INTEREST RATE MODELING AND A TIME SERIES MODEL FOR
INTEREST RATE MODELING AND A TIME SERIES MODEL FORFUNCTIONAL DATAA DISSERTATIONSUBMITTED TO THE DEPARTMENT OF STATISTICSAND THE COMMITTEE ON GRADUATE STUDIESOF STANFORD UNIVERSITYIN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR THE DEGREE OFDOCTOR OF PHILOSOPHYChung Kwan PongAugust 2010
2010 by Chung Kwan Pong. All Rights Reserved.Re-distributed by Stanford University under license with the author.This work is licensed under a Creative Commons AttributionNoncommercial 3.0 United States 3.0/us/This dissertation is online at: http://purl.stanford.edu/nv255rn6736ii
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.Tse Lai, Primary AdviserI certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.Paul SwitzerI certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.Guenther WaltherApproved for the Stanford University Committee on Graduate Studies.Patricia J. Gumport, Vice Provost Graduate EducationThis signature page was generated electronically upon submission of this dissertation inelectronic format. An original signed hard copy of the signature page is on file inUniversity Archives.iii
AbstractIn finance, an interest rate derivative is a financial instrument where the underlying asset is aninterest rate at which payments are made based on a notional amount. A common approach to priceinterest rate derivatives is through the use of interest rate models. However, a drawback with thisapproach is that calibration of interest rate models does not involve the interest rate being modeled.Hence, calibrated models may not be good representations of interest rates and may not producereliable derivative prices.To deal with the issue, we propose a time series modeling approach to analyze interest rates,specifically, the zero-coupon yield curves. In this approach, yield curves are modeled as functionaldata and we introduce models that are based on the well-known autoregressive model in time seriesanalysis. The objective of this approach is to understand the dependency of the yield curves onhistorical data and to predict future yield curves before they are observed.The proposed models are illustrated and compared with the time series of US Treasury zerocoupon yield curves. We explore how individual models perform during different times in an economiccycle. We also propose a way to predict future caplet prices by combining yield curve predictionusing functional time series models and historical implied volatilities of caplets. The time seriesapproach that we propose are shown to work well against existing models such as the Hull-Whitemodel.iv
AcknowledgementsI would like to express my sincere appreciation to my advisor, Professor Tze Leung Lai, for hisimmense support throughout the course of my PhD career. He has given me insightful commentsand advices on my research. I thank him for his patience and encouragement. I would not be ableto complete my thesis without the guidance from him.I would like to thank Professor Paul Switzer for giving valuable suggestions on my researchand future direction. I also thank my oral examination committee members, Professors GuentherWalther, Yinyu Ye, and Antoine Toussaint, who provided constructive feedback on my research.I am grateful to many friends at Stanford. Specifically, I must thank Ling Chen and Kevin Sunfor discussing research problems with me. I also thank Zhen Wei, Nicholas Johnson, Zongming Ma,Baiyu Zhou, Shaojie Deng, Wai Wai Liu, and Feng Zhang for their time with me.On a more personal note, my girlfriend Yi Fang Chen has given me strength and motivation. Ithank her for her patience and understanding even during hard times of this study. I would alsolike to thank my parents for their support and encouragement. Their confidence in me has inspiredme to push limits and reach my full potential. I am grateful to have such a wonderful family thatcontinues to support me in my future endeavors.v
ContentsAbstractivAcknowledgementsv1 Introduction11.1Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2Functional Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21.3Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 Interest Rate Markets2.114Elements of Interest Rate Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . .42.1.1Bond Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42.1.2LIBOR and Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . .52.1.3Interest Rate Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62.1.4Caps and Floors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72.1.5Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72.2Yield Curve Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82.3Stochastic Interest Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92.3.1Short Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92.3.2Heath-Jarrow-Morton (HJM) Framework . . . . . . . . . . . . . . . . . . . .132.3.3LIBOR Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13Parameter Estimation in Interest Rate Models . . . . . . . . . . . . . . . . . . . . .152.4vi
