Technische Universität München. Zentrum Mathematik. A Fractional Heath-Jarrow-Morton Approach For Interest Rate Markets

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1 Technische Universiä München Zenrum Mahemaik A Fracional Heah-Jarrow-Moron Approach For Ineres Rae Markes Diplomarbei von Parick Peer Hargu Themenseller/in: Prof. Dr. Claudia Klüppelberg Bereuer/in: Holger Fink Abgabeermin: 16. Sepember 21

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3 Hiermi erkläre ich, dass ich die Diplomarbei selbssändig angeferig und nur die angegebenen Quellen verwende habe. Garching, den 16. Sepember 21

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5 Acknowledgemens Foremos, I would like o express my sincere graiude o Prof. Dr. Claudia Klüppelberg for giving me he opporuniy o wrie his ineresing hesis a he Chair of Mahemaical Saisics. I am very hankful for her experised and valuable suggesions. I also hank Holger Fink for his excellen menoring. I really appreciae his commimen and his suppor for my hesis. This hesis benefied a lo of our frequen exchange of ideas, our numerous discussions and he insighs he provided me wih. Finally, I hank my parens for heir enduring suppor and encouragemen. i

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7 Conens Abbreviaions Noaions vii ix 1 Inroducion 1 2 Preliminaries Ineres-Rae Markes Fracional Brownian Moion Inegral Represenaions of Fracional Brownian Moion Inegraion Wih Respec o Fracional Brownian Moion Shor-Rae Models The Vasicek Model The Cox-Ingersoll-Ross Model The Hull-Whie Model Conclusion The Heah-Jarrow-Moron Model The Se-Up Arbirage Free Bond Pricing Conclusion The Fracional Heah-Jarrow-Moron Model The Se-Up Arbirage Free Bond Pricing Conclusion Simulaions of Ineres-Rae Models Simulaions of a Fracional Brownian Moion Simulaions of Sochasic Differenial Equaions Simulaions of HJM Bond Prices Conclusion Summary 81 Bibliography 83 iii

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9 Lis of Figures 2.1 Zero curve Zero curve and forward curve Simulaions of fracional Brownian moion Fracional Vasicek ineres-rae dynamics for various Hurs parameers Fracional HJM ineres-rae dynamics for various Hurs parameers Fracional and classical HJM dynamics compared Bond price simulaion FBm bond price vs. Bm bond price v

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11 Abbreviaions Bm fbm H-sssi a.s. sde CIR HJM Brownian moion fracional Brownian moion H-self similar wih saionary incremens almos surely sochasic differenial equaion Cox-Ingersoll-Ross Heah-Jarrow-Moron vii

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13 Noaions P(, T) bond price a ime wih mauriy T B () money marke accoun a ime Z (T) discouned bond price a ime wih mauriy T R(, T) zero rae a ime wih mauriy T r() shor rae a ime f(, T) forward rae a ime wih mauriy T T upper limi of rading inerval log, exp naural logarihm, exponenial funcion sup supremum {B()} R Brownian moion H Hurs parameer {B H ()} R fracional Brownian moion Var, Cov variance, covariance P, Q real-world measure, risk-neural measure E, E Q expecaion, expecaion wih respec o he risk-neural measure d = equaliy in disribuion sign signum funcion 1 B indicaor funcion of he se B χ B indicaor funcion of he se B used for rading sraegy L p (R) space of p-inegrable funcions in R a b minimum of a, b R x greaes ineger no exceeding x R C 2 (R, R) space of all wo imes coninuously differeniable funcions f : R R C space of all real-valued coninuous funcions on a meric space Dom(A) domain of he se A [X, X] quadraic variaion of he sochasic process X square inegrable kernel funcion K H ix

