essays on pricing fixed income derivatives and risk management

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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

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UMI Number: 9988858 Copyright 2000 by Zhang, Jun All rights reserved ® UMI UMI Microform9988858

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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.D program 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.D students for being my friends and cheering me up through the ups and downs of the Ph.D program

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ABSTRACT

ESSAYS ON PRICING FIXED INCOME DERIVATIVES AND RISK MANAGEMENT

SEPTEMBER 2000

JUN ZHANG, B.S., HUAZHONG UNIVERSITY OF SCIENCE AND TECHNOLOGY M.S., UNIVERSITY OF MASSACHUSETTS AMHERST

Ph.D 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

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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

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TABLE OF CONTENTS

ACKNOWLEDGEMENTISS 0 ccc 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 22.0 ce cee nnn cece nce cc nccccvcccnnsvvcesnncvccccsvennccesnuvenncvenswcesancusesend

I FI ÔÔÔ ‹

1 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 Hài

4 ï 5

2.1 Entrodtction 2.2 2 cece ecw cence weet ecnneccccenewecnneunseercasncentteecesnsseesses) 22 ae ng oe Derivation Fatt HSER KEKE EKER HEE HESEFESEERAR KERR HH HHAH RRR HEHE RE ceuuuacucsneccccanveccneseusvened

2.2.1 Price and Duration Relationships -.~-~ Ð

2.2.2 Pricing and Durations of Caps, Floors, Collars, Floaters,

and SŠWAapS - cà ky ceereveeeeesevee 2

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7.3 Numerical Simulations .cccces ec cccne cence cece cnneercanvcsseucccsnnnne L®

2 ï 22

2.4 ConCÏUSIODS -.- cv nkevereeeeeeeeeseveeee-c 22

3 GENERALIZED DURATION VECTOR MODELS FOR MANAGING

3.1 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 3Ợ

3.2 The Generalized Duraton Vector Models 32

3.2.1 Review of M-Square Model and M-Vector ModelL 3

3.2.2 The Generalized Duration Vector Models .5 re bo

3.3 Empirical T 3

2.3 pric ests wes nh 'ÔÔ

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4 RECOMBINING FORWARD RATE TREES — A NEW APPROACH TO PRICING INTEREST RATE CONTINGENT CLAIMS UNDER

HIM MODEL WTTH DETEERRMTNISTIC VOLATTLTTTES 49 4.1 Introduction .2 0 c eee ce cee nen ce ee ne ne cee nveveennewetreerrcesreee ke eseree 49 4 ; | 33 42 Review of HJM Forward Rate Model Seep vecnvcnccensceucceerssess 53 *» ~~ đc

4.3 The MiodelÌ -~ che tr nh vn nhe —

44 Numerical Stimulatons ~ con em ke rời 64

~ + ~ —_

45 Extension to Multi-Factor Mods - «cc sec sex 65

46 Conclusions ke eeeeee ¬ ¬— OS

5 PRICING LONG-DATED CAPS, FLOORS, AND COLLARS

UNDER PROPORTIONAL VOLATILITY HJM FORWARD RATE

MODELS — ,Ơ "¬¬ 74 5.1 mỗtroduciion ni nh nàn KÝ Km nh Hư vi vi vi 74

5.2 Maturity-Specific Equivalent Martingale Measure in the

Numeraire Economy ccccecnccceceercensecceesreccccrcceercesnseassenened 5.3 Proportional Volatlity HIM Models - cà cà ceee enero ee nies 81 5.3.1 The Simple Proportional Volatility HJM Model .81 5.3.2 Hybrid Proportional Volaulty HIM Miodel 84 5.4 Recombining Trees for Pricing European Calls and Puts Based on

Proportional Volatility HJM Models cà eeằe«eee 85

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LIST OF TABLES Table 3.1 4.1 4.2 3.1

Deviation of Actual Values from Target Values for D' Strategy

Deviation of Actual Values from Target Values for D* Strategy

Deviation of Actual Values from Target Values for DỶ Strategy

Deviation of Actual Values from Target Values for D* Strategy

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

45

47

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LIST OF FIGURES

Figure

2.1 Duration of Cap vs Cap Rate and Duration of Floor vs Floor Rate .24

2.2 Duration of Collar vs Cap Rate and Floor Rate 25

2.3 Durations of Cap, Floor and Collar vs Interest Rate Volatility 26

2.4 Durations of Cap, Floor and Collar vs Instantaneous Interest Rate 27

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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