3 A Functional Time Series Approach163.1Yield Curves as Functional Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183.2Review of Time Series Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183.3Autoregressive Functional Time Series Models . . . . . . . . . . . . . . . . . . . . . .203.4Estimation of the Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .213.4.1Discretizing the Domain of Definition of the Functions . . . . . . . . . . . . .223.4.2Basis Functions Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . .233.5Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .263.6Basis Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .273.7Autoregressive Functional Exogenous Model (ARFX) . . . . . . . . . . . . . . . . . .313.7.1Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .323.7.2Solving for the Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .323.7.3Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34Pointwise Autoregressive Functional Model (PARF) . . . . . . . . . . . . . . . . . .353.8.1Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .353.8.2Solving for the Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .363.8.3Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .373.84 Empirical Study394.1Yield Curve Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .394.2Yield Curve Prediction Performance . . . . . . . . . . . . . . . . . . . . . . . . . . .404.2.1Benchmark Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .434.2.2Autoregressive Functional Models. . . . . . . . . . . . . . . . . . . . . . . .454.3Principal Component Analysis on the Yield Curves . . . . . . . . . . . . . . . . . . .514.4Caplet Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .524.5Caplet Pricing Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .544.5.1Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .544.5.2Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55Caplet Price Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .554.6.1554.6Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vii
4.6.2Prediction Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .575 Summary62A Splines64A.1 B-splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66A.2 Spline Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67A.3 Spline Smoothing in Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . .68References69viii
List of Tables4.1One-day ahead prediction error of the yield curve of several models. Error is calculatedas the sum of the absolute difference between the forecast and the observed yieldcurves over maturities 0.25, 0.5, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. ARF (B.S.) is theARF model with basis selection. PARF (B) and PARF (D) correspond to the PARFmodel where the coefficients are approximated by the basis function or discretizationapproaches, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .484.2A Summary of windows 1-25 (stable economic period) of Table 4.1. . . . . . . . . . .504.3A Summary of windows 26-39 (unstable economic period) of Table 4.1. . . . . . . . .514.4Loadings of the first 3 principal components when PCA is applied to the differencedyield curves during 3 periods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.552RMSE of one-day ahead prediction of caplets with strike 4% and maturities less thanor equal to 5 years. “ARF (B.S.)” refers to ARF basis selection; and “PARF (B)”refers to PARF solved using basis functions approach. . . . . . . . . . . . . . . . . .4.6A summary of caplet prediction RMSE in windows 1-25 (stable economic period) ofTable 4.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.75961A summary of caplet prediction RMSE in windows 26-39 (unstable economic period)of Table 4.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ix61
List of Figures3.1U.S. Treasury zero-coupon bond yield curve on the first trading day of each of years2005-2009 obtained by the nonparametric splines approach. . . . . . . . . . . . . . .4.119U.S. Treasury zero-coupon bond yield curves from Jun 28, 2004 to Mar 10, 2009.The top graph is the original data and the bottom graph is obtained by applyingnonparametric splines method on the original data. . . . . . . . . . . . . . . . . . . .4.2Differenced U.S. Treasury zero-coupon bond yield curves from Jun 28, 2004 to Mar10, 2009. Obtained by differencing the smoothed yield data of Figure 4.1. . . . . . .4.34142Daily caplet price data with strike 4% and maturities less than or equal to 5 yearsfrom Jun 28, 2004 to Mar 10, 2009. . . . . . . . . . . . . . . . . . . . . . . . . . . . .53A.1 B-spline basis functions on the interval [0,10] with knots 1, 2, . . . , 9. Left figure showslinear (order 2) basis functions; right figure shows quadratic (order 3) basis functions. 67x
Interest Rate Modeling and a Time Series Model forFunctional DataChung Kwan PongAugust 24, 2010