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15 1 Inroducion One dollar oday is beer han one dollar omorrow. And one dollar omorrow is obviously beer han one dollar in a year. The quesion ha capures us is wha we should pay oday for a guaraneed cash paymen of one dollar a some specified ime in he fuure. This is he quesion we deal wih when pricing a zero-coupon bond. I has been one of he main challenges in ineres-rae heory o find he driving facors of hese zero-coupon bond prices. Therefore many differen ineres-rae models evolved such as he models by [Vasicek [1977]], [Brennan and Schwarz [1979]] or [Ho and Lee [1979]] and many modificaions have been made as well, e.g. by [Cox e al. [1985]]. Anoher imporan reason for developing ineres-rae models is he pricing of ineres rae derivaives, i.e. financial insrumens whose payoffs depend in some way on he level of he underlying ineres raes. The noional amoun of ineres rae derivaives globally ousanding a he end of 29 increased by 6% from he year before o an esimaed $426.8 rillion afer a year of a decline due o he financial crisis. 1 Hence he ineres rae derivaives marke can be considered he world s bigges marke. Ineres rae derivaives are more difficul o evaluae han equiy or foreign exchange derivaives which is due o a number of reasons. Firs of all he behaviour of an individual ineres rae is more complicaed han ha of a sock price or an exchange rae, i.e. ineres raes are driven by macroeconomic facors such as gross domesic producs or volailiies, which exhibi long-range dependence. We refer o [Henry and Zaffaroni [23]] for empirical evidence of his finding and will ge back o i in deail laer on. Secondly, for many producs i is necessary o develop a model ha describes he enire zero-coupon yield curve in order o valuae hem. Moreover, he volailiies of differen poins on he yield curve are differen. And mos obviously, ineres raes are used boh for discouning and as he underlying defining he payoff of he derivaive, which is differen o he valuaion of sock opions for example. Due o hese complexiies new approaches for ineres-rae models had o be developed. For insance, amongs oher models, [Heah e al. [1992]] came up wih a more general and unifying framework for ineres-rae models. The aim of his hesis is o implemen he above menioned long-range dependence of ineres raes ino he Heah-Jarrow-Moron ineres-rae model. The commonly used Brownian moion does no reflec his long-range dependence due o is independen incremens. Therefore we will embed fracional Brownian moion ino a Heah-Jarrow-Moron model following an approach by [Ohashi [29]] in 1 Numbers from [Inernaional Swaps and Derivaives Associaion [29]] 1

16 1 Inroducion order o capure his dependence in our model. A line of argumenaion for his approach will be given. Therefore our mahemaical focus is argeed a fracional Brownian moion, where a lo of heory exiss for, e.g. by [Samorodnisky and Taqqu [2]], [Pipiras and Taqqu [2]] or [Duncan e al. [22]]. We will work hrough his heory always focused on he purpose of ineres-rae modelling. There has already been some research done wih embedding fracional Brownian moion ino ineres-rae models such as in [Fink e al. [21], Secion 4] for a fracional Vasicek model. Even for a Heah-Jarrow-Moron model oher fracional approaches exis, for insance in [Gapeev [24]] whereas his paper only focuses on he Markovian case. This hesis is organized as follows. We will sar wih some basic definiions used in he course of his hesis and inroduce he imporan noions for ineres-rae markes in Secion 2.1. In Secion 2.2 we will give a definiion of fracional Brownian moion and analyze is mos relevan properies. Moreover we come up wih an inegraion heory wih respec o fracional Brownian moion, which is differen o ordinary sochasic calculus and a bi more sophisicaed. We will build our fracional Heah-Jarrow-Moron model upon hose preliminaries laer on. We will inroduce some famous ineres-rae models in Chaper 3, i.e. shor-rae models ha have been developed some ime ago already, bu which are sill imporan for ineres-rae modelling heory. Hereby we will ge an idea of wha ineres-rae modelling is abou and where problems may arise. Furhermore, his illusraes he evoluion of hose ineres-rae models. Addiionally o our summary in he end, we will always provide a conclusion for every chaper in order o poin ou he main resuls. Aferwards we will go ino deail wih he Heah-Jarrow-Moron model in Chaper 4, a model ha akes on a differen and more general approach by modelling forward raes. We will give a closed form soluion of bond prices in Secion 4.1 and derive a no-arbirage condiion in Secion 4.2. This chaper is a very imporan sep in order o undersand he differences o he shor-rae models menioned above and for providing he basics and a horough background for he fracional Heah-Jarrow-Moron model in Chaper 5, our main focus of his hesis. In Chaper 5 we sar wih he se-up of he fracional Heah-Jarrow-Moron model in order o derive a closed form soluion for he bond price process. We will need a lo of mahs in order o come up wih a no-arbirage framework and he change of measure for deriving he no-arbirage drif condiion, which is a lo more sophisicaed han in he classical case as i will urn ou. Consequenly, his will enable us o sae he bond price in a condiional expecaion form as well. In Chaper 6 we will run simulaions for he pahs of fracional Brownian moion, he underlying sochasic differenial equaions of some ineres-rae models and finally for he bond prices, oo. This will necessiae mahemaical prework as well and we will need o specify our economic environmen. The purpose of his chaper is o illusrae our resuls. Therefore we will presen several graphs for models under differen assumpions. 2

