PRICING INFLATION-INDEXED SWAPS AND SWAPTIONS USING AN HJM MODEL

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2 PRICING INFLATION-INDEXED SWAPS AND SWAPTIONS USING AN HJM MODEL A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF APPLIED MATHEMATICS OF MIDDLE EAST TECHNICAL UNIVERSITY BY ZEYNEP CANAN TEMİZ IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN FINANCIAL MATHEMATICS DECEMBER 29

3 Approval of he hesis: PRICING INFLATION-INDEXED SWAPS AND SWAPTIONS USING AN HJM MODEL submied by ZEYNEP CANAN TEMİZ in parial fulfillmen of he requiremens for he degree of Maser of Science in Deparmen of Financial Mahemaics, Middle Eas Technical Universiy by, Prof. Dr. Ersan Akyıldız Direcor, Graduae School of Applied Mahemaics Assis. Prof. Dr. Işıl EROL Head of Deparmen, Financial Mahemaics Assoc. Prof. Dr. Azize Hayfavi Supervisor, Insiue of Applied Mahemaics, METU Examining Commiee Members: Prof. Dr. Gerhard Wilhelm Weber Insiue of Applied Mahemaics, METU Assoc. Prof. Dr. Azize Hayfavi Insiue of Applied Mahemaics, METU Assoc. Prof. Dr. C. Coşkun Küçüközmen Insiue of Applied Mahemaics, METU Assoc. Prof. Dr. Gül Ergün Deparmen of Saisics, Haceepe Universiy Assis. Prof. Dr. Ömür Uğur Insiue of Applied Mahemaics, METU Dae:

4 I hereby declare ha all informaion in his documen has been obained and presened in accordance wih academic rules and ehical conduc. I also declare ha, as required by hese rules and conduc, I have fully cied and referenced all maerial and resuls ha are no original o his work. Name, Las Name: ZEYNEP CANAN TEMİZ Signaure : iii

5 ABSTRACT PRICING INFLATION-INDEXED SWAPS AND SWAPTIONS USING AN HJM MODEL Temiz, Zeynep Canan M.S., Deparmen of Financial Mahemaics Supervisor : Assoc. Prof. Dr. Azize Hayfavi December 29, 54 pages Inflaion-indexed insrumens provide a real reurn and proec invesors from he erosion of he purchasing power of money. Hence, inflaion-indexed markes grow very fas day by day. In his hesis, we focus on pricing of he inflaion-indexed swaps and swapions which are he mos liquid derivaive producs raded in he inflaion-indexed markes. Firsly, we review he Hull-Whie exended Vasicek model in he HJM framework. Then, we use his model o price inflaion-indexed swaps. Also, pricing of inflaion-indexed swapions is given using Black s marke model. Keywords: Inflaion-indexed swaps, swapions, HJM framework, Hull-Whie exended Vasicek model iv

6 ÖZ ENFLASYONA ENDEKSLİ SWAP VE SWAP ÜZERİNE YAZILAN OPSİYONLARIN HJM MODELİ KULLANILARAK FİYATLANDIRILMASI Temiz, Zeynep Canan Yüksek Lisans, Finansal Maemaik Bölümü Tez Yöneicisi : Doç. Dr. Azize Hayfavi Aralık 29, 54 sayfa Enflasyona endeksli ensrümanlar reel geiri kazanma imkanı sağlar ve yaırımcıları paranın alım gücünde meydana gelen aşınmadan korur. Bu nedenle, enflasyona endeksli piyasalar her geçen gün hızlı bir şekilde büyümekedir. Bu çalışmada, enflasyona endeksli piyasalarda en liki ürev ürünleri olan enflasyona endeksli swap ve enflasyona endeksli swap üzerine yazılan opsiyonların fiyalandırılması üzerinde çalışılmışır. İlk olarak, Hull-Whie genişleilmiş Vasicek modeli HJM çerçevesinde incelenmişir. Daha sonra, enflasyona endeksli swapları fiyalamak için bu model kullanılmışır. Ayrıca, enflasyona endeksli swap üzerine yazılan opsiyonları fiyalama formülü Black piyasa modeli kullanılarak elde edilmişir. Anahar Kelimeler: Enflasyona endeksli swap, Enflasyona endeksli swap üzerine yazılan opsiyon, HJM çerçevesi, Hull-Whie genişleilmiş Vasicek modeli v

7 To my family vi

8 ACKNOWLEDGMENTS Firsly, I would like o express my deepes graiude o my supervisor Assoc. Prof. Dr. Azize Hayfavi for paienly guiding, moivaing and encouraging me hroughou his sudy. I am graeful o Prof. Dr. Gerhard Wilhelm Weber, Assis. Prof. Dr. Ömür Uğur, Assoc. Prof. Dr. Gül Ergün and Assoc. Prof. Dr. Coşkun Küçüközmen for heir help during he correcion of he hesis. I acknowledge my deb and express my hearfel hanks o Mehme Özgür Kulu for sharing hardes imes, being wih me all he way, lisening o me all he ime and paienly moivaing me. I am very hankful o İbrahim Ehem Güney for answering all of my quesions paienly, for his suppor and helpful commens. I am graeful o my friends; Aysun Civil, Fikre Çirkin and Begüm Anlar for heir invaluable friendship and for heir suppor. Finally, I wish o express my sincere graiude o my family; o my admirable faher Hasan, o my lovely moher Haice, o my dear siser Burçin and o my dear broher Mura Can for heir love, for always rusing and supporing me. vii

9 TABLE OF CONTENTS ABSTRACT ÖZ iv v DEDICATION vi ACKNOWLEDGMENTS TABLE OF CONTENTS vii viii LIST OF TABLES x CHAPTERS 1 INTRODUCTION Inflaion Inflaion-linked Securiies Inflaion-linked Derivaives Inflaion-indexed Swaps and Swapions REVIEW OF LITERATURE PRELIMINARIES Fundamenals of Mahemaical Finance Bonds and Ineres Raes Change of Numeraire Zero-Coupon Bond as Numeriare in Forward Measure HEATH-JARROW-MORTON FRAMEWORK The HJM Forward-Rae Dynamics HJM Drif Condiion HJM Under Risk-Neural Measure THE HULL-WHITE EXTENDED VASICEK MODEL IN THE HJM FRAME- WORK viii

10 5.1 The Model in he HJM Framework Foreign-Currency Analogy Forward Measure PRICING INFLATION-INDEXED SWAPS AND SWAPTIONS Inflaion-Indexed Swaps Zero-Coupon Inflaion-Indexed Swaps Year-on-Year Inflaion-Indexed Swaps Inflaion-Indexed Swapions Year-on-Year Inflaion-Indexed Swapions Zero-Coupon Inflaion-Indexed Swapions CONCLUSION REFERENCES ix

