## Pricing and Modeling of Bonds and Interest Rate Derivatives

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

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 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) 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 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 nance. 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 aect the value of these variables and cause extensive loss to financial institutions and investors, and could potentially
affect 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 for the inconsistency of interest rates, which have been the subject of extensive research and generate lots of chaos in the financial world. To mitigate against this inconsistency, financial analysts develop an instrument to hedge this risk and speculate 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 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 n 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 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 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 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.