17 2 Preliminaries 2.1 Ineres-Rae Markes We will sar off wih some basic knowledge of ineres-rae markes and inroduce erminology we will need hroughou his hesis. The guaraneed cash paymen of one dollar in he fuure is one of he basic asses in he ineres-rae marke. I is called a zero-coupon bond. A he evaluaion ime R + we have o pay a price P(, T) o receive ha one dollar a he mauriy dae T R +, T, denoed in years. Zero-coupon bonds are raded in face value, also called nominal value. Zero-coupon refers o he fac ha here will be no paymens during he lifeime of his conrac. Conversely, in ineres-rae markes here are so-called coupon bonds as well. For hose he holder of he coupon bond receives some specified periodic paymens from he issuer of he coupon bond during he lifeime of he bond - he coupons. In European markes coupons are usually paid once a year, whereas in he Unied Saes coupon bonds may have semi-annual paymens. Throughou his hesis we will focus on he pricing of zero-coupon bonds since coupon bonds can be considered as a porfolio of many zero-coupon bonds. In order o illusrae his le C(T i ), i = 1,..., N, N N, wih T 1 < T 2 <... < T N = T, denoe he coupons of he bond paid a ime T i and T he mauriy of he bond. For his maer we denoe he price of he coupon bond by P C (, T). Then we can wrie P C (, T) = N C(T i )P(, T i ), where he las coupon paymen C(T N ) includes he face value of he bond as well. The ime T refers o he period of ime o T and is called ime o mauriy. Moreover we focus on he case of non-defaulable bonds, assuming he issuer of he bond can always mee is liabiliies and will no go bankrup. Formally his can be expressed by P(T, T) = 1. We will ouch on he much more sophisicaed case of defaulable bonds in brief laer on. Now we will sae some imporan definiions, we will need in he following chapers of his hesis. Definiion 2.1 (The shor rae). The shor rae is he ineres rae a ime for an infiniesimal period of ime given he limi exiss, which is why we refer o i as he insananeous shor rae. Formally his is log P(, + ) r() := R(, ) := lim, ց 3

18 2 Preliminaries where R(, T) is he zero rae (or spo rae) ha denoes he appreciaion of one uni a ime in an inerval [, T] and P(, T) is he price of he zero-coupon bond a ime wih mauriy T and face value (or liabiliy) L = 1. [Zags [27]] We repea ha he erm insananeous refers o he fac ha a ime one borrows money a a cerain ineres rae and pays i back jus one insan laer. Remark 2.2. The mapping T R(, T) is he zero curve (or spo curve), which describes he evoluion of spo raes for differen mauriies. The mapping T P(, T) is called discoun curve a ime. So, a simple approach o a bond price would be P(, T) = e R(,T)(T ). The zero curve is called normal if is slope is posiive, i.e. he mapping T R(, T) is increasing whereas he zero curve is called inverse if is slope is negaive, i.e. he mapping T R(, T) is decreasing. A fla zero curve would be characerized by a slope of zero. Figure 2.1: Zero curves of he German Bundesbank a differen imes in hisory 2 We observe many normal zero curves in Figure 2.1 excep of he yellow curve in 199, which can be considered fla and he green curve in 1991, which is inverse. This developmen was caused by he German reunificaion, an exraordinary economic circumsance. 2 Numbers from [Deusche Bundesbank] 4