11 LIST OF TABLES TABLES Table 1.1 Some counries in which indexed public secor bonds have been issued.. 5 x

12 CHAPTER 1 INTRODUCTION People are saving heir money for fuure consumpion. The savers are concerned only wih he real purchasing power of heir savings. If individuals can be cerain ha deferred consumpion is equivalen o more consumpion or similarly corporaions can be sure ha he fuure real value of heir capial is increased, hen hese would provide a powerful incenive o save. The uncerainy of fuure changes in he level of prices and is effec in he purchasing power of money is one of he forces ha is boosing he inflaion-indexed markes. Inflaion linked derivaives marke has grown so rapidly ha, by mid-23, in some markes has become a significan proporion of ha in he underlying bond markes. As is he case wih all ypes of derivaives, inflaion indexed derivaives are designed o help fill he gaps in he marke for he underlying securiies. The marke paricipans sill hink he inflaion linked securiies marke as having a large poenial marke growh worldwide. The main objecives of his hesis are o search he finance lieraure for he sudies of inflaion linked securiies, o review Heah-Jarrow-Moron framework, o price inflaion-indexed swaps using Hull-Whie exended Vasicek model and o price inflaion-indexed swapions using he defined inflaion-indexed swap marke model. The organizaion of his hesis is as follows: In he firs chaper, we give he definiion of inflaion, used inflaion indexes, and an overview for he inflaion-indexed securiies and derivaives which are he concern of his hesis. In Chaper 2, we presen a review for he sudies abou pricing or he working principles of inflaion-indexed securiies. In Chaper 3, he fundamenal definiions and heorems of mahemaical finance, bonds, ineres raes and change of numeriare echniques are given. Chaper 4, reviews he HJM framework. In Chaper 5, Hull-Whie exended Vasicek model is inroduced in he HJM framework, hen he dynamics 1

13 in he real economy and also in he forward measure are given. In Chaper 6, he derivaion for prices of inflaion-indexed swaps using he model given in chaper 5 and prices of inflaionindexed swapions using he Black s marke model are presened. Chaper 7, concludes he hesis wih a shor summary. 1.1 Inflaion Inflaion is defined as an increase in he level of prices in an economy, and herefore in effec wih he real value of money. If prices are no fixed over ime, he value of money will floa, usually upwards, bu rarely he prices decrease, hen i is called deflaion. The reasons behind inflaion are complex; so in he lieraure here are various heories using macroeconomic and microeconomic analysis. Some facors ha cause inflaion can be lised as follows: The level of moneary demand in he economy is one of he main reasons of inflaion. When a he curren price level, demand for he goods and services in he economy is greaer han he economy s abiliy o produce hem; inflaion ends o rise. A rise in producion coss or labor coss can also lead o inflaion. If raw maerials increase in price or workers demand wage increases, hese lead o an increase in he cos of he produc. Then, o provide sabiliy in profis, he companies choose o pass hese coss o heir cusomers, finally he price of final produc increases and inflaion occurs. Anoher cause of inflaion is inernaional lending and naional debs. As counries borrow money, prices rise o cope wih heir debs and ineress. Inflaion can be measured in differen ways. The commonly used measures for inflaion are: Consumer Price Index (CPI: measures prices of a selecion of goods and services purchased by a consumer. The baske includes hundreds of hings; from basic iems o new producs. Prices are colleced every monh over he counry and all hese prices are combined o produce an index of prices. Then he inflaion rae is he percenage rae of change of a price index over a period, generally welve monhs. Producer Price Index (PPI: which measures changes in prices received by domesic producers for heir oupu. This differs from he CPI in ha profis and axes may cause he amoun received by he producer o differ from wha he consumer paid. There is a 2

14 delay beween an increase in he PPI and any final increase in he CPI. Producer price index measures he pressure being pu on producers by he coss of heir raw maerials. This could be passed on o consumers or i could be absorbed by profis, or could be compensaed by increasing produciviy. Core Price Index: Because food and oil prices can change quickly due o changes in supply and demand condiions in he food and oil markes, i can be difficul o deec he rend in price levels when hose are included. Therefore a measure of core inflaion is repored which removes he mos volaile componens (such as food and oil from a price index like CPI. Because core inflaion is less affeced by shor run supply and demand condiions in markes, cenral banks rely on i o beer measure he inflaionary impac of curren moneary policy. Wholesale Price Index (WPI: is he price of a represenaive baske of wholesale goods. This index focuses on he price of goods raded beween corporaions raher han goods bough by consumers which is measured by he CPI. The purpose of he WPI is o monior price movemens ha reflec supply and demand in indusry, manufacuring and consrucion. GDP deflaor: is a measure of he price of all he goods and services included in Gross Domesic Produc (GDP. Since almos all price indexed securiies are linked o CPI, consumer prices are he only concern of his hesis. 1.2 Inflaion-linked Securiies Indexaion firsly used in he beginning of he 18h cenury. Deacon, Derry and Mirfenderesky presen he hisory of indexaion: In 177, Bishop William Fleewood produced a sudy ino he erosion of he purchasing power of money, for a fellowship esablished in 145 whose membership was resriced o hose wih an annual income of less han 5. He examined changes in he prices of corn, mea, drink, cloh beween and he found ha here had been a huge increase in he prices. He concluded ha he accepance rule for he fellowship mus be he real annual income of an individual would be less han 5, no his 3

15 nominal annual income. Then in 1742, he Sae of Massachuses issued bills of public credi which were linked o he cos of silver in he London Exchange. Since he price of silver appreciaed more rapidly han he general price level, in 1747 he Parliamen passed a law elling ha a larger group of commodiies should be used for indexaion in fuure deb. Afer his law, he Sae of Massachuses issued Depreciaion Noes in 178 for soldiers during he American Revoluion and herefore a baske of goods was firsly defined. Economiss made so many researchs abou indexaion and hey published books and aricles wih developing concep of indexaion. Sir George Shuckburgh Evelyn (1798, Joseph Lowe (1822 and G. Povle Scrope (1833 were firs economiss who discussed he consrucion of an index o represen he general level of prices. In 1875, W. Sanley Jevons suggesed o use gold prices for indexaion and old he benefis of indexaion in his works. Alfred Marshall (1886, John Maynard Keynes (1924, Richard Musgrave, Milon Friedman and Rober Barro were also srong supporers of he concep o index wages and financial insrumens. Alhough so many invesigaions were made for indexaion earlier, indexed deb was imporan in financial markes in he second half of he 2h cenury. Table 1.1 shows some counries issued indexed public secor bonds and which index hey used for hese bonds 1. Inflaion indexed securiies are insrumens ha proec he invesors from changes in he general level of prices in he economy. Since inflaion indexed securiies have an inflaion adjusmen, heir yields are considered o be real yields. In hese securiies, he securiy holder earns his real reurn plus he inflaion realized over he life of he securiy. Bu in nominal securiies, he reurn ha he invesor ges by holding his securiy o he mauriy is equal o is yield. The yield on a nominal securiy has wo componens: real yield and inflaion expecaion over he life of he securiy. Since he invesors have differen expecaions of fuure inflaion, boh nominal and real markes have some invesors. For example; in Turkish bond marke, he difference beween he yields on he 4-year Treasury and 4-year Treasury inflaion proeced bonds is 4, 62 %. If an invesor expecs inflaion o be higher han 4, 62 % over he life of he bonds which is 4 year here, hen ha invesor would prefer owning he inflaion proeced bond. On he conrary, if he invesor expecs inflaion o be lower han 4, 62 % over 4 years, [9]. 1 Mark Deacon, Andrew Derry and Dariush Mirfendereski. Inflaion Indexed Securiies. Wiley-Finance, 24 4