Chapter 1Introduction1.1MotivationIn finance, an interest rate derivative is a financial instrument where the underlying asset is aninterest rate at which payments are made based on a notional amount. The price of an interest ratederivative depends on the level of the interest rate and its expected change in the future. To pricean interest rate derivative, a common approach is to define the future evolution of the interest ratesusing an interest rate model. There are three main types of interest rate models. A short rate modeldescribes the short rate; an HJM model describes the instantaneous forward rate; and a marketmodel describes the forward rate. These interest rate models are based on some parameters whichare solved by a process called calibration. Calibration makes sure that the interest rate modelsproduce prices that are close to the market prices of some interest rate derivatives. These modelparamters are then used to price other interest rate derivatives.A drawback of this approach is that the interest rate that is modeled by the interest rate modelis never used in the calibration procedure. Hence, the calibrated model may not give a very goodrepresentation of the dynamics of the interest rate, and hence the prices it produces may be unreliable. In this thesis, we propose a time series approach to analyze the time series of zero-couponyield curves. It aims to understand the dependence structure of the yield curve on past data and tomake accurate forecasts of future yield curves. In the model, yield curves are treated as functional1
CHAPTER 1. INTRODUCTION2data, where each curve is regarded as a functional observation.In Chapter 4, we illustrate the use of this modeling approach by looking at how well it canpredict the yield curve on future periods. We then propose to use these predictions in combinationwith the implied volaitlities of caplets to predict the prices of future caplets. Not only is this timeseries approach of predicting caplet prices shown to work well against pricing by calibrating interestrate models, but treating yield curves as functional data is seen to have an advantage because theforecasts of the model are also functional. When calculating caplet prices, points on the yield curveat maturities where no observations are recorded are needed.1.2Functional DataA set of data can be considered functional if observations have common underlying functional structure such as curves or surfaces. For example, the weights of an individual recorded at different timesis a functional observation because it can be considered as a function of the relationship betweenweight and time. If the weight information is recorded for more than one individual, it is considereda functional data set. A functional time series is a set of functional data where the observationsare taken at successive times. In Chapters 3 and 4, the time series of yield curves is treated asa functional time series. We assume that the functional observations are spaced at uniform timeintervals.Normally, functional data are collected in discrete form rather than in functional form becauseit is not possible to record and store an infinite number of points. In order to analyze them infunctional form, these data are first converted into functions via means such as spline or kernelsmoothing. The time series of yield curves is treated in the same way. Section 2.2 discusses someparametric and nonparametric methods that estimate the zero-coupon yield curve from the pricesof a number of bonds. This also makes it possible to convert the zero-coupon yield curve to theforward rate curve, which involves taking derivatives to the curve.Analyzing data in functional form is not a new idea. Ramsay and Silverman (2005) describeseveral statistical methods that can be used to analyze functional data such as functional principalcomponent analysis, functional canonical correlation analysis, and functional linear regression.
CHAPTER 1. INTRODUCTION1.33Outline of the ThesisThis thesis is organized as follows. Chapter 2 provides some background information about interestrate markets. In particular, it describes some popular interest rate derivatives such as swaps, caps,floors, and swaptions, and interest rate models that include short rate models, HJM models, andmarket models. Chapter 3 provides functional time series models that are used to model the timeseries of yield curves. The functional autoregressive model and the pointwise autoregressive functional model are presented. They are solved by first assuming the functional coefficients are linearcombinations of basis functions and then using least squares method. Chapter 4 provides empiricalstudies that illustrate how the functional time series models can be applied to real world data. Thefirst study compares the prediction accuracy of the one-day ahead out-of-sample forecast of severaltime series models. The second study shows how the results of the first study can be used to predictfuture caplet prices. Chapter 5 provides a summary of the main ideas and results of the thesis.