19 2.1 Ineres-Rae Markes Remark 2.3. Moreover he shor rae can be viewn in a differen way, ha is log P(, ) : = log P(, T) T T T= log P(, T + ) log P(, T) = lim ց T= log P(, T + ) = lim log P(, T) ց + lim ց T= log P(, + ) = lim = r(), ց where he las equaion sems from he fac ha he second par of he sum equals zero since P(, ) = 1. Definiion 2.4 (The forward shor rae). The insananeous forward shor rae (from now on only referred o as forward rae) is he ineres rae for an infiniesimal period of ime a ime T measured a ime T, i.e. log P(, T + ) log P(, T) f(, T) := lim = ց T T= log P(, T). (2.1) The mapping T f(, T) is called forward curve. Obviously r() = f(, ) holds. [Zags [27]] There is also a differen, more inuiive approach o he forward rae in [Zags [22], chaper 4]. We imagine a conrac (a so-called forward zero-coupon bond) in which, a ime, we agree a no cos o exchange a zero-coupon bond a a fuure ime T 1 wih mauriy T 2 T 1 for a cash paymen denoed by P(, T 1, T 2 ). The quesion is how large he price P(, T 1, T 2 ) has o be. Therefore we come up wih a simple no-arbirage argumen. We sell a number P(, T 1, T 2 ) of he zero-coupon bonds wih mauriy T 1 a ime and we agree o inves a he fuure ime T 1 he amoun P(, T 1, T 2 )P(, T 1 ), ha we receive from his sale, in a zero-coupon bond wih mauriy T 2. By a simple no-arbirage reasoning his porfolio has o be idenical o a zero-coupon bond wih mauriy T 2. Oherwise here would be an opporuniy for a risk-less profi. Therefore he price of he porfolio has o equal he price of he T 2 -zero-coupon bond a all imes. Formally his is P(, T 1, T 2 )P(, T 1 ) = P(, T 2 ) P(, T 1, T 2 ) = P(, T 2) P(, T 1 ). (2.2) Analogously o he zero rae in Definiion 2.1 we denoe he corresponding forward zero rae by R(, T 1, T 2 ). Then we know P(, T 1, T 2 ) = e R(,T 1,T 2 )(T 2 T 1 ) R(, T 1, T 2 ) = log P(, T 1, T 2 ) T 2 T 1, 5

20 2 Preliminaries which yields by 2.2 R(, T 1, T 2 ) = log P(, T 2) log P(, T 1 ) T 2 T 1. We le T 2 T 1 approach zero and T 1 = T o come up wih he insananeous forward shor rae log P(, T + ) log P(, T) f(, T) := R(, T, T) := lim = log P(, T). T Remark 2.5. If he zero curve is normal hen he forward curve lies above he zero curve, since i has o balance he gap beween shor mauriy and long mauriy (see Figure 2.2 below). Conversely, if he zero curve is inverse hen he forward curve lies below he zero curve. Moreover he zero curve and he forward curve coincide for = T. Formally his can be easily verified by f(, T) = T log P(, T) = T ( ) R(, T)(T ) = R(, T) + (T ) R(T, ), T where R(T, ) is posiive for a normal zero curve and negaive for an inverse zero curve T respecively. For = T we obviously ge f(t, T) = R(T, T) = r(t). Figure 2.2: Zero curve and forward curve 3 In Figure 2.2 we can easily see how forward rae and zero rae drif apar, hen move back owards each oher and finally coincide a = T in he case of a normal zero curve. In he following chapers we will see ha boh he shor rae r() and he forward rae f(, T) are saring poins for many ineres rae models in order o describe he evoluion of ineres raes over ime ha is he erm srucure of ineres raes. 3 Figure from [Murray Sae Universiy] 6

21 2.2 Fracional Brownian Moion 2.2 Fracional Brownian Moion In his secion we will give some imporan mahemaical background knowledge - definiions and conclusions we will need hroughou his hesis. The main pars of his secion on fracional Brownian moion are based on he chaper abou self-similar processes in [Samorodnisky and Taqqu [2]]. Definiion 2.6 (Fracional Brownian moion). A fracional Brownian moion (fbm) {B H ()} R wih Hurs parameer H (, 1) is a Gaussian zero-mean process wih B H () =, saionary incremens and covariance funcion R H ( 1, 2 ) := Cov ( B H ( 1 ), B H ( 2 ) ) = 1 2( 1 2H + 2 2H 1 2 2H ) Var(B H (1)), (2.3) for 1, 2 R. From now on we will always deal wih he sandard fracional Brownian moion, ha is Var(B H (1)) = 1. In his case and for H = 1 we ge a sandard Brownian moion, since 2 he incremens will be independen in his case. Recall ha a real valued process {X()} R has saionary incremens if for h R + we have {X( + h) X(h)} R d = {X() X()} R. There are some imporan properies of fracional Brownian moion, which we will discuss in he following. Definiion 2.7 (Self-similariy). A real valued process {X()} R is self-similar wih index H > (H-ss) if for all a > he finie-dimensional disribuions of {X(a)} R are idenical o he finie-dimensional disribuions of {a H X()} R, i.e. if for any d 1, 1,..., d R and any a > ( X(a1 ),..., X(a d ) ) d = ( a H X( 1 ),..., a H X( d ) ). (2.4) Lemma 2.8. (i) For every H-self-similar process X we have X() = a.s., since for each a > we ge X() = X(a) d = a H X(). (ii) Every H-self similar process X wih saionary incremens (H-sssi) is symmeric, i.e. X( ) = X( ) X() d = X() X() (i) = X() for all R. (2.5) Theorem 2.9. Fracional Brownian moion exiss and is he only Gaussian process ha is self-similar wih index H (, 1) and has saionary incremens. Proof. Exisence: Le X be a zero-mean Gaussian random variable whose characerisic funcion is given by ϕ X (θ) := E[e iθx ] = exp( σ 2 θ 2 ), θ R, σ R +. 7