16 Table 1.1: Some counries in which indexed public secor bonds have been issued Counry I ssue Dae Index used Ausralia Consumer prices 1991 Average weekly earnings Ausria 1953 Elecriciy prices 23- Consumer prices Brazil Wholesale prices General prices 22- Consumer prices Canada Consumer prices France 1952, 1973 Gold price 1956 Level of indusrial producion 1956 Average value of French securiies 1957 Price of equiies Domesic consumer prices 21- European consumer prices Germany 22, 23 European consumer prices Ialy 1983 Deflaor of GDP a facor cos 23- European consumer prices Souh Africa 2- Consumer prices Turkey Wholesale prices Consumer prices USA 1742, 178 Commodiy prices Consumer prices 5

17 hen he invesor would prefer owning he nominal bond. Inflaion-linked securiies have some drawbacks for invesors. In mos counries, he CPI for a given monh is announced in he middle of he following monh. So, he cash flows of indexed securiies can no be adjused by inflaion up o he momen which hey are paid. There mus be a lag beween he acual movemens in he price index and he inflaion adjusmens o he cash flows. When inflaion is volaile, hese lags can be problem for invesors. Anoher facor is ha he inflaion-linked securiies are axed disadvanageously. Since cash flows are axed on a nominal (no real basis, he real reurns of hese insrumens afer ax are uncerain. Also, inflaion indexed securiies have a less liquid marke han nominal securiies. In spie of hese drawbacks, invesors prefer inflaion indexed securiies for heir porfolios. Because, here is no asse class which is able o provide such a proecion agains he erosion of purchasing power and since hese securiies are generally purchased by buy-and-hold invesors, he lower liquidiy problem is unlikely o be significan. Inflaion linked securiies have advanages for he issuers a he same ime. The main advanage of issuance inflaion indexed securiies by governmens is ha i allows o reduce he cos of financing. Cos savings can be in several ways. Firsly, if invesors are willing o pay premium for proecion agains inflaion, hen his premium will be refleced in a lower yield paid by he governmen on he insrumens ha provide such proecion. Also, if inflaion over he life of he securiy urns ou o be lower han he marke had expeced a he ime of issuance, hen indexed deb again provides cheaper funding han convenional deb. Second advanage of indexing he governmen s deb is ha i allows a more precise maching of he governmen s asses and liabiliies. 1.3 Inflaion-linked Derivaives Derivaives in he inflaion marke provide many opporuniies o invesors. They mee he needs of invesors ha he bond marke unable o provide. When invesor demands and issuer needs can no be mached, inflaion linked derivaives can be used o remove he mismaches. Derivaives help o mach he mauriy, frequency of cash flows, size, index and iming. 6

18 1.3.1 Inflaion-indexed Swaps and Swapions A swap is a ransacion in which wo counerparies agree o pay each oher a series of cash flows over a specified period of ime. The four ypes of swaps are currency swaps, ineres rae swaps, equiy swaps and commodiy swaps. In an inflaion indexed swap, a leas one of he cash flows is ied o inflaion. A receiver swap is a swap where he holder a each paymen dae receives a fixed amoun and pays a floaing amoun which is he inflaion rae in his hesis. A payer swap is a swap where he holder receives he inflaion rae and pays a fixed rae. A swapion is an opion o ener ino a swap a a pre specified dae for a specified swap rae. The righ o ener ino a swap paying a fixed rae is called a payer swapion, and he righ o ener ino a swap receiving a fixed rae is called a receiver swapion. An inflaion indexed swapion is a swapion where he underlying swap is an inflaion indexed swap. 7

19 CHAPTER 2 REVIEW OF LITERATURE There are so many researches, aricles and books abou inflaion linked securiies in he finance lieraure. Advanages and disadvanages of inflaion indexed securiies for issuers and invesors, pricing of hese insrumens are invesigaed around he world, bu in Turkey here is a lile work abou his subjec. One of he early sudies abou inflaion derivaives was made by Hughson [21]. He inroduced a mehodology based on he foreign-currency analogy. Here nominal asses are hough of as domesic asses, real asses as foreign asses and he consumer price index is hough as a kind of exchange rae o deermine he nominal payou of a real bond a mauriy. In his aricle, he assumed a complee marke wih no arbirage. For consumer price index, real and nominal bond prices, Heah-Jarrow-Moron model are used. Pricing formulas are given for inflaion linked derivaives, bu all he formulas are given in he closed form. Jarrow and Yıldırım [22] developed a hree facor HJM model in order o price reasury inflaion proeced securiies (TIPS and opions which are linked o he inflaion index. They consider a cross-currency economy under no-arbirage assumpion. They assumed ha he volailiies of all asse prices and he consumer price index are deerminisic. When he bond prices are Gaussian, hey obained he pricing formulas and apply he HJM model o price a call opion linked o he CPI-U 1 index. Belgrade-Benhamou-Koehler [4] inroduced a new marke model o obain a link beween zero coupons and year-on-year swaps which is disregarded in he Jarrow-Yıldırım model. Their model is robus and simple which has only few parameers. The main hypohesis is ha he marke model for inflaion considers forward inflaion index reurn as a diffusion wih 1 Consumer price index for all urban consumers. 8

