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**ABSTRACT**

Let P(t; T) denote the price of a zero-coupon bond at initial time t with maturity

T, given the stochastic interest rate (rt)t2R+ and a Brownian ltration fFt : t 0g.

Then,

P(t; T) = EQ

h

e

R T

t r(s)ds j Ft

i

under some martingale (risk-neutral) measure Q. Assume the underlying interest

rate process is solution to the stochastic dierential equation (SDE)

dr(t) = (t; r(t))dt + (t; rt)dW(t)

where (Wt)t2R is the standard Brownian motion under Q, with (t; rt) and (t; rt)

of the form, (

(t; rt) = a br

(t; rt) =

p

2

where r(0); a; b and are positive constants.

Then, the bond pricing PDE for P(t; T) = F(t; rt) written as

(t; rt)

@

@x

F(t; rt) +

@

@t

F(t; rt) +

1

2

2 @2

@x2F(t; rt) r(t)F(t; rt) = 0

subject to the terminal condition F(t; rt) = 1 which yield the Riccati equations,

8<

:

dA(s)

ds = aB(s) +

2

2

B(s)2

dB(s)

ds = bB(s) 1

with solution of the PDE in analytical form as the Price for zero-coupon bond is

given by,

P(t; T) = exp [A(T t) + B(T t)rt]

where,

A(Tt) =

2 ab

b3

eb(Tt)+

2

4b3

e2b(Tt)+

2 2ab

2b2

(Tt)+

4ab 32

4b3

B(T t) = 1

b

eb(Tt) 1

vii

**TABLE OF CONTENTS**

Epigraph iii

Dedication iv

Acknowledgement v

Abstract vii

1 Introduction

1

1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Background

7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Key concepts of bonds . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Fixed and oating coupon bonds . . . . . . . . . . . . . . . . 12

2.2.3 Interest Rate Derivatives [20] . . . . . . . . . . . . . . . . . . 15

3 Stochastic Processes [4] 21

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Classes of Stochastic Processes . . . . . . . . . . . . . . . . . 22

3.2.2 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.3 Filtration and Adapted Process . . . . . . . . . . . . . . . . . 22

3.3 Martingale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Brownian Stochastic Integral . . . . . . . . . . . . . . . . . . . . . . 27

3.5 Stochastic Dierential Equation (SDE) . . . . . . . . . . . . . . . . 31

3.5.1 It^o formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

viii

3.5.2 Existence and uniqueness of solution . . . . . . . . . . . . . . 34

4 Pricing of bonds and interest rate derivatives 38

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1.1 Basic Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1.2 Financial Market . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.3 Contingent claim, arbitrage and martingale measure . . . . . 40

4.2 Martingale Pricing Approach . . . . . . . . . . . . . . . . . . . . . . 49

4.2.1 Valuation of Interest rate Derivatives . . . . . . . . . . . . . 50

4.3 PDE Pricing Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.1 Bond Pricing using PDE . . . . . . . . . . . . . . . . . . . . . 56

5 Modelling of Interest Rate Derivatives and Bonds 58

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2 Short Rate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3 Vasicek Model [25] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.3.1 Pricing of zero-coupon bonds . . . . . . . . . . . . . . . . . . 70

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

ix

**CHAPTER ONE**

In Financial Mathematics, one of the most important areas of research where considerable

developments and contributions have been recently observed is the pricing

of interest rate derivatives and bonds. Interest rate derivatives are nancial instruments

whose payo is based on an interest rate. Typical examples are swaps,

options and Forward Rate Agreements (FRA’s). The uncertainty of future interest

rate movements is a serious problem which most investors (commission broker and

locals) gives critical consideration to, before making nancial decisions. Interest

rates are used as tools for investment decisions, measurement of credit risks, valuation

and pricing of bonds and interest rate derivatives. As a result of these, the

need to profer solution to this problem, using probabilistic and analytical approach

to predict future evolution of interest need to be established.

Mathematicians are continually challenged to real world problems, especially in

nance. To this end, Mathematicians develop tools to analyze; for example, the

changes in interest rates corresponding to dierent periods of time. The tool designed

is a mathematical representation to replicate and solve a real world problem.

These models are designed to produce results that are suciently close to reality,

which are dependent on unstable real life variables. In rare situations, nancial

models fail as a result of uncertain changes that aect the value of these variables

and cause extensive loss to nancial institutions and investors, and could potentially

aect the economy of a country.

Interest rates depends on several factors such as size of investments, maturity

date, credit default risk, economy i.e in ation, government policies, LIBOR (London

Inter Bank Oered Rate), and market imperfections. These factors are responsible

1

for the inconsistency of interest rates, which have been the subject of extensive research

and generate lots of chaos in the nancial world. To mitigate against this

inconsistency, nancial analysts develop an instrument to hedge this risk and speculate

the future growth or decline of an investment. A nancial instrument whose

payo depends on an interest rate of an investment is called interest rate derivative.