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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 However there exist other non-standard Taylor

series expansions which may allow better approximation to the bond return function The second essay of this dissertation explores a class of polynomial Taylor series expansions for approximating the bond return function, extending the work of Fong and Fabozzi [1985] and Nawaikha and Chambers [1997] In our generalized duration vector models

mt

the mth order risk measure is given as the weighted average of the terms (g(t)— g(H))

where g(t) is chosen from a class of fiinctions of the form r* with @ ranging from 0.25

to 1.5 Our empirical tests confirm that immunization results can be improved for madels

g(t) =¢* with a less than | when higher order generalized duration vectors are used

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Jarrow, and Morton [1992] introduce a family of forward rate processes which are consistent with the initial term structure of interest rates and can be used to price interest

rate contingent claims The HJM model ts very general and it accommodates a wide

spectrum of interest rate processes appealing to the practitioners Unfortunately a

significant problem in application of the HJM models is that the evolution of the forward

rate trees is path-dependent except for a few special cases Generally, in the case of

deterministic volatility HJM models closed form solutions exist for European options

But for pricing American options numerical approach must be used Amin [1991] solves

the non-recombining tree problem for the deterministic volatility one-factor HJM model by changing the step size along the tree However Amin’s approach cannot be

generalized to multi-factor models The third essay of this dissertation introduces a new

methodology to build recombining trees for the deterministic volatility HJM models which can be easily extended to multi-factor cases We observe that there is no drift term

in Merton’s [1973] PDE for pricing bond options Hence we start with the volatility

terms of HJM forward rate process and ignore the drift term Since HJM drift restriction

does not have to be met, Nelson-Ramaswamy [1990] transformation can be used to make

the forward rate tree recombining We show that the option prices obtained from our

recombining trees satisfy Merton’s PDE when step size approaches zero Numerical simulations give evidence that this approach is efficient in pricing both European and American contingent claims

Deterministic volatility HJM models have two important limitations First

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upon the level of interest rates Proportional volatility HIM models overcome these limitations However no closed-form solutions exist even for pricing European options under these models The fourth essay of this dissertation obtains computationally efficient trees for pricing European options under two types of proportional volatility HJM models namely simple proportional volatility model and hybrid proportional volatility model Since a default-free interest rate cap (floors) can be modeled as a string of European put (call) options on increasing maturity pure discount bonds with increasing option expiration dates, our method can be used to efficiently price long-dated caps and floors To achieve this goal, we first construct a numeraire economy in which European options are priced using a maturity-specific equivalent martingale measure We show that for the two types of proportional volatility HJM models European option prices are independent of the forward rate drift under the maturity-specific equivalent martingale measure Numerical simulation results confirm that the recombining trees obtained using our approach both converge faster and are computationally more efficient than the non-

recombining trees Our method is particularly beneficial when used to price long-dated

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CHAPTER 2

Pricing and Duration of Floaters and Interest Rate Swaps with Embedded Options

2.1 Introduction

The increased volatility of interest rates in the [970s gave rise to the increase use of debt securities whose coupon payments adjust or are reset with changes in interest rates at pre-specified time intervals The increasing demand for hedging interest rate risk has led to a rapid growth in the interest rate swap market Variable rate instruments such as floating rate bonds and interest rate swaps usually have embedded options such as caps and floors Good investment and hedging decisions require an understanding of the

interest rate risk characteristics of these complex instruments

This chapter combines Briys et al.’s approximation of valuing caps and floors with the extended Vasicek term structure model to derive pricing and duration formulas

for floating rate bonds and interest rate swaps with embedded options Using this

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A floating rate bond with an embedded collar is a portfolio of a naked floating rate bond and an interest rate collar An interest swap is a portfolio of a fixed rate bond and a floating rate bond A collar is a portfolio of a cap and a floor In absence of default

risk interest rate volatility is the only source of risk for these securities which can be

measured by durations and other interest rate risk measures

The traditional Macaulay's duration model is based upon restrictive assumptions of flat yield curve with parallel and infinitesimal yield curve shifts Since 1970s, a stream of research has generalized the traditional duration model to muiti-factor duration