20 deerminisic volailiy srucure. Their model is more suied in markes here is enough informaion from zero coupon and year-on-year swaps and also i is compuaionally inensive. Mercurio [28] sudied on pricing of zero coupon inflaion indexed swaps, year-on-year inflaion indexed swaps, inflaion indexed caples and floorles. Firsly, he swaps are priced using he Jarrow-Yıldırım model wih Hull-Whie paramerizaion and hen he inroduced wo differen marke models for pricing swaps. The difference beween wo marke models is ha he dependence on he volailiy of real raes which may be hard o esimae. Also, he finally esed he performance of he models using he Euro inflaion indexed swaps marke daa. The oher sudy on pricing inflaion indexed derivaives was made by Malvaez [26]. The aim of his work is o apply he Jarrow-Yıldırım model for pricing inflaion indexed derivaives, especially swaps, European and Bermudan swapions. He also esed he performance of he model calibraing o he Mexican marke daa, esimaed parameers of he models and gave he analysis of resuls for European swapion. Hinnerich [18] suggesed an exended HJM model o price inflaion linked swaps. The model includes jump componens, also consumer price index and he forward raes are allowed o be driven by boh Wiener process and a general marked poin process. Volailiies of all asse prices and he consumer price index are deerminisic wih respec o he Wiener process and he poin process. The imporan difference in he aricle is ha derivaives are priced in HJM framework wihou assuming he foreign-currency analogy. Inflaion indexed swap marke model is inroduced o price inflaion indexed swapions. Therefore, his work proved he validiy of foreign-currency analogy. Dodgson and Kainh [1] used a correlaed Hull-Whie model o price inflaion derivaives. They also priced complex derivaives using Mone Carlo sampling. Inflaion swaps, opions, cap and floors are included in his work wih a wo process shor rae model. Sewar [32] sared o his work giving he working principles of inflaion indexed markes. Then he reviewed he wo currency Heah-Jarrow-Moron framework and he derived he prices of inflaion indexed derivaives using he Hull-Whie exended Vasicek model. He finished his work calibraing o marke daa. Deacon, Derry and Mirfendereski [9] wroe a book abou inflaion indexed securiies. This book ells us hese securiies and he markes rading hese insrumens around he world in 9

21 deail. The problems in indexaion, ax regulaions, advanages and disadvanages of issuing or invesing in indexed securiies, he principles of indexed bonds and derivaives marke are included in he book. Mauri and Mercurio [27] reviewed he approach given by Jarrow-Yıldırım o model inflaion and nominal raes, o price inflaion indexed swap and hey inroduced a consan volailiy LIBOR marke model o price inflaion. Cosanini, D Ippolii and Papi [8] worked on valuaion of inflaion derivaives wih payoff depending on European inflaion, on European Cenral Bank official ineres rae and on he shor erm ineres rae a imes beween he presen and he mauriy dae. They invesigaed he relaionship beween hese hree quaniies in a sochasic ime seing for his reason. The las sudy abou inflaion derivaives has made by Leung and Wu [25]. In he aricle, hey presen exended Heah-Jarrow-Moron model in erms of coninuous compounding nominal and inflaion forward raes. They price several inflaion derivaives excep swapions under his model. Then hey inroduced a lognormal model for displaced forward inflaion raes for simple compounding and use his model o price inflaion indexed swapion. They conclude wih calibraion resuls of he marke model. Garcia and Rixel [11] presen he inflaion linked bonds from a cenral bank perspecive. They gave he developmen of inflaion-linked bond markes, he argumens for and agains issuing inflaion-linked bonds boh from he perspecive of he issuer and he invesor and finally hey old he uses of inflaion-linked bonds o show invesors inflaion expecaions and he oulook for economic growh. European Cenral Bank s experiences are used in his work. The only deailed work in Turkey is made by Tekmen [33]. In his work, hisory of inflaion linked bonds in Turkey and oher bond markes around he world are old in deail. He gave he advanages of issuing inflaion indexed bond and a regression analysis is included in he sudy. 1

22 CHAPTER 3 PRELIMINARIES In his chaper, some basic definiions and heorems ha we need in laer chapers are reviewed. 3.1 Fundamenals of Mahemaical Finance Definiions and heorems in his secion are mainly aken from Lamberon and Lapeyre [24], Björk [6], Shreve [31], Brigo and Mercurio [7] and Proer [3]. Definiion Consider he probabiliy space (Ω, A, P. A filraion (F is an increasing family of σ - algebras included in A. The σ- algebra F represens he informaion available a ime. A process (X is adaped o (F, if for any, X is F - measurable. Definiion A probabiliy measure Q is called a maringale measure or risk neural measure if he following condiion holds: S = R EQ [S 1 ]. Proposiion The marke model is arbirage free if and only if here exiss a maringale measure Q. Definiion A Brownian moion is a real-valued, coninuous sochasic process (X wih independen and saionary incremens. 11

23 Coninuiy: P - a.s. he map s X s (w is coninuous. Independen incremens: If s hen X X s is independen of F s = σ (X u, u s. Saionary incremens: If s hen X X s and X s X have he same probabiliy law. Definiion A Brownian moion is sandard if X = P - a.s. E(X = Var(X = E(X 2 =. Definiion Le (Ω, F, P be a probabiliy space, le T be a fixed posiive number and le (F, T, be a filraion of sub σ - algebras of F. An adaped sochasic process (M, T, is : a maringale, if E(M F s = M s for all s T, I has no endency o rise or fall. a submaringale, if E(M F s M s for all s T, I has no endency o fall, i may have a endency o rise. a supermaringale, if E(M F s M s for all s T, I has no endency o rise, i may have a endency o fall. Theorem Le (Ω, F, P be a probabiliy space. Le P be anoher probabiliy measure on (Ω, F ha is equivalen o P and le Z be an almos surely posiive random variable ha relaes P and P. Then Z is called he Radon-Nikodym derivaive of P wih respec o P, and we wrie Z = d P dp. Theorem (Girsanov Theorem Le (W T be a Brownian moion on a probabiliy space (Ω,F,P and le (F T be a filraion for his Brownian moion. 12

24 Le (θ T be an adaped measurable process saisfying θ2 s ds < a.s. and such ha he process (Z T defined by Z = exp ( θ u dw u 1 2 θ 2 u du is a maringale. Then, under he probabiliy measure P wih densiy Z(T relaive o P, he process defined by is a sandard Brownian moion. W( = W( + θ u du Theorem (Girsanov Theorem in Corollary Case Le X be a measurable process adaped o he naural filraion. Define Z = ε(x = exp (X 1 2 X, X where ε(x is he Doléans-Dade exponenial of X wih respec o W. If Z is a maringale, hen a probabiliy measure P can be defined on (Ω,F such ha we have Radon-Nikodym derivaive d P dp F = Z = ε(x. Then, if X is a coninuous process and W is Brownian moion under measure P, hen W = W [W, X] is a Brownian moion under P. Definiion Le (Ω, F,(F, P be a filered probabiliy space and (W be an F -Brownian moion. (X T is an R-valued Io process if i can be wrien as P a.s. T X = X + K s ds + H s dw s, (3.1 where X is F - measurable. (K T and (H T are F -adaped processes. 13