Interest rate derivatives are the most common derivatives that have been traded

in the nancial markets over the years. According to [17] interest rate derivatives can

be divided into dierent classications; such as interest rate futures and forwards,

Forward Rate Agreements(FRA’s), caps and oors, interest rate swaps, bond options

and swaptions. Generally, investors who trade on derivatives are categorised

into three groups namely: hedgers, speculators and arbitrageurs. Hedgers are risk

averse traders who uses interest rate derivatives to mitigate future uncertainty and

inconsistency of the market, while speculators use them to assume a market position

in the future, thereby trading to make gains or huge losses when speculation

fails. Arbitrageurs are traders who exploit the imperfections of the markets to take

dierent positions, thereby making risk less prots.

Investors minimize risk of loss by spreading their investment portfolio into different

sources whose returns are not correlated. Due to uncertainties in the market,

investing in dierent portfolios of bonds, stocks, real estate and other nancial securities

reduces risk and provides nancial security. Many investors hold bonds in

their investment portfolios without knowing what a bond is and how it works. A

bond is a form of loan to an entity (i.e nancial institution, corporate organization,

public authorities or government for a dened period of time where the lender

(bond holder) receives interest payments (coupon) annually or semiannually from

the (debtor) bond issuer who repay the loaned funds (Principal) at the agreed date

of refund (maturity date). Bonds are categorized based on the issuer, considered

into four groups: corporate bonds, government bonds (treasury), municipal bonds

also called mini bonds and agency bonds.

Bonds are risk-free kind of investment compared to stock, for instance treasury

bonds commonly called T-bills are credit default risk free investments, since the

bonds are issued by the government, also the mini bond are free of federal or State

taxes. Investing in bonds, preserve capital and yield prot with a predictable income

stream from such indenture and bond can even be sold before maturity date. Al-

2

though bonds carry also risk, such as credit default risk, interest rate risk, liquidity

risk, exchange rate risk, economic risk and market uctuations risk. Understanding

the characteristics of each kind of bond can be used to control exposure to these

forms of risk.

Bonds and shares have a similar property of price uctuation, for bonds interest

rate has an inverse relationship with bond price: when bond price goes up, interest

rate go down and when bond price go down, interest rate go up. Investors who trade

on bonds frequently ask brokers this question:

What is the total return on a bond and the current market value of

the bond ?

For example, an investor who buys a bond from a secondary market at a discount

(price below the bond’s price) and collects coupons on same bond and at the

maturity date, would collect same par value of the bond, but while holding the bond

before the maturity date, suppose the interest rate of same bond in the market increases,

which result to depreciation of value of the bond below the discount prize

that he bought the bond. At this stage, the investor wants to sell his old bond to

obtain the bond with higher interest rate and consult his investment broker from

whom he bought the bond with same question.

The investment broker analysis to decide expected return and market value of

the bond is determined using suitable models for the pricing of bonds and other

forms of interest rate derivatives.

1.1 Literature Review

In the past three decades, there have been a phenomenal growth in the trading of

interest rate derivatives, leading to a surge in research on derivative pricing theory.

Even before the upsurge of active trading of derivatives, considerable research had

been devoted to the valuation of interest rate. Several models of the term structure

have been proposed in the literatures. Examples are Black’s Scholes (1973), Dothan

(1978), Brennan and Schwartz (1979), Richard (1979), Langetieg (1980), Courtadon

(1982), Cox, Ingersoll, Ross (1985b), Ho and Lee (1986), Longsta (1989), Longsta

and Schwartz (1992) and Koedijk, Nissen, Schotman, and Wol (1997).

3

All these models have the advantage that they can be used to value interest rate

derivatives in a consistent way. Most practitioners often use Black’s Scholes (1973)

model for valuing options on commodity futures where forward bond prices rather

than forward interest rates are assumed to be lognormal. Elliot and Baier (1979)

in their work, studied six dierent econometric interest rate models to explain and

predict interest rates, tested the accuracy of the models tted to US monthly data

over a sample period of 7 years. The results obtained indicated that four out of the

models predict current interest rates movements quite accurately but their ability to

forecast future interest rates by applying actual information is seemed to be inaccurate.

Further work done by Brennan and Schwartz (1982) focused on modelling and

pricing of US government bonds from 1948 to 1979, with the objective of evaluating

the ability of the pricing model to detect underpriced or overpriced bonds. In their

result obtained over this period, it indicates no relationship between future values

of the short term interest rate and the long term interest rate, signifying that the

valuation model is consistent for short periods of time.

To improve previous valuation models, Cox, Ingersoll and Ross (1985) developed

an intertemporal general equilibrium asset pricing model to study the term

structure of interest rates. The model takes into consideration key factors for determining

the term structure of interest rates; which include anticipation of future

events, risk preferences, investment alternatives and preferences about the timing

of consumption. In contribution, these model is able to eliminate negative interest

rates in Vasicek (1977) models and able to predict how changes in a diverse range

of underlying variables will aect the term structure.