22 2 Preliminaries The finie-dimensional disribuions of a Gaussian process {X()} R saisfy ( E[e i m j=1 θ jx( i ) ] = exp 1 m m m A( j, k )θ j θ j + µ( j )θ j ), 2 j=1 k=1 where θ 1,...θ m R, m 1, µ( j ) is a real-valued funcion for all j = 1,..., m, R and {A( 1, 2 ) : 1, 2 R} is non-negaive definie. Conversely, o each µ and A corresponds a Gaussian process wih µ being is mean and A being is auocovariance funcion. Now fix < H < 1: Since he funcion { 1 2H + 2 2H 1 2 2H : 1, 2 R} is non-negaive definie (for a proof see [Samorodnisky and Taqqu [2], Lemma 2.1.8]), here exiss a Gaussian process {X()} R wih mean zero and covariance funcion R H ( 1, 2 ) = 1 2( 1 2H + 2 2H 1 2 2H ), (2.6) 1, 2 R. We sill need o show ha his process is self-similar: We ge Cov ( X(a 1 ), X(a 2 ) ) = 1 ( a1 2H + a 2 2H a 1 a 2 2H ) 2 = a2h ( 1 2H + 2 2H 1 2 2H ), 2 where he self-similariy index is H := 2H >. The mean is zero on boh sides. Since he Gaussian disribuion is compleely deermined by is mean and covariance, we can conclude ha his process is self-similar. Similarly he saionariy of he incremens can be shown: Cov ( X( 1 + h) X(h), X( 2 + h) X(h) ) = 1 ( 1 + h 2H h 2H 1 2 2H ) 2 1 ( 1 + h 2H + h 2H 1 2H ) 1 h 2 2( 2H h 2H 2 2H ) + 1 ( h 2H + h 2H ) = ( 2H + 2 2H 1 2 2H ) = Cov ( X( 1 ) X(), X( 2 ) X() ) Since he mean is zero in eiher case, he same reasoning for a Gaussian disribuion as for self-similariy yields saionariy of he incremens. Uniqueness: Le {Y ()} R be anoher H-sssi Gaussian process and Var(Y (1)) = 1 as for he sandard fracional Brownian moion. We uilize hese properies o derive he covariance as E[Y ( 1 )Y ( 2 )] = 1 ( E[Y 2 ( 1 )] + E[Y 2 ( 2 )] E[(Y ( 1 ) Y ( 2 )) 2 ] ) 2 = 1 ( E[Y 2 ( 1 )] + E[Y 2 ( 2 )] E[(Y ( 1 2 ) Y ()) 2 ] ) 2 = 1 2( 1 2H + 2 2H 1 2 2H ). j=1 8

