Project Abstract

<p> </p><p>Let P(t; T) denote the price of a zero-coupon bond at initial time t with maturity<br>T, given the stochastic interest rate (rt)t2R+ and a Brownian ltration fFt t 0g.<br>Then,<br>P(t; T) = EQ<br>h<br>eô€€€<br>R T<br>t r(s)ds j Ft<br>i<br>under some martingale (risk-neutral) measure Q. Assume the underlying interest<br>rate process is solution to the stochastic dierential equation (SDE)<br>dr(t) = (t; r(t))dt + (t; rt)dW(t)<br>where (Wt)t2R is the standard Brownian motion under Q, with (t; rt) and (t; rt)<br>of the form, (<br>(t; rt) = a ô€€€ br<br>(t; rt) =<br>p<br>2<br>where r(0); a; b and are positive constants.<br>Then, the bond pricing PDE for P(t; T) = F(t; rt) written as<br>(t; rt)<br>@<br>@x<br>F(t; rt) +<br>@<br>@t<br>F(t; rt) +<br>1<br>2<br>2 @2<br>@x2F(t; rt) ô€€€ r(t)F(t; rt) = 0<br>subject to the terminal condition F(t; rt) = 1 which yield the Riccati equations,<br>8&lt;<br><br>dA(s)<br>ds = aB(s) +<br>2<br>2<br>B(s)2<br>dB(s)<br>ds = ô€€€bB(s) ô€€€ 1<br>with solution of the PDE in analytical form as the Price for zero-coupon bond is<br>given by,<br>P(t; T) = exp [A(T ô€€€ t) + B(T ô€€€ t)rt]<br>where,<br>A(Tô€€€t) =</p><p>2 ô€€€ ab<br>b3</p><p>eô€€€b(Tô€€€t)+</p><p>ô€€€<br>2<br>4b3</p><p>eô€€€2b(Tô€€€t)+</p><p>2 ô€€€ 2ab<br>2b2</p><p>(Tô€€€t)+<br>4ab ô€€€ 32<br>4b3<br>B(T ô€€€ t) = 1<br>b<br>ô€€€<br>eô€€€b(Tô€€€t) ô€€€ 1</p><p>vii</p> <br><p></p>

Project Overview

<p> In Financial Mathematics, one of the most important areas of research where considerable<br>developments and contributions have been recently observed is the pricing<br>of interest rate derivatives and bonds. Interest rate derivatives are nancial instruments<br>whose payo is based on an interest rate. Typical examples are swaps,<br>options and Forward Rate Agreements (FRA’s). The uncertainty of future interest<br>rate movements is a serious problem which most investors (commission broker and<br>locals) gives critical consideration to, before making nancial decisions. Interest<br>rates are used as tools for investment decisions, measurement of credit risks, valuation<br>and pricing of bonds and interest rate derivatives. As a result of these, the<br>need to profer solution to this problem, using probabilistic and analytical approach<br>to predict future evolution of interest need to be established.<br>Mathematicians are continually challenged to real world problems, especially in<br>nance. To this end, Mathematicians develop tools to analyze; for example, the<br>changes in interest rates corresponding to dierent periods of time. The tool designed<br>is a mathematical representation to replicate and solve a real world problem.<br>These models are designed to produce results that are suciently close to reality,<br>which are dependent on unstable real life variables. In rare situations, nancial<br>models fail as a result of uncertain changes that aect the value of these variables<br>and cause extensive loss to nancial institutions and investors, and could potentially<br>aect the economy of a country.<br>Interest rates depends on several factors such as size of investments, maturity<br>date, credit default risk, economy i.e in ation, government policies, LIBOR (London<br>Inter Bank Oered Rate), and market imperfections. These factors are responsible<br>1<br>for the inconsistency of interest rates, which have been the subject of extensive research<br>and generate lots of chaos in the nancial world. To mitigate against this<br>inconsistency, nancial analysts develop an instrument to hedge this risk and speculate<br>the future growth or decline of an investment. A nancial instrument whose<br>payo depends on an interest rate of an investment is called interest rate derivative.<br>Interest rate derivatives are the most common derivatives that have been traded<br>in the nancial markets over the years. According to [17] interest rate derivatives can<br>be divided into dierent classications; such as interest rate futures and forwards,<br>Forward Rate Agreements(FRA’s), caps and oors, interest rate swaps, bond options<br>and swaptions. Generally, investors who trade on derivatives are categorised<br>into three groups namely: hedgers, speculators and arbitrageurs. Hedgers are risk<br>averse traders who uses interest rate derivatives to mitigate future uncertainty and<br>inconsistency of the market, while speculators use them to assume a market position<br>in the future, thereby trading to make gains or huge losses when speculation<br>fails. Arbitrageurs are traders who exploit the imperfections of the markets to take<br>dierent positions, thereby making risk less prots.<br>Investors minimize risk of loss by spreading their investment portfolio into different<br>sources whose returns are not correlated. Due to uncertainties in the market,<br>investing in dierent portfolios of bonds, stocks, real estate and other nancial securities<br>reduces risk and provides nancial security. Many investors hold bonds in<br>their investment portfolios without knowing what a bond is and how it works. A<br>bond is a form of loan to an entity (i.e nancial institution, corporate organization,<br>public authorities or government for a dened period of time where the lender<br>(bond holder) receives interest payments (coupon) annually or semiannually from<br>the (debtor) bond issuer who repay the loaned funds (Principal) at the agreed date<br>of refund (maturity date). Bonds are categorized based on the issuer, considered<br>into four groups: corporate bonds, government bonds (treasury), municipal bonds<br>also called mini bonds and agency bonds.