Pricing And Modeling Of Bonds And Interest Rate Derivatives

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(T􀀀t) =

2 􀀀 ab
b3

e􀀀b(T􀀀t)+

􀀀
2
4b3

e􀀀2b(T􀀀t)+

2 􀀀 2ab
2b2

(T􀀀t)+
4ab 􀀀 32
4b3
B(T 􀀀 t) = 1
b
􀀀
e􀀀b(T􀀀t) 􀀀 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

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