25 T K s ds < P a.s. T H s 2 ds < P a.s. Theorem (Io-Doeblin Formula for an Io-process Le (X T be an Io process X = X + K s ds + H s dw s and f be a wice coninuously differeniable funcion, hen where, by definiion Likewise, if (, x f (X = f (X + f (X s dx s + 1 f (X s d X, X s (3.2 2 X, X = H 2 s ds. f (, x is a funcion which is wice differeniable wih respec o x and once wih respec o, and if hese parial derivaives are coninuous wih respec o (,x, Io formula becomes f (, X = f (, X + f s (s, X s ds + f x (s, X s dx s f xx (s, X s d X, X s. (3.3 Proposiion (Inegraion by Pars Formula Le (X and (Y be wo Io processes such ha and X = X + K s ds + H s dw s Y = Y + K s ds + H s dw s, hen X Y = X Y + X s dy s + Y s dx s + X, Y (3.4 wih X, Y = H sh s ds. 14

26 Ornsein-Uhlenbeck process The Ornsein-Uhlenbeck process, also known as he mean-revering process, is a sochasic process r given by he following sochasic differenial equaion: dr = θ(µ r d + σdw. For he soluion of his equaion, we apply Io s lemma o he funcion: f (r, = r e θ, d f (r, = θr e θ d + e θ dr = e θ θµd + σe θ dw. Inegraing from o, we ge r e θ = r + and, hen, e θs θµ ds + r = r e θ + µ(1 e θ + σe θs dw s σe θ(s dw s. Definiion The Markov propery means ha he fuure behaviour of he process (X afer depends only on he value X and ha is no influenced by he hisory of he process before. An F -adaped process (X saisfies he Markov-propery if, for any bounded Borel funcion f and for any s and such ha s, we have E( f (X F s = E( f (X X s. Theorem Le X = Z + b(s, X s ds + and r(s,x be a non-negaive measurable funcion. For >s, P a.s. E(e s r(u,x u du f (X F s = E(e 15 s r(u,xs,x u σ(s, X s dw s du f (X s,x x = X s.

27 Remark 3.1 If b and σ are independen of x and f is a bounded measurable funcion, hen In ha case for >s, E( f (Xs+ s,x = E( f (X,x. E(e s r(x u du f (X F s = E(e s r(x,x u du f (X s,x x = X s. 3.2 Bonds and Ineres Raes In his secion, we presen some known definiions and resuls following Brigo and Mercurio [7] and Björk [6]. Definiion (Bank accoun/money-marke accoun Le r be a posiive funcion of ime. The value of a bank accoun a ime is defined by B( = exp( r s ds. Then, he bank accoun saisfies he following differenial equaion: db( = r B( d, B( = 1. Definiion (Discoun facor The discoun facor D(, T beween wo insans and T is he amoun a ime, ha is equivalen o one uni of currency payable a ime T, and i is given by D(, T = B( T B(T = exp( r s ds. Definiion (Zero-coupon bond A T-mauriy zero-coupon bond is a conrac ha guaranees is holder he paymen of one uni of currency a ime T, wih no inermediae paymens. The conrac value a ime < T is denoed by P(,T. Clearly, P(T, T = 1 for all T. Definiion The simple forward rae for [S, T] conraced a is defined as P(, T P(, S L(; S, T = (T S P(, T. 16

28 Definiion The simple spo rae for [S, T] is defined as P(S, T 1 L(S, T = (T S P(S, T. Definiion The coninuously compounded forward rae for [S, T] conraced a is defined as R(; S, T = log P(, T log P(, S. T S Definiion The coninuously compounded spo rae R(S, T, for he period [S, T] is defined as R(S, T = log P(S, T. T S Definiion The insananeous forward rae wih mauriy T, conraced a, is defined by f (, T = log P(, T. T Definiion The insananeous shor rae a ime, is defined by r( = f (,. Lemma For s T, T P(, T = exp( f (, s ds. 3.3 Change of Numeraire This secion deals wih change of numeraire echniques in risk neural and forward measure following Brigo and Mercurio [7], Björk [6] and Shreve [31]. Definiion A numeriare is any sricly posiive, non-dividend paying asse in which oher asses are denominaed. 17

29 Theorem (Sochasic represenaion of asses Le N be a sricly posiive price process for a non-dividend paying asse, eiher primary or derivaive, in he mulidimensional marke model. Then here exiss a vecor volailiy process such ha ν( = (ν 1 (,.., ν d ( dn( = R(N(d + N(ν(d W(. This equaion is equivalen o each of he following equaions: d(d(n( = D(N(ν( d W(, D(N( = N( exp( N( = N( exp( ν(u d W(u 1 2 ν(u d W(u + ν(u 2 du, (R(u 1 2 ν(u 2 du. According o Girsanov Theorem, we can use he volailiy vecor of N( o change he measure. Define W (N j ( = and a new probabiliy measure P (N (A = ν j (u du + W j (, j = 1,..., d 1 D(TN(Td P, f oralla F. N( A Theorem (Change of risk-neural measure Le S( and N( be he prices of wo asses denominaed in a common currency and le σ( = (σ 1 (,.., σ d ( and ν( = (ν 1 (,.., ν d ( denoe heir respecive volailiy vecor processes: d(d(s ( = D(S (σ(d W(, d(d(n( = D(N(ν(d W(. Take N( as he numeriare, so he price of S( becomes S (N ( = S ( N(. Under he measure P (N, he process S (N ( is a maringale. Moreover, ds (N ( = S (N ([σ( ν(]d W (N (. 18

30 3.3.1 Zero-Coupon Bond as Numeriare in Forward Measure A zero-coupon bond is an asse and herefore he discouned bond price D(P(, T mus be a maringale under he risk neural measure P. According o sochasic represenaion of asses heorem[31], here is a volailiy process σ (, T for he bond such ha d(d(p(, T = σ (, TD(P(, Td W(. Definiion Le T be a fixed mauriy dae. We define T-forward measure P T by for he bond such ha P T (A = 1 D(Td P, f oralla F. P(, T A The T-forward measure corresponds o aking as numeriare N( = P(, T. According o change of risk neural measure heorem, he process W T ( = σ (u, T du + W( is a Brownian moion under P T. Furhermore, under he T-forward measure, all asses denominaed in unis of he zero-coupon bond mauring a ime T are maringale. In oher words, T-forward prices are maringales under he T-forward measure P T. 19