<br>Bonds are risk-free kind of investment compared to stock, for instance treasury<br>bonds commonly called T-bills are credit default risk free investments, since the<br>bonds are issued by the government, also the mini bond are free of federal or State<br>taxes. Investing in bonds, preserve capital and yield prot with a predictable income<br>stream from such indenture and bond can even be sold before maturity date. Al-<br>2<br>though bonds carry also risk, such as credit default risk, interest rate risk, liquidity<br>risk, exchange rate risk, economic risk and market uctuations risk. Understanding<br>the characteristics of each kind of bond can be used to control exposure to these<br>forms of risk.<br>Bonds and shares have a similar property of price uctuation, for bonds interest<br>rate has an inverse relationship with bond price: when bond price goes up, interest<br>rate go down and when bond price go down, interest rate go up. Investors who trade<br>on bonds frequently ask brokers this question:<br>What is the total return on a bond and the current market value of<br>the bond ?<br>For example, an investor who buys a bond from a secondary market at a discount<br>(price below the bond’s price) and collects coupons on same bond and at the<br>maturity date, would collect same par value of the bond, but while holding the bond<br>before the maturity date, suppose the interest rate of same bond in the market increases,<br>which result to depreciation of value of the bond below the discount prize<br>that he bought the bond. At this stage, the investor wants to sell his old bond to<br>obtain the bond with higher interest rate and consult his investment broker from<br>whom he bought the bond with same question.<br>The investment broker analysis to decide expected return and market value of<br>the bond is determined using suitable models for the pricing of bonds and other<br>forms of interest rate derivatives.<br>1.1 Literature Review<br>In the past three decades, there have been a phenomenal growth in the trading of<br>interest rate derivatives, leading to a surge in research on derivative pricing theory.<br>Even before the upsurge of active trading of derivatives, considerable research had<br>been devoted to the valuation of interest rate. Several models of the term structure<br>have been proposed in the literatures. Examples are Black’s Scholes (1973), Dothan<br>(1978), Brennan and Schwartz (1979), Richard (1979), Langetieg (1980), Courtadon<br>(1982), Cox, Ingersoll, Ross (1985b), Ho and Lee (1986), Longsta (1989), Longsta<br>and Schwartz (1992) and Koedijk, Nissen, Schotman, and Wol (1997).<br>3<br>All these models have the advantage that they can be used to value interest rate<br>derivatives in a consistent way. Most practitioners often use Black’s Scholes (1973)<br>model for valuing options on commodity futures where forward bond prices rather<br>than forward interest rates are assumed to be lognormal. Elliot and Baier (1979)<br>in their work, studied six dierent econometric interest rate models to explain and<br>predict interest rates, tested the accuracy of the models tted to US monthly data<br>over a sample period of 7 years. The results obtained indicated that four out of the<br>models predict current interest rates movements quite accurately but their ability to<br>forecast future interest rates by applying actual information is seemed to be inaccurate.<br>Further work done by Brennan and Schwartz (1982) focused on modelling and<br>pricing of US government bonds from 1948 to 1979, with the objective of evaluating<br>the ability of the pricing model to detect underpriced or overpriced bonds. In their<br>result obtained over this period, it indicates no relationship between future values<br>of the short term interest rate and the long term interest rate, signifying that the<br>valuation model is consistent for short periods of time.<br>To improve previous valuation models, Cox, Ingersoll and Ross (1985) developed<br>an intertemporal general equilibrium asset pricing model to study the term<br>structure of interest rates. The model takes into consideration key factors for determining<br>the term structure of interest rates; which include anticipation of future<br>events, risk preferences, investment alternatives and preferences about the timing<br>of consumption. In contribution, these model is able to eliminate negative interest<br>rates in Vasicek (1977) models and able to predict how changes in a diverse range<br>of underlying variables will aect the term structure.<br>The inconsistency in the volatility parameters of dierent models is a concern<br>to practitioners on the choice of suitable model for dierent situations. As a result<br>there have been extension of existing models as new improved models to replace<br>the old ones. In 1990, John Hull and Alan White extended interest rate models of<br>Vasicek (1977) and Cox, Ingersoll, and Ross (1985b) so that they are consistent with<br>both the current term structure of interest rates and either the current volatilities<br>of all spot interest rates or the current volatilities of all forward interest rates.