Pricing and Modeling of Bonds and Interest Rate Derivatives
Chapter One
PREAMBLE OF THE STUDY
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 financial instruments whose payoff 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 financial 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 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
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
Let (Ω, A, F, P) be a filtered probability space with a finite number of stochastic process S0, S1 . . . SN under the assumption that the stochastic process are semimartingales. Using this setup, we consider N + 1 traded assets where Si(t) is the price of one unit of asset i at time t with S0 as the numeraire asset (asset that define
the units in which security prices are measured).
Definition 4.1.1
(1) A portfolio (trading strategy) is a (N + 1) component, locally bounded and predictable vector process of the form:
Chapter Five
Modelling of Interest Rate Derivatives and Bonds
Introduction
In the previous chapters, we discussed some approaches to the pricing of interest rate derivatives and bonds. This chapter presents a model for the pricing of interest rate derivatives and bonds.
There are several models, for the pricing and hedging of interest rate derivatives and bonds. Some of the models are widely used in practice. In this work, we would give an analysis of a model for the valuation of bonds and interest rate derivatives.
Short Rate Model
The short rate model is the oldest existing interest rate model, it assumes that the
1) short rate r at time t is the instantaneous spot rate r(t), under a risk neutral measure Q and satisfies the SDE
dr(t) = α(t, rt)dt + σ(t, rt)dW(t) (5.2.1)
2) drift α and the volatility σ satisfy the usual Lipschitz and boundedness conditions for the existence and uniqueness of a strong solution of the SDE.
By assumption (1) and the existence of a risk-neutral measure Q, the arbitrage-free price at time t of a contingent claim with payoff X(T) at time T is given by
Π(T, X) = S0(t)E
Q
X(T)
S0(T)
|Ft
(5.2.2)
thus, the price at time t of a zero coupon bound maturing at time T is given as
Chapter Six
Conclusion
Pricing and modeling of interest rate derivatives and bonds have proven to be a complex area where extensive research needs to be done, to develop consistent techniques (models) using theoretical and numerical tools.
In this work, we have described approaches for the valuation of interest rate derivatives and bonds.
Bibliography
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