**Pricing and Modeling of Bonds and Interest Rate Derivatives**

**TABLE OF CONTENTS**

Epigraph iii

Dedication iv

Acknowledgement v

Abstract vii

1 Introduction

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

2 Background

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

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

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

2.2.2 Fixed and coating 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

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

**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 he RtT r(s)ds j Fti

under some martingale (risk-neutral) measure Q. Assume the underlying interest rate process is solution to the stochastic differential 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, p 2

( (t; rt) =

(t; rt) = a br

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

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

@ @ 1 2 @2

(t; rt) F (t; rt) + F (t; rt) + F (t; rt) r(t)F (t; rt) = 0

2

@x @t 2 @x

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

8 dA(s) = aB(s) + 2 B(s)2

dB(s) 2

= bB(s)

< ds 1

ds

:

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,

2 ab 2 2 2ab 4ab 3 2

A(T t) = e b(T t)+ e 2b(T t)+ (T t)+

b3 4b3 2b2 4b3

B(T t) = 1 e b(T t) 1

**Chapter One**

**Introduction**

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 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) give 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 proffer 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 de-signed 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 an etc. the value of these variables and cause extensive loss to financial institutions and investors and could potentially an ect 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 action, government policies, LIBOR (London Inter Bank O ered Rate), and market imperfections. These factors are responsible for the inconsistency of interest rates, which have been the subject of extensive re-search and generate lots of chaos in the financial world. To mitigate against this inconsistency, financial analysts develop an instrument to hedge this risk and spec-ulate the future growth or decline of an investment. A financial 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 financial markets over the years. According to [17] interest rate derivatives can be divided into different classifications, such as interest rate futures and forwards, Forward Rate Agreements (FRA’s), caps and oors, interest rate swaps, bond op-tions 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 different positions, thereby making risk less pro ts.

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 different portfolios of bonds, stocks, real estate, and other financial securities reduces risk and provides financial 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 financial institution, corporate organization, public authorities, or government for a de ned 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 pro t with a predictable income stream from such indenture and bond can even be sold before maturity date. Although 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 dis-count (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 in-creases, 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.

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