interest rate swaps with collars.
It also presents comparative statics analyses of the duration risk measure of these securities with respect to the underlying variables such as the cap rate. the floor rate, the interest rate volatility, and the level of interest rates. We find that the magnitude of the durations of caps, floors. and collars are very large due to the implicit leverage in these securities.
Duration and immunization have long been used by fixed income portfolio managers and financial institutions to control interest rate nsk. Researchers have derived second and higher order extensions to the duration model. The M-square model of Fong and Fabozzi [1985] and the M-vector model of Nawalkha and Chambers [1997] are based upon higher order Taylor series expansions of the bond return function around a planning horizon H. These models improve the immunization performance considerably over the traditional Macaulay duration model.
there exist other non-standard Taylor series expansions which may allow better approximation to the bond return function.