Chapter 2Interest Rate MarketsThis chapter provides some financial background of the interest rate markets. Section 2.1 providessome basic concepts and definitions of popular interest rate derivatives including bonds, LIBOR,forward rates, interest rate swaps, caps, floors, and swaptions. Section 2.2 describes methods that canbe used to estimate the zero-coupon yield curve (as known as the zero curve) from given default-freebond prices. Section 2.3 describes several types of models in the literature that model the futureevolution of interest rates. Short rate models describe the short rate; HJM models describe theinstantaneous forward rate; the LIBOR market model describes the forward LIBOR. Section 2.4discusses how the paramters in the interest rate models can be estimated.2.12.1.1Elements of Interest Rate MarketsBond MarketIn finance, a zero-coupon bond with face value 1 is a debt security that pays 1 unit of currency at thematurity date. A coupon bond is like a zero-coupon bond but with interest payments at specifiedtimes before the maturity. In the United States, these debt securities issued by the US Treasury aredivided into 3 categories. Treasury bills (T-bills) are zero-coupon bonds with maturity less than 1year. Treasury notes (T-notes) are semiannual coupon bonds with maturity between 1 and 10 years.Treasury bonds (T-bonds) are semiannual coupon bonds with maturity longer than 10 years.4
CHAPTER 2. INTEREST RATE MARKETS5Denote the price at time t of a zero-coupon bond with face value 1 and maturity T by P (t, T ).It is clear that P (T, T ) 1. For a coupon bond with face value 1 and coupon payments ci at timesT1 T2 · · · Tn T , it can be realized as a sum of zero-coupon bonds and its price is given byn 1Xci P (t, Ti ) (1 cn )P (t, Tn ).(2.1)i 1Under continuous compounding, the yield-to-maturity (or simply yield) y of such a bond is definedto be the solution of the following equationbond price n 1Xci ey(Ti t) (1 cn )e y(Tn t) .i 1The spot rate is the yield of a zero-coupon bond (under continuous compounding) and is given byR(r, T ) log P (t, T ).T tThe relationship between interest rates and their maturities is called the term structure.2.1.2LIBOR and Forward RatesLIBOR stands for London InterBank Offered Rate. It is a daily reference rate based on the interestrates that banks in the London wholesale money market charge for borrowing funds to each other. Itis an annualized, simple interest rate that will be delivered at the end of a specified period. DenoteLIBOR by F (t, T ), where t is the current time and T is the maturity. Then it is defined asF (t, T ) 1T t 1 1 .P (t, T )A forward rate agreement (FRA) is a contract today t for a loan between T1 and T2 . It givesits holder a loan at time T1 , with a fixed simple interest rate for the period T2 T1 , to be paid attime T2 in additional to the principal. The rate agreed in a FRA is called the forward rate and is
CHAPTER 2. INTEREST RATE MARKETS6denoted by F (t, T1 , T2 ). Its value is defined asF (t, T1 , T2 ) 1T2 T1 P (t, T1 ) 1 .P (t, T2 )(2.2)The instanteneous forward rate at time t with maturity T is denoted by f (t, T ) and is given byf (t, T ) log P (t, T ). TFrom this definition, the zero-coupon bond price P (t, T ) can be expressed as( ZP (t, T ) exp )Tf (t, u)du .(2.3)tThe short rate r(t) is defined asr(t) lim f (t, T ) f (t, t).T t(2.4)It represents the interest rate at which a loan is made for an infinitesimally short period of timefrom time t.2.1.3Interest Rate SwapsAn interest rate swap is a contract between two parties in which they agree to exchange one streamof cash flows based on fixed interest rate κ with another stream based on variable interest rate,which is usually taken to be LIBOR. Let t T0 T1 · · · Tn , where t is the current timeand T1 , . . . , Tn are times at which payments occur. Tn is called the maturity of the swap. Often,Ti Ti 1 δ and δ can be 1, 1/2, or 1/4 year. If t T0 , then the swap is called a forward swap.Let N be the notional of the swap. Then at each time Ti , i 1, . . . , n, one party pays the otherparty a fixed amount N δκ and receives a floating amount N δF (Ti 1 , Ti ). The fixed interest rate κis called the swap rate. Its value at time t is denoted by rswap (t, T0 , Tn ) and is defined as follows:rswap (t, T0 , Tn ) P (t, T0 ) P (t, Tn )Pn.δ i 1 P (t, Ti )