23 2.2 Fracional Brownian Moion The firs sep is a simple rearrangemen. The second sep is due o he saionary incremens where we use he resul of Lemma 2.8, (i) for a H-self-similar process, ha is Y () = a.s. The las sep uilizes he H-self similariy of Y and E[Y 2 (1)] = 1. Moreover we will have o calculae he mean of Y (): Since Y () is H-sssi and Y () =, we know ha E[Y (1)] = E[Y (2) Y (1)] = 2 H E[Y (1)] E[Y (1)] = (2 H 1)E[Y ()]. We can conclude E[Y (1)] = and hence E[Y ()], because E[Y ( 1)] = E[Y (1)] due o Lemma 2.8, (ii) and E[Y ()] = H E[Y (sign())], which is obvious by self-similariy. So all H-sssi Gaussian processes have he covariance funcion from above and mean zero. For a given H hese processes only differ by a muliplicaive consan. This proves ha Y () d = B H () and so uniqueness is proved. Remark 2.1. The incremens of fracional Brownian moions are called fracional Gaussian noise. One can show ha for H ( 1, 1) an fbm displays long-range dependence, ha 2 is is auocovariance funcion γ B H(h) := Cov(B H ( + h), B H ()) decreases so slowly a large lags ha h= γ B H(h) = as γ BH(h) holds. Inuiively, when long-range dependence is presen, high-lag correlaions may be individually small, bu heir cumulaive effec is significan. Proof: Considering he covariance funcion R H ( 1, 2 ) he auocovariance funcion of an fbm is given by γ B H(h) = 1 2( h 1 2H 2 h 2H + h + 1 2H). Define he funcion g(x) = (1 x) 2H 2+(1+x) 2H and noe ha γ B H(h) = 1 2 h2h g(1/h), for h 1. Using a Taylor expansion a he origin of g(1/h) o he second degree one can see ha γ B H(h) H(2H 1)h 2H 2 for h and so for H > 1 he series 2 h= γ BH(h) obviously diverges. Long-range dependence is one of he key facs why fracional Brownian moions are more reasonable for financial modelling compared o ordinary Brownian moions in some areas. Dependen on he daa he independen incremens of Brownian moions may no be very realisic when observing ime series ino he pas. In empirical sudies of financial ime series, for insance, [Mandelbro [1997]] demonsraed ha log-reurns exhibi his long-range dependence, a finding ha is very conroversial and he suppors solely. More ineresingly for our examinaion laer on, [McCarhy e al. [24]] and many ohers have deeced long-range dependence for macroeconomic daa, i.e. ineres raes or volailiies. This is where our new modelling approach for ineres-rae markes in Chaper 5 applies. Remark Insead of analyzing a sochasic process in he ime domain, processes can also be analyzed in he so-called frequency or specral domain. A saionary ime-domain series can be ransformed ino a frequency-domain series wihou loss of informaion by he so-called Fourier ransform, defined by ˆf() = 1 2π f(x)e ix dx [Rudin [25]]. This means ha he ime-domain series is perfecly recovered from he frequency-domain series 9

24 2 Preliminaries by he so-called inverse Fourier ransform. A deerminisic funcion or a realizaion of a sochasic process can be hough of o consis of rigonomeric funcions wih differen frequencies. The informaion o which exen each frequency is presen in he signal is hen summarized in he so-called specral densiy, which is defined as he square of he magniude of he Fourier ransform, ha is Φ(θ) = 1 2π f(x)e iθx dx 2. So i capures he frequency conen of sochasic processes and helps idenify periodiciies. In he case of long-range dependence he specral densiy increases like a power funcion a low frequencies and explodes a he origin. Moreover, as Theorem 2.9 poins ou, here are alernaive ways o define a fracional Brownian moion, which we will summarize in he following corollary: Corollary Fix < H < 1 and le σ 2 = E[X 2 (1)] = 1. The following saemens are equivalen: (i) X(), R, is Gaussian and H-self-similar wih saionary incremens (ii) X(), R, is an fbm wih self-similariy index H (iii) X(), R, is Gaussian, has mean zero and covariance Cov(X( 1 ), X( 2 )) = 1 2( 1 2H + 2 2H 1 2 2H), 1, 2 R. 1

25 2.2 Fracional Brownian Moion Inegral Represenaions of Fracional Brownian Moion In he following we will presen wo inegral represenaions of a fracional Brownian moion - he ime represenaion and he specral represenaion. Each represenaion will ake he inegral form f (x)m(dx) bu wih is own se of deerminisic funcions f, R, and is own random measure M. For our means i is sufficien o consider M as a Brownian moion B for he ime represenaion and a Gaussian measure B for he specral represenaion, respecively. These inegral represenaions are anoher way o characerize fracional Brownian moion, bu more imporanly we will need hem in order o define an inegraion heory wih respec o fracional Brownian moion, which we will ouline in he subsequen subsecion The ime represenaion This represenaion is also called he moving average represenaion of fracional Brownian moion. Proposiion Le (Ω, F) be a measure space and le B be a sandard Brownian moion defined on R. Le H (, 1). Then he sandard fracional Brownian moion B H (), R has he inegral represenaion B H () d = 1 C 1 (H) where C 1 (H) = ( ) ((1 + s) H 1 2 s H 1 2) 2 ds H ( ) (( s)+ ) H 1 2 (( s)+ ) H 1 2 db(s), R, See [Samorodnisky and Taqqu [2], Prop ] for a proof. The specral represenaion This inegral represenaion is also known as he harmonizable represenaion and is of he form f (x) M(dx), where f is a complex and deerminisic funcion and M is a specific complex measure. We will focus on a simplified special case which will be sufficien for our purposes. We will inegrae wih respec o a complex Gaussian measure B = B 1 + ib 2 such ha B 1 (A) = B 1 ( A), B 2 (A) = B 2 ( A) and E[B 1 (A)] 2 = E[B 2 (A)] 2 = 1 A, for 2 a Borel se A of finie Lebesgue measure A. Proposiion Le < H < 1. Then he sandard fracional Brownian moion {B H (), R} has he inegral represenaion ( where C 2 (H) = B H () d = 1 C 2 (H) π HΓ(2H) sin(hπ) )1 2. For a proof of his see Samorodnisky and Taqqu [2]. e ix 1 x (H 1 2 ) d ix B(x), R, (2.7) 11