31 CHAPTER 4 HEATH-JARROW-MORTON FRAMEWORK Since shor rae models have some drawbacks, such as an exac calibraion o he iniial curve of discoun facors and a clear undersanding of he covariance srucure of forward raes are boh difficul o achieve, various auhors ried o propose models alernaive o shor rae models. In 1986, Ho and Lee modeled he evoluion of he enire yield curve in a binomial-ree seing. Then in 1992, Heah, Jarrow and Moron developed a quie general framework for modeling of ineres rae dynamics in coninuous ime. By choosing he insananeous forward raes as fundamenal quaniies o represen he yield curve, hey derived an arbirage free framework where he forward rae dynamics are fully specified hrough heir insananeous volailiy srucures. In oher words, hey showed ha here is a relaionship beween he drif and volailiy parameers of he forward rae dynamics in an arbirage free marke. In his chaper, we review he HJM forward-rae dynamics and give he arbirage-free condiion under an objecive and a risk-neural measure. 4.1 The HJM Forward-Rae Dynamics Le f (, T, T T, be he iniial forward rae curve which is known a ime. In he HJM model, he forward rae a laer imes for invesing a laer imes T is given by f (, T = f (, T + α(u, T du + σ(u, T dw(u. (4.1 When we differeniae wih respec o he variable, he variable T is being held consan, he 2

32 above equaion can be wrien as d f (, T = α(, T d + σ(, T dw(, T. (4.2 Here W(u is a Brownian moion under he measure P. The processes α(, T and σ(, T may be random and for each fixed T, hey are adaped processes in -variable. We assume ha he forward rae is driven by a single Brownian moion. The differenial of T f (, udu is given by d( T T f (, udu = f (, d d f (, udu. (4.3 From he definiion of he insananeous shor rae, we know ha r( = f (,. Using his and forward rae dynamics in differenial form, we have T d( f (, udu = r(d T [α(, u d + σ(, u dw(]du. (4.4 Define α (, T = T α(, udu, (4.5 and σ (, T = T σ(, udu. (4.6 By he Fubini heorem, we ge T d( and using (4.5 and (4.6, we have T T f (, u du = r(d α(, u du d σ(, u du dw( T d( f (, u du = r(d α (, T d σ (, T dw(. (4.7 Le g(x = e x. We know T P(, T = exp( T f (, u du = g( 21 f (, udu.

33 By Io-Doeblin formula, he dynamics of he bond prices is given by dp(, T = g ( T + 1 T 2 g ( f (, u du d( T f (, udu T f (, u du [ d( f (, u du ] 2 = P(, T [ r(d α (, T d σ (, T dw( ] P(, Tσ (, T 2 d = P(, T [ r( α (, T σ (, T 2 ] d σ (, TP(, T dw(. ( HJM Drif Condiion We will show ha here is no opporuniy for arbirage by rading he zero-coupon bonds wih mauriy T for all T [, T] in he HJM model. According o he firs fundamenal heorem of asse pricing, if a marke model has a riskneural probabiliy measure, hen i does no admi arbirage. So, we need a probabiliy measure P under which discouned bond prices are maringales. Since D(P(, T = exp( dd( = d(exp( r(u dup(, T, T r(u du = r(d( d and by using Io s inegraion by pars formula, he differenial form of he discouned bond price is given by d(d(p(, T = dd(p(, T + D(dP(, T = r(d(p(, T d + D(dP(, T = D(P(, T [ ( α (, T σ (, T 2 d σ (, T dw( ]. (4.9 22

34 We wan o wrie he erm in square brackes as σ (, T [Θ( d + dw( ] and we can apply Girsanov s heorem o change o a probabiliy measure P, where W( = is a Brownian moion. Θ(u du + W( Then he discouned bond price formula becomes d(d(p(, T = D(P(, Tσ (, T d W(. (4.1 I is obvious ha D(P(, T is a maringale under he measure P. Now, o find Θ( we have o solve he equaion ( α (, T σ (, T 2 d σ (, T dw( = σ (, T (dw( + Θ( d and, equivalenly, ( α (, T σ (, T 2 = σ (, T Θ(. (4.11 Here, he process Θ( is he marke price of risk and here are infiniely many soluions of hese marke price of risk equaions for each mauriy T [, T]. Bu in our case, since we assume here is only one Brownian moion, here is only one process Θ(. When we differeniae he equaions (4.5 and (4.6 wih respec o T we have; and T α (, T = α(, T T σ (, T = σ(, T. Then we differeniae (4.11 wih respec o T and we ge and, equivalenly, α (, T + σ (, Tσ(, T = σ(, TΘ( α(, T = σ(, T [ σ (, T + Θ( ]. (4.12 Then we can wrie he Heah-Jarrow-Moron no-arbirage condiion: 23

35 Theorem (HJM Drif Condiion A erm srucure model for zero coupon bond prices of all mauriies in (, T] and driven by a single Brownian moion does no admi arbirage if here exiss a process Θ( such ha α(, T = σ(, T [ σ (, T + Θ( ] (4.13 holds for all T T. Here, α( and σ( are he drif and he diffusion, respecively, of he forward rae, σ (, T = T σ(, udu and Θ( is he marke price of risk. 4.3 HJM Under Risk-Neural Measure Under he risk-neural measure, he local rae of reurn has o be equal o he shor rae r. So we ake Θ( = in HJM drif condiion equaion (4.13 under he maringale measure. Theorem (HJM Drif Condiion Under Risk-Neural Measure Under he risk-neural measure P, he processes α(, T and σ(, T saisfy he following equaion, for every and every T : α(, T = σ(, T σ (, T. (4.14 Then he forward rae dynamics under he risk-neural measure is given by d f (, T = σ(, T σ (, T d + σ(, T d W(. (4.15 The zero-coupon bond price dynamics under he risk-neural measure P is as follows dp(, T = r(p(, T d σ (, TP(, T d W(, (4.16 where W( is a Brownian moion under P, r( = f (, and σ (, T = T And, finally, he discouned bond prices saisfy σ(, udu. d(d(p(, T = σ (, TP(, T d W(, (4.17 where D( = exp( r(udu is he discoun process. 24

36 CHAPTER 5 THE HULL-WHITE EXTENDED VASICEK MODEL IN THE HJM FRAMEWORK The poor fiing of he iniial erm-srucure of ineres raes implied by he exising models was realized by some auhors and hey ried o propose exogenous erm-srucure models as opposed models ha endogenously produce he curren erm-srucure of raes. Ho and Lee (1986 have been he firs o propose a model like his. Their model was based on he assumpion of a binomial ree governing he evoluion of he enire erm-srucure of raes and is coninuous-ime limi. Because of he lack of mean reversion in he shor rae dynamics, heir model canno be regarded as a proper exension of he previous models. Then, in 199, Hull and Whie inroduced a ime-varying parameer in he Vasicek model and he shor rae process for he Hull-Whie exended Vasicek model is given by dr( = (θ( a(r( d + σ( dw(, where θ, a and σ are deerminisic funcions of ime. Such a model can be fied o he erm-srucure of ineres raes and he erm-srucure of spo or forward rae volailiies. Bu o do an exac calibraion o he curren yield curve, his model has some drawbacks: Since some marke secors are less liquid, he volailiies ha are quoed in he marke canno be informaive or reliable. Fuure volailiy srucures implied by his model are unlikely o be realisic in ha hey do no conform o ypical marke shapes. 25