CHAPTER 2. INTEREST RATE MARKETS2.1.47Caps and FloorsAn interest rate caplet with reset date T and settlement date T δ is a European call optionon LIBOR, that pays its owner at T δ if the rate exceeds the strike rate K in the amount ofδ max(0, F (T, T δ) K).An interest rate cap is a series of caplets. Let T0 T1 · · · Tn be future dates and K be thestrike rate. Then at each time Ti , i 1, . . . , n, the cap pays (Ti Ti 1 ) max(0, F (Ti 1 , Ti ) K).Interest rate caps are designed to provide insurance against the interest rate of a floating rate loanrising above a certain level K.Floorlet and floor are European put option counterparts of caplet and cap, respectively. Usingthe same notations, a floorlet pays δ max(0, K F (T, T δ)), and a floor is a series of floorlets andpays δ max(0, K F (Ti 1 , Ti )) at time Ti for i 1, . . . , n.In the market, caplets and caps (as well as floorlets and floors) are quoted in terms of theirimplied volatilities. Their prices are obtained by plugging the implied volatilities into Black (1976)capletiformula. Suppose the implied volatility at time t for the ith caplet is σt. Then its price attime t is given bycapleti p(Ti Ti 1 )P (t, Ti ) Black(F (t, Ti 1 , Ti ), K, σtTi 1 t),(2.5)where Black(L, K, σ) LΦlog(L/K) σ σ2 KΦlog(L/K) σ σ2 ,(2.6)and Φ is the standard normal cumulative distribution function. The relationship between the capcapprice and the cap implied volatility σtis given byn Xcap p(Ti Ti 1 )P (t, Ti ) Black F (t, Ti 1 , Ti ), K, σtTi 1 t .i 12.1.5SwaptionsA swaption with strike κ is an option that grants its owner the right to enter into a swap withfixed rate κ at the maturity T0 of the swaption. Suppose the payments of the swap occur at timesT1 , . . . , Tn . The duration of the swap Tn T0 is called the tenor of the swaption. The payoff of the
CHAPTER 2. INTEREST RATE MARKETS8swaption at maturity T0 is given byδ max{0, rswap (T0 , Tn ) κ}nXP (T0 , Ti ).i 12.2Yield Curve EstimationThe term structure of interest rate can be described by the price of a zero-coupon bond withface value 1 versus its maturity. Alternatively, it can be described by the yield curve, which is therelationship between the yield of a zero-coupon bond and its maturity. The yield curve is usually, butnot always, an increasing function of time t. Its shape and level reveal conditions in the economyand the financial markets, and so yield curves are monitored closely by economists and marketpractitioners.To estimate the yield curve at current time 0, one takes a set of n reference default-free bondssuch as the US Treasury bonds, which is seen in (2.1) to be dependent on the function P (0, ·). Aparametric or nonparametric model is assumed on P (0, T ) and least squares regression is used toestimate the parameters. That is, the following quantity is minimized over the parameters of thePn2model:i 1 (Bi B̂i ) , where Bi is the observed price of the ith bond, and B̂i is model price ofthe ith bond.A parametric model on P (0, T ) assumes certain form on it so that by varying the parameters,the model is able to reproduce the the shapes seen in historical yield curves: increasing, decreasing,flat, humped, and inverted. For example, Nelson and Siegel (1987) assume the following model onthe instanteneous forward rate.n son sosf (0, s) β0 β1 exp β2 exp ,τττwhich implies (by (2.3))noP (0, t) exp β0 t (β1 β2 )τ (1 e t/τ ) tβ2 e t/τ .Svensson (1994) generalized the Nelson-Siegel model by assuming the instantenous forward rate has
CHAPTER 2. INTEREST RATE MARKETS9the following form: sssssf (0, s) β0 β1 exp β2 exp β3 exp ,τ1τ1τ1τ2τ2which is capable of producing additional U and humped shapes for P (0, t):noP (0, t) exp β0 t (β1 β2 )τ1 (1 e t/τ1 ) tβ2 e t/τ1 β3 τ2 (1 e t/τ2 ) tβ3 e t/τ2 .The nonparametric approach to estimate the yield curve is to express the function P (0, T ) usingspline basis functions. The regression parameters can then be estimated using ordinarily least squaresmethod. We will use this approach in Chapter 4 with a spline function called B-spline. Details ofB-spline is given in Appendix A.2.3Stochastic Interest Rate ModelsInterest rate models are mathematical models that describe the dynamics of interest rates in thefuture. These models are used to price interest rate derivatives that depend on future interest rates.The quantities that are modeled can be different in different interest rate models. This section givesa brief description of three kinds of interest rate models: short rate models, HJM framework, andLIBOR market model, where the quantities of interest are short rate, instantaneous forward rate,and forward LIBOR, respectively.2.3.1Short Rate ModelsShort rate models are stochastic models that prescribe the dynamics of the short rate rt , which wasdefined in (2.4). It can be shown that in the absence of arbitrage, under some technical conditions,the price of a zero-coupon bond is related to short rate by the following formula:"( ZP (t, T ) E exp tT)rs ds#rt r .