26 2 Preliminaries Inegraion Wih Respec o Fracional Brownian Moion In our subsequen analysis of he fracional Heah-Jarrow-Moron model in chaper 5 we will make use of he ime represenaion when we come up wih an inegraion approach wih respec o fracional Brownian moion. These insighs are based on [Pipiras and Taqqu [2]]. In order o explain he difficuly of defining a sochasic inegral wih respec o fracional Brownian moion we conras i wih he sandard Brownian moion case. Therefore we define E as he se of all elemenary funcions f(u) := n f k 1 [uk,u k+1 )(u), u R, (2.8) k=1 where f k and u k < u k+1 are real numbers. Moreover we need o define an inegral of f E wih respec o he fbm B H for H (, 1) as I H (f) := f(u)db H (u). Then denoe he closed span of B H by R Sp(B H ) := {X : I H (f n ) L2 X for some (f n ) E }. An elemen X Sp(B H ) is a zero-mean Gaussian random variable wih variance Var(X) = lim n Var(I H (f n )). Le f X denoe he equivalence class of sequences of elemenary funcions (f n ) such ha I H (f n ) L2 X and wrie he inegral wih respec o fbm on he real line as X = f X db H. (2.9) R We recall ha he characerizaion for he sandard Brownian moion B 1 2 simplifies due o is independen incremens and so Var(I 1 2(f)) = R f2 (u)du, f E. Hence, if (f n ) E and if I 1 2(f n ) converges o X Sp(B 1 2) in he L 2 -sense, here is a unique funcion f X L 2 (R) such ha Var(X) = lim Var(I 1 2 (fn )) = lim fn n n R 2 (u)du = fx 2 (u)du, R due o he fac ha L 2 (R) is a complee space. So, in conras o he general case in (2.9), X Sp(B 1 2) can more easily be characerized by a single funcion f X L 2 (R) as X = f X db 1 2 (u). (2.1) The crucial fac is ha for X, Y Sp(B 1 2) we can sae ha E[XY ] = f X (u)f Y (u)du R R 12

27 2.2 Fracional Brownian Moion and herefore we can say ha Sp(B 1 2) and L 2 (R) are isomeric, i.e. here is a linear and injecive mapping beween he spaces ha preserves he inner producs [Kallsen [27], Theorem 5.1.2]. Therefore our goal is o find a Hilber space C of funcions on he real line ha is isomeric o Sp(B H ) as well in order o come up wih an inegral form in he spiri of (2.1). When we proceed o our fracional Heah-Jarrow-Moron model in Chaper 5 we will only consider he long-range dependen case where 1/2 < H < 1 and ha is why we can focus on his case. Unforunaely, spaces isomeric o Sp(B H ) could no be found, ye. Bu we can come up wih spaces ha are isomeric o linear subspaces of Sp(B H ). Therefore we cie a proposiion ha shows us how o consruc hose classes of inegrands: Proposiion Le E be he se of elemenary funcions as in (2.8). Le I H (f) := R f(u)dbh (u) be an inegral of f E wih respec o he fbm B H for H (, 1). Suppose ha C is a se of deerminisic funcions on he real line such ha (i) C is an inner produc space wih an inner produc (f, g) C, for f, g C, (ii) E C and (f, g) C = E[I H (f)i H (g)], for f, g E and (iii) he se E is dense in C. Then here is an isomery beween he space C and a linear subspace of Sp(B H ), which is an exension of he mapping f I H (f), for f E. Moreover C is isomeric o Sp(