The inconsistency in the volatility parameters of dierent models is a concern

to practitioners on the choice of suitable model for dierent situations. As a result

there have been extension of existing models as new improved models to replace

the old ones. In 1990, John Hull and Alan White extended interest rate models of

Vasicek (1977) and Cox, Ingersoll, and Ross (1985b) so that they are consistent with

both the current term structure of interest rates and either the current volatilities

of all spot interest rates or the current volatilities of all forward interest rates.

Chan, Karolyi, Longsta and Sanders (1992) compare eight models of short term

interest rate within same framework to determine which model best ts the short

term Treasury bill yield data. A comparison of these models indicates that models

4

which best describe the dynamics of interest rates over time are those that allow the

conditional volatility of interest rate on the level of the interest rates. It is found

that of Vasicek and Cox-Ingersoll-Ross Square Root models, perform poorly in comparison

with Dothan and Cox-Ingersoll-Ross Variable Rate models.

Ho and Lee (1986) pioneered a new approach by showing how an interest-rate

model can be designed so that it is consistent with any specied initial term structure.

Their work has been extended by a number of researchers, including Black,

Derman, and Toy (1990), Dybvig (1988), and Milne and Turnbull (1989). Heath,

Jarrow, and Morton (HJM) (1987) present a general multifactor interest rate model

consistent with the existing term structure of interest rates and any specied volatility

structure. In the extensions, they use forward rate instead of bond prices, incorporate

continuous trading and replace the one factor model of Ho-Lee with multiple

random factors, broadening insight into the theoretical and pratical approach. The

HJM model provide practitioners with a general framework within which a no arbitrage

model can be developed for the pricing and hedging of interest rate derivatives

and bonds. For this reason, it was widely accepted by both the academics and practitioners.

Although it has aws with dierence in dimension with short rate models,

positive probability of instantaneous forward rate and recovery of caplets, which led

to the use of Monte Carlo Simulation Method named after Monte Carlo, which is a

time consuming approach used in rare cases when other options fail.

Longsta and Schwartz (1992) develop a two factor general equilibrium model

of the term structure of interest rates. The model is applied to derive closed form

expressions for discount bond prices and discount bond option prices. Factors used

are the short term interest rate and volatility of short term interest rates. The model

is able to determine the value of interest rate contingent claims as well as hedging

strategies of interest rate contingent claims. The model demonstrates advantages

over two factor models which include endogenous determination of interest rate risk

and a simplied version of the term structure of interest rates. Johansson (1994)

models a continuous time stochastic process on short term interest rates based on

sample results of the average interest rate for overnight loans on the interbank market

for the ve largest Swedish banks from 1986 to 1991. Results suggest that accuracy

on parameters is dependent on sample^as time length. Brenner, Harjes and Kroner

(1996) also analyze two dierent interest rate models; LEVELS and GARCH models

to develop an alternative class of model which improve on the inadequacy of the two

5

models. By comparison, LEVELS models put much emphasis on the dependence

of volatility on interest rate levels and neglect serial correlation in variances, while

GARCH models depend extensively on serial correlation in variances and neglect

the relationship between interest rates and volatility.

Furthermore, Koedijk, Nissen, Schotman, and Wol (1997) compare their model

against a single factor model, GARCH model, and to a level GARCH model for one

month Treasury bill rates. Quasi-maximum likelihood method was used to estimate

these models with results that demonstrate both models determines interest rate

volatility whereas GARCH models are non stationary in variance. Also In 1997

Brace, Gatarek and Musiela (BGM) presented a suitable approach that solves the

HJM problems. Further research explored this approach to develop new inventive

models suitable for pricing interest rate derivatives and models of these forms are

called LIBOR market models. In 2001, Linus Kajsajuntti considered pricing of interest

rate derivatives with the LIBOR Market Model. Treepongkaruna and Gray

(2003) compares various interest rate derivatives by applying closed form solutions,

a trinomial tree procedure and a Monte Carlo simulation technique and also provide

an accurate description on how to use Monte Carlo simulation to value interest rate

derivatives when the short rate follows arbitrary time series process.

Recent years have seen considerable contribution to interest rate derivatives and

bonds due to its market demand. In [12], the Fourier transform approach was applied

in the pricing of interest rate derivatives based on a technique introduced by Lewis

(2001) for equity options. In the books of James and Webber (2000), Hunt and

Kennedy (2000), Rebonato (2002), Cairns (2004) and Peter-Kohl Landgraf (2007),

extensive work was done on these subject relating dierent models and suitable

techniques to relatively price and hedge interest rate derivatives and bonds, which

serve as a guide for further research to solve the problem of pricing and hedging

these products.

6