models, where the higher order duration measures are used to hedge the effects of non- parallel and non-infinitesimal term structure shifts To deal with the non-parallel vield curve shifts researchers have intraduced methods such as “key rate durations” (Ho [1992)}), and “risk point method” (Dattatreya and Fabozzi [1995]) These methods generally divide the yield curve into maturity intervals and evaluate the change of cash flows of bond portfolio and bond options corresponding to shifts in interest rates at each

maturity interval This approach allows managers to assess a portfolio’s joint exposure to the key rates and predicts the portfolio’s response to yield curve change using multiple

durations “Value at Risk” (Ho, Eng, and Chen [1996]) is a variance/covariance approach to estimate the risk exposure of a bond portfolio position “Principal component method” (Golub and Tilman [1997]) uses factor analysis to pinpoint the risk of a bond portfolio

Over the last few decades another stream of research has focused on modeling the term structure of interest rates Some well-known term structure models are those

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Jarrow, and Morton [HJM 1992] Despite the rapid development in both research areas only a few studies in the existing duration literature have incorporated continuous time term structure models

For bonds with embedded options the yield curve shifts make future cash flows become uncertain Recently researchers have proposed the risk measure “effective duration” which takes into account the change of cash flows caused by a yield change Yet most of the “effective duration” methods do not utilize continuous time term structure models Dunetz and Mahoney [1988], Jamshidian and Zhu [1988], Chance [1990] among others has analyzed the duration and other interest rate risk characteristics of callable bonds and non default-free bonds However, their approach does not apply directly to the interest rate risk analysis of bond options since the model of Dunetz and Mahoney [1988], Jamshidian and Zhu [1988] are not consistent with Merton’s [1973]

stochastic interest rate option pricing framework and the model of Chance [1990]

assumes that the asset underlying the option has a zero duration Sobti and Sykes [1993] analyze the risk management of floating rate CMOs (Collateralized Mortgage

Obligations) with an interest rate cap, but no specific term structure model is assumed

Nawalkha [1995] assumes a general form of forward rate process and provides a continuous-time approach to the traditional duration vector model and derives the

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neutral portfolios They also divide the term structure of interest rates into intervals called buckets, and give a formal description of bucket hedging: for each bucket a bucket factor is derived from the original factor (for example if there are nm buckets for a two-factor model, then there are 2” bucket factors) The change of portfolio value is obtained by ageregating the value changes caused by all the bucket factors

This chapter 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 It derives the duration measures for a stream of fixed income derivatives by incorporating the extended Vasicek term structure model (Hull and White [1990b]) which is consistent with the initial yield curve In an insightful study, Bryis, Grouhy and Schobel [1991] show that a cap can be interpreted as a string of put options and a floor as a string of call options on increasing maturity zero coupon bonds Combining Bryis et al.’s approximation and the extended Vasicek term structure model, this chapter calculates the durations of floaters and swaps with an embedded collar at the initial time The effects of yield curve shifts and potential change of cash flows due to embedded options are captured in the derivation of prices and durations under the extended Vasicek term structure model

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The rest of the chapter is organized as follows: Section 2.2 gives the relationship among the prices and durations of fixed rate bonds naked floating rate bonds caps floors collars floating rate bonds with collars and interest rate swaps with collars Expressions are then derived for the prices and durations of these securities Section 2.3 presents numerical simulations and graphically Ulustrates how the durations of these instruments change with the cap rate the floor rate the interest rate volatilitv and the level of interest rates Section 2.4 concludes the chapter

2.2 Derivation

2.2.1 Price and Duration Relationships

A floating rate bond is a variable rate bond with coupon payments reset

periodically to the current short term rate or other reference rate Usually the floating rate

bond comes with an interest rate cap or an interest rate floor or both An interest rate cap is an agreement between the issuer and the buyer of floating rate bond to limit the issuer's floating interest payment to a specific rate which is called the cap rate [f the market rate exceeds the cap rate at the coupon resetting date the bondholder only gets the interest payment corresponding to the cap rate on the next coupon resetting date An interest rate floor is an agreement between the issuer and buyer of floating rate bond to insure that the buyer gets a minimum interest payment corresponding to the floor rate’ A collar consists ofa cap and a floor which means that the floating rate bond buyer gets variable interest

payments restricted in a band

' “The interest rate cap or floor agreements are not necessarily between the bond issuer and the bond buyer