37 Because of hese drawbacks, in 1994 he Hull-Whie inroduced he following exension of Vasicek model and he dynamics of he shor rae is given by dr( = (θ( ar( d + σ dw(, where a and σ are posiive consans, and θ is chosen so as o exacly fi he erm-srucure of ineres raes being currenly observed in he marke. Since he dynamics of his model allow for he derivaion of explici prices for derivaive securiies, we concenrae on his model in he following chaper. This chaper reviews he model which is analyzed by Sewar [32]. The firs secion of his chaper describes how he Hull-Whie exended Vasicek model can be defined in he HJM framework. Forward rae and bond price dynamics by using his model are given in his secion. The nex secion inroduces he inflaion index and he real economy ino he model. Foreign-currency analogy inroduced by Jarrow and Yıldırım [22] is used in his secion. In he las secion, he dynamics under he forward measures are given. 5.1 The Model in he HJM Framework The sochasic differenial equaion for he shor rae in he exended Vasicek model is given by dr( = (θ( a(r( d + σ( dw(, (5.1 where W( is a Brownian moion under he maringale measure. When wriing he equaion (5.1 in he form ( θ( dr( = a( a( r(d + σ(dw(, (5.2 we ge he Ornsein-Uhlenbeck process. To solve his equaion, le X( = r(e a(s ds. (5.3 Applying Io s lemma o (5.3, dx( = e a(s ds dr( + r(e a(s ds a( d. 26

38 Using (5.2 in he above equaion, we ge ( ] θ( dx( = e [a( a(s ds a( r( d + σ( dw( dr( + r(e a(s ds a( d = e a(s ds θ( d + e a(s ds σ( dw(. (5.4 Inegraing (5.4 from o, we obain Le α(= r(e a(s ds = r( + u e a(s ds θ(u du + a(s ds. Then he soluion of (5.1 is given by r(e α( = r( + r( = e α( [r( + e α(u θ(u du + e α(u θ(u du + u e a(s ds σ(u dw(u. e α(u σ(u dw(u ] e α(u σ(u dw(u. (5.5 Hence, T r(s ds = T e α(s [r( + s ] T [ s ] e α(u θ(u du ds + e α(s e α(u σ(u dw(u ds and by he Fubini heorem we have T r(s ds = T e α(s [r( + s We know ha he bond price formula is and he bank accoun is given by ] T [ T ] e α(u θ(u du ds + e α(s ds e α(u σ(u dw(u. u [ ( T ] P(, T = E exp r(s ds F ( B( = exp r(s ds. Since W( is a Brownian moion under he maringale measure, he process P(,T B( maringale under his measure. P(, T B( ( = exp mus be a [ ( T ] r(s ds E exp r(s ds F. (5.6 By he Markov propery, we can wrie P(, T B( ( = exp [ ( r(s ds E exp 27 θ ] r(s ds,

39 where θ = T. By Laplace ransform, we can calculae he above expecaion: E (e ( ( θ θ r(s ds = exp E r(s ds + 12 ( θ var r(s ds. Firsly, and, hen, ( θ E r(s ds ( θ var r(s ds = E( = + θ θ θ ( θ = E e α(s [r( + [ θ u e α(s [r( + ( θ [ θ = E s ] e α(u θ(u du ds ] e α(s ds e α(u σ(u dw(u. s ] e α(u θ(u du ds. ( θ 2 r(s ds E r(s ds u ] 2 e α(s ds e α(u σ(u dw(u. Le φ = e α(u du and g =e α( σ(. Then, using hese and he calculaed expecaion, we can wrie (5.6 as ( P(, T = P(, T exp (φ T φ u g u dw(u 1 (φ T φ u 2 g 2 u du. B( 2 (5.7 From he definiion of he insananeous forward rae, we ge f (, T = log P(, T T = P(, T log B( T B( = P(, T log B( log T T B( = P(, T log T B(. 28

40 Taking he logarihms and differeniaing wih respec o T, we ge f (, T = ( log P(, T + (φ T φ u g u dw(u + 1 T T 2 = f (, T + φ T T g u dw(u + φ T T Under he maringale measure, dynamics of he forward rae, i holds φ T d f (, T = g T dw( + φ T T (φ T φ u g 2 u du (φ T φ u g 2 u du. (5.8 (φ φ T d. Then he exended Vasicek model is an HJM model wih he volailiy of he insananeous forward rae given by σ(, T = g φ T T Zero coupon bond price formula P(, T = exp = e α( σ( ( T e α(u du T = σ(e α( α(t. ( T f (, s ds. When we use he equaion (5.8, he dynamics of he zero coupon bond in erms of he exended Vasicek parameers under he maringale measure is represened by [ T ( P(, T = exp f (, s + φ s g u dw(u + φ s (φ s φ u g 2 u du s s By he Fubini heorem, we obain P(, T = exp( T exp( [ exp and aking he inegrals we ge ( P(, T P(, T = P(, exp f (, s ds exp f (, s ds ( T ( exp (φ T φ φ u g 2 u du [ ( T ] φ s s (φ s φ u ds g 2 u du, ] φ s s ds g u dw(u ] ds. ( 1 (φ T φ g u dw(u exp 2 (φ T φ (φ T + φ g 2 u du = P(, T P(, exp[ (φ T φ ( g u dw(u (φ T + φ 2φ u g 2 u du]. (5.9 29

41 Now, we derive he dynamics of he zero coupon bond in erms of exended Vasicek parameers when discouned by he zero coupon bond P(, T under he T-forward measure. Under he T-forward measure P T, he expression P(,S P(,T is a maringale. By equaion (5.9, he bond prices are given by P(, S = P(, S P(, exp[ (φ S φ ( g u dw(u and P(, T = (φ S + φ 2φ u g 2 u du]. P(, T P(, exp[ (φ T φ ( g u dw(u Then, he expression P(,S P(,T is: P(, S P(, T (φ T + φ 2φ u g 2 u du]. = P(, S P(, T exp( (φ S φ T + (φ 2 T φ2 S g 2 u du + 2 = P(, S P(, T exp( (φ S φ T 1 2 (φ S φ T g u dw(u (φ S φ T φ u g 2 u du g u dw(u (φ T + φ S 2φ u g 2 u du. When we change he measure from risk neural o T-forward measure, by Girsanov s Theorem we can use volailiy of he bond price dynamics (5.9: P(, S P(, T = P(, S P(, T exp( (φ S φ T where W T (u is a Brownian moion under P T. g u dw T (u 1 2 (φ S φ T 2 g 2 u du, (5.1 As we inroduced before, ε is he Doléans Dade exponenial and ε(x is defined by ( ε(x = exp X 1 2 X, X. (5.11 Then he equaion (5.1 can be rewrien by using Doléans Dade exponenial: P(, S P(, T = P(, S ( P(, T ε (φ S φ T g u dw T (u. (5.12 3