<br>Chan, Karolyi, Longsta and Sanders (1992) compare eight models of short term<br>interest rate within same framework to determine which model best ts the short<br>term Treasury bill yield data. A comparison of these models indicates that models<br>4<br>which best describe the dynamics of interest rates over time are those that allow the<br>conditional volatility of interest rate on the level of the interest rates. It is found<br>that of Vasicek and Cox-Ingersoll-Ross Square Root models, perform poorly in comparison<br>with Dothan and Cox-Ingersoll-Ross Variable Rate models.<br>Ho and Lee (1986) pioneered a new approach by showing how an interest-rate<br>model can be designed so that it is consistent with any specied initial term structure.<br>Their work has been extended by a number of researchers, including Black,<br>Derman, and Toy (1990), Dybvig (1988), and Milne and Turnbull (1989). Heath,<br>Jarrow, and Morton (HJM) (1987) present a general multifactor interest rate model<br>consistent with the existing term structure of interest rates and any specied volatility<br>structure. In the extensions, they use forward rate instead of bond prices, incorporate<br>continuous trading and replace the one factor model of Ho-Lee with multiple<br>random factors, broadening insight into the theoretical and pratical approach. The<br>HJM model provide practitioners with a general framework within which a no arbitrage<br>model can be developed for the pricing and hedging of interest rate derivatives<br>and bonds. For this reason, it was widely accepted by both the academics and practitioners.<br>Although it has aws with dierence in dimension with short rate models,<br>positive probability of instantaneous forward rate and recovery of caplets, which led<br>to the use of Monte Carlo Simulation Method named after Monte Carlo, which is a<br>time consuming approach used in rare cases when other options fail.<br>Longsta and Schwartz (1992) develop a two factor general equilibrium model<br>of the term structure of interest rates. The model is applied to derive closed form<br>expressions for discount bond prices and discount bond option prices. Factors used<br>are the short term interest rate and volatility of short term interest rates. The model<br>is able to determine the value of interest rate contingent claims as well as hedging<br>strategies of interest rate contingent claims. The model demonstrates advantages<br>over two factor models which include endogenous determination of interest rate risk<br>and a simplied version of the term structure of interest rates. Johansson (1994)<br>models a continuous time stochastic process on short term interest rates based on<br>sample results of the average interest rate for overnight loans on the interbank market<br>for the ve largest Swedish banks from 1986 to 1991. Results suggest that accuracy<br>on parameters is dependent on sample^as time length. Brenner, Harjes and Kroner<br>(1996) also analyze two dierent interest rate models; LEVELS and GARCH models<br>to develop an alternative class of model which improve on the inadequacy of the two<br>5<br>models. By comparison, LEVELS models put much emphasis on the dependence<br>of volatility on interest rate levels and neglect serial correlation in variances, while<br>GARCH models depend extensively on serial correlation in variances and neglect<br>the relationship between interest rates and volatility.<br>Furthermore, Koedijk, Nissen, Schotman, and Wol (1997) compare their model<br>against a single factor model, GARCH model, and to a level GARCH model for one<br>month Treasury bill rates. Quasi-maximum likelihood method was used to estimate<br>these models with results that demonstrate both models determines interest rate<br>volatility whereas GARCH models are non stationary in variance. Also In 1997<br>Brace, Gatarek and Musiela (BGM) presented a suitable approach that solves the<br>HJM problems. Further research explored this approach to develop new inventive<br>models suitable for pricing interest rate derivatives and models of these forms are<br>called LIBOR market models. In 2001, Linus Kajsajuntti considered pricing of interest<br>rate derivatives with the LIBOR Market Model. Treepongkaruna and Gray<br>(2003) compares various interest rate derivatives by applying closed form solutions,<br>a trinomial tree procedure and a Monte Carlo simulation technique and also provide<br>an accurate description on how to use Monte Carlo simulation to value interest rate<br>derivatives when the short rate follows arbitrary time series process.<br>Recent years have seen considerable contribution to interest rate derivatives and<br>bonds due to its market demand. In [12], the Fourier transform approach was applied<br>in the pricing of interest rate derivatives based on a technique introduced by Lewis<br>(2001) for equity options. In the books of James and Webber (2000), Hunt and<br>Kennedy (2000), Rebonato (2002), Cairns (2004) and Peter-Kohl Landgraf (2007),<br>extensive work was done on these subject relating dierent models and suitable<br>techniques to relatively price and hedge interest rate derivatives and bonds, which<br>serve as a guide for further research to solve the problem of pricing and hedging<br>these products.<br>6 <br></p>