CHAPTER 2. INTEREST RATE MARKETS10Hence, future bond prices can be calculated if the dynamics of the short rate is specified. Interest ratemodels with different forms have been proposed in the literature and this section describes someof them. In the following models, Wt denotes a standard Brownian motion under a risk-neutralprobability measure.Vasicek ModelUnder the Vasicek model (Vasicek, 1977), the short rate rt is modeled asdrt a(b rt )dt σdWt .The Vasicek model was the first one to capture the mean-reverting property of the interest rate.Mean reversion refers to the fact that interest rate flutuates around some level and this property iswhat sets interest rates apart from other financial prices such as stock prices. In the model, b is thelong-term mean level of rt , a is the speed of reversion, and σ is the volatility.Under the Vasicek model, the price of a zero-coupon bond has the following closed-form expression:α(t, T )e β(t,T )rt , σ2σ2 2α(t, T ) expb 2 [β(t, T ) (T t)] β (t, T ) ,2a4a 1 β(t, T ) 1 e a(T t) .aP (t, T ) Cox-Ingersoll-Ross (CIR) ModelCox, Ingersoll, and Ross (1985) modify the Vasicek model due to the fact that the short rate processrt in the Vasicek model can become negative. The CIR model assumes the following dynamics forthe short rate rt : drt a(b rt )dt σ rt dWt .This model is also a mean-reverting process. The square root ensures that rt stays nonnegative.
CHAPTER 2. INTEREST RATE MARKETS11Under the CIR model, the price of a zero-coupon bond has the following closed-form expression:where h P (t, T ) α(t, T ) β(t, T ) α(t, T )e β(t,T )rt , 2ab/σ2 2he(a h)(T t)/2,2h (a h)[e(T t)h 1] 2 e(T t)h 1,2h (a h)[e(T t)h 1]a2 2σ 2 .Hull-White ModelVasicek and CIR models have only a few number of paramters and they usually can not reproducethe initial term structure exactly. A model that can fit the initial term structure perfectly is theHull-White model, which was introduced by Hull and White (1990) and has a more general formthan the Vasicek model:drt (bt art )dt σdWt .(2.7)Under this model, the price of a zero-coupon bond is given by α(t, T )e β(t,T )rt , P (0, T ) σ 2 (1 e 2at ) 2α(t, T ) exp β(t, T ) log P (0, t) β (t, T ) ,P (0, t) t4a 1 β(t, T ) 1 e a(T t) .aP (t, T )Also, the price of a caplet at current time 0 with strike K on the LIBOR F (T, T δ) is given by (1 δK)P (0, T ) 1P (0, T δ)Φ( d2 ) Φ( d1 ) ,1 δKP (0, T )(2.8)
CHAPTER 2. INTEREST RATE MARKETS12wherelogd1 hP (0,T δ)P (0,T ) (1 δK)i Σ Td1 Σ T ,r1 e aδ 1 e 2aTσ.a2aT d2 Σ 1 Σ T,2Some Other Short Rate ModelsThe Vasicek, CIR, and Hull-White models belong to a more general type of models called affinemodels. Affine models consist of the class of diffusion models for rt which have the nice propertythat the zero-coupon bond price P (t, T ) has explicit solutions as P (t, T ) e A(t,T ) B(t,T )rt . Duffieand Kan (1996) showed that a necessary and sufficient condition for a model to be affine is that ithas the following form:drt (bt βt rt )dt at αt rt dWt ,where b, β, a, α are deterministic functions of time. It can be shown that A(t, T ) and B(t, T ) satisfythe following system of two ordinary differential equations:αt 2B (t, T ) 1,2B 0 (t, T ) βt B(t, T ) A0 (t, T )1 bt B(t, T ) at B 2 (t, T ),2with terminal conditions B(T, T ) 0 and A(T, T ) 0. In order to solve for A and B, the strategyis to first solve for B and integrate it into A’s differential equation to get A. However, in generalthere are no closed-form solutions for A and B. Numerical methods are needed to solve for theseordinary differential equations.Some examples of affine models are given below: Ho-Lee model: drt θt dt σdWt . Extended CIR model: drt (bt βrt )dt σ rt dWt .
CHAPTER 2. INTEREST RATE MARKETS2.3.213Heath-Jarrow-Morton (HJM) FrameworkThe HJM framework was proposed by Heath, Jarrow, and Morton (1992). It is a general framework that models the dynamics the instantaneous forward rate f (t, T ). The motivation behind thedevelopment o
3 A Functional Time Series Approach 16 . interest rate at which payments are made based on a notional amount. The price of an interest rate derivative depends on the level of the interest rate and its expected change in the future. To price an interest rate derivative, a common approach is to de ne the future evolution of the interest rates .
changes on the exchange rate. However, changes on exchange rate cause changes in the local interest rate while changes on the foreign interest rates do not cause changes in the local interest rate. In addition, changes on both the exchange rate and foreign interest rate jointly do cause changes on the local interest rate. Finally changes on
interest rate. An interest rate future is actively used to hedge against future interest rate movement, i.e., so-called interest rate market risk. Due to the varying feature of the underlying interest rates, the way to calculate the price and to quote the interest rate future varies a lot.
while, the difference between the ex ante real interest rate—the nominal interest rate minus expected infla-tion—and the equilibrium real interest rate is defined as the real interest rate gap. In the new Keynesian model, the real interest rate (RIR hereafter) gap is central to the determination of output and inflation.
Lecture Notes: Interest Rate Theory Foreword Goals I Basic concepts of stochastic modeling in interest rate theory. I "No arbitrage"as concept and through examples. I Concepts of interest rate theory like yield, forward rate curve, short rate. I Spot measure, forward measures, swap measures and Black's formula. I Short rate models I A ne LIBOR models I Fundamentals of the SABR model
Interest rate (§ 230.2(o)) An interest rate is the annual rate of interest paid on an account and does not reflect compounding. For purposes of the account disclosures in section 230.4(b)(1)(i), the interest rate may, but need not, be referred to as the "annual percentage rate" in addition to being referred to as the "interest rate."
The theory of compound interest handles this problem by assuming that the interest earned is automatically reinvested. With compound interest the total investment of principal and interest earned to date is kept invested at all times. A constant rate of compound interest implies a constant effective rate of interest, and, moreover, that .
14 D Unit 5.1 Geometric Relationships - Forms and Shapes 15 C Unit 6.4 Modeling - Mathematical 16 B Unit 6.5 Modeling - Computer 17 A Unit 6.1 Modeling - Conceptual 18 D Unit 6.5 Modeling - Computer 19 C Unit 6.5 Modeling - Computer 20 B Unit 6.1 Modeling - Conceptual 21 D Unit 6.3 Modeling - Physical 22 A Unit 6.5 Modeling - Computer
to AGMA 9 standard, improved the quality and performance of the QE range. Today, the QE Vibrator not only meets industry expectations, but will out-perform competitive models when correctly selected and operated in line with the information given in this brochure. When a QE Vibrator is directly attached to a trough it is referred to as a “Brute Force” design. It is very simple to calculate .