42 5.2 Foreign-Currency Analogy In his secion, he real economy and he inflaion index are inroduced ino he model ha we have presened in he previous chaper. Under he no-arbirage assumpion, a cross-currency economy as also known foreign-currency analogy is considered. In he foreign currency analogy, he nominal ineres raes correspond o he ineres raes in he domesic economy, he real ineres raes correspond o he ineres raes in he foreign economy and he inflaion index corresponds o he exchange rae beween wo economies. So, every asse in he real economy can be convered ino an asse in he nominal economy using he inflaion index. We defined he noaion used in he model as in Jarrow and Yıldırım model [22]: r is used for real, n is used for nominal and I is used for he inflaion index. (Ω, F, P is he probabiliy space where Ω is a sae space, F is a σ-algebra and P is he probabiliy measure on (Ω, F. {F, [,T]} is he filraion generaed by hree Brownian moions (W n (, W r (, W I ( : [, T]. W n (, W r ( and W I ( are sandard Brownian moions iniialized a zero and he correlaions beween hem are given by: dw n (dw r ( = ρ nr d, dw n (dw I ( = ρ ni d, dw r (dw I ( = ρ ri d. I( is he inflaion index a ime, which is considered as CPI in his hesis. P n (, T is he ime price of a nominal zero-coupon bond ha pays ou one uni of nominal currency a mauriy T. P r (, T is he ime price of a real zero-coupon bond ha pays ou one uni of real currency a mauriy T. 31

43 f k (, T is he ime forward rae for dae T where k {r, n}: d f k (, T = α k (, Td + σ k (, TdW k (, ( T P k (, T = exp f k (, udu. r k ( is he ime insananeous shor rae where k {r, n}: r k ( = f k (,. B k ( is he ime money marke accoun value for k {r, n}: ( B k ( = exp r k (udu. Every nominal radable asse when discouned by he nominal money marke accoun mus be a maringale under he measure P n. So, for he process I(B r (, here mus exis a process σ I ( such ha I(B r ( B n ( ( = I(Br ( B n ε σ I (s dw I (s ( ( = I(ε σ I (s dw I (s, (5.13 where W I (s is a Brownian moion under he nominal risk neural measure P n, and we know B r ( = B n ( = 1. When we normalize (5.13 by I(, we have he Radon-Nikodym densiy of he P r measure wih respec o he P n, measure and i is given by Z( = I(Br ( I(B n ( ( = ε σ I (sdw I (s. (5.14 Also; from he definiion of he maringale measure P n, I(P r (, T is a maringale under P n measure when discouned by he nominal money marke accoun B n (. We know from he previous secion ha: P(, T B( ( = P(, T exp (φ T φ u g u dw(u (φ T φ u g 2 u du.

44 Using his and he Hull-Whie paramerizaion which is inroduced in he previous secion, he dynamics is given by I(P r (, T B n ( = I(P r (, T ( ε (φ r T φr ug r u dw r (u + σ I (udw I (u. (5.15 We know ha Heah, Jarrow and Moron approach models he insananeous forward rae as: f (, T = f (, T + Le A(, T = α(u, T du + T α(, s ds, σ(u, T dw(u. Σ(, T = T σ(, s ds. Then he zero-coupon bond price P(, T can be expressed in erms of he forward rae: ( T P(, T = exp f (, u du ( T ( = exp f (, s + = = ( P(, T P(, exp ( T σ(u, s dw(u + σ(u, s ds dw(u ( P(, T P(, exp (Σ(u, T Σ(u, dw(u α(u, s du ds ( T α(u, s ds du (A(u, T A(u, du. And he money marke accoun B( can be wrien in erms of he forward rae as: ( ( B( = exp r(s ds = exp f (s, s ds ( = exp = = ( f (, s + ( 1 P(, exp ( 1 P(, exp ( s u σ(u, s dw(u + s σ(u, s ds dw(u + α(u, s du ds ( Σ(u, dw(u + A(u, du. 33 u α(u, s ds du

45 Then, he bond price discouned by he money marke accoun is given by P(, T B( Hence, we can also wrie (5.15 as: ( = P(, T exp Σ(u, T dw(u A(u, T du. I(P r (, T B n ( = I(P r (, T ( ε Σ r (s, T dw r (s + σ I (sdw I (s. ( Forward Measure In his secion, he dynamics in he forward measure and mainly he Radon-Nikodym densiy of he real forward measure P r,t wih respec o he nominal forward measure P n,t are given. P r (,S P r (,T is a maringale in he real T-forward measure Pr,T. We know from he firs secion ha: P(, S P(, T = P(, S P(, T exp( (φ S φ T g u dw T (u 1 2 (φ S φ T 2 g 2 u du. Using his and he Hull-Whie paramerizaion, he dynamics of his raio is given by P r (, S P r (, T = Pr (, S P r (, T ε ( (φ rs φrt g r u dw r,t (u. (5.17 where W r,t (u is a Brownian moion under he real T-forward measure P r,t. Le = T in (5.17. The real zero-coupon bond P r (, T pays 1 uni of real currency a mauriy T. Hence, he price of he real zero-coupon bond a ime T which pays 1 uni of real currency a ime S can be wrien as P r (T, S = Pr (, S P r (, T T ε ( (φ rs φrt g r u dw r,t (u. (

46 By Girsanov s Theorem, i is known ha he volailiy erm is no affeced by he change of measure, only he drif erm changes. Then (5.17 and (5.18 can equally be wrien as shown in he previous secion: P r (, S P r (, T = Pr (, S P r (, T ( ε (Σ r (s, S Σ r (s, T dw r,t (s. (5.19 And when = T i is given by P r (T, S = Pr (, S P r (, T ( T ε (Σ r (s, S Σ r (s, T dw r,t (s. (5.2 Also, I(P r (, S is a maringale under he nominal T-forward measure P n,t when discouned by he nominal zero-coupon bond price P n (, T. So; he dynamics is given by I(P r (, S P n (, T = I(Pr (, S P n (, T ( ε σ I (s dw I,T (s + Σ r (s, S dw r,t (s Σ n (s, T dw n,t (s, (5.21 where W I,T (s, W r,t (s and W n,t (s are correlaed P n,t Brownian moions. Hence, he Radon-Nikodym densiy of he real T-forward measure P r,t wih respec o he nominal T-forward measure P n,t is given by ( Z(, T = ε σ I (s dw I,T (s + Σ r (s, T dw r,t (s Σ n (s, T dw n,t (s. (