Pricing and Modeling of Bonds and Interest Rate Derivatives Chapter One PREAMBLE OF THE STUDY

Mathematicians are continually challenged to real world problems, especially in finance. To this end, Mathematicians develop tools to analyze; for example, the changes in interest rates corresponding to different 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 sufficiently close to reality, which are dependent on unstable real life variables. In rare situations, financial models fail as a result of uncertain changes that affect the value of these variables and cause extensive loss to financial institutions and investors, and could potentially
affect the economy of a country.

Chapter Two
Background
 Introduction
Financial mathematics employs different terms, notations, theories and theorems derived from concepts in both mathematics and finance. In this chapter, we introduce terminologies useful in our work, derived from concepts of mathematics and finance giving their mathematical interpretation and notation.
Definition

BOND
A bond is a form of loan to an entity (i.e financial institution, corporate organization, public authorities or government) for a defined 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)
 Key concepts of bonds
We shall employ the following terms in the description of bonds.
1. Face value or Par
Face value or Par is the amount a bond holder receives at the maturity date of the bond.
2. Coupon
Coupon is the amount the bond holder receives annually or semiannually from the bond issuer as compensation for holding the bond.
3. Coupon rate
Coupon rate is the agreed rate of interest payment on the par value.
4. Maturity date
Maturity date (T) is the date of contract expiration.
5. Time to maturity
Time to maturity (T − t) is the amount of time (in years) from the present time t to the maturity time T > t.
6. Discount
Discount (D) is the purchase price of a bond in the secondary market, below the face value.
7. Premium
Premium(P) is the purchase price of a bond in the secondary market above the face value.
Definition 

Bank Account
Let B(t) denote the value of a bank account at time t ≥ 0. Then B(t) is assumed to satisfy the initial value ordinary differential eqaution:
(
dB(t) = r(t)B(t)dt
B(0) = 1.
(2.2.1)
where r(t) is a positive function of time, called the interest rate at time t.
Then
B(t) = exp Z t
0
r(s)ds
, (2.2.2)
If r(t) is a constant interest rate,
B(t) = e
rt (2.2.3)
The bank account at time t is related to the bank account at a future time T by
B(t) = B(T)exp
−
Z T
t
r(s)ds

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Chapter Three
Stochastic Processes [4]
 Introduction
In this chapter, we introduce our readers to concepts of stochastic process that are useful in our work, we assume that notions of probability theory and stochastic processes [4] are already known to the reader.
 Stochastic Processes
Let (Ω, A, P) be a probability space.
A stochastic process is a collection of parametrized random variables {Xt}t∈I indexed by a time interval I = [0, ∞) defined on the probability space and assuming values  in R
d
, i.e Xt
: Ω → R
d
The value of the stochastic process Xt at ω ∈ Ω is denoted by Xt(ω) or X(t, ω).
Remark 3.2.1
1. Xt (ω) represents the result at time t of the possible outcome ω in the sample
space (Ω).
2. For fixed ω ∈ Ω, the map
t → X(t, ω) ∈ R
d
; t ∈ I
is called a sample path or trajectory of Xt

Chapter Four
Pricing of bonds and interest rate derivatives
Introduction
In this chapter, we highlight two main approaches, the martingale approach and the approach based on the use of partial differential equations, for deriving the formula for pricing bonds and interest rate derivatives. The martingale method is based on the risk valuation principle, which uses the theory of martingales to establish the price of a derivative security.
Basic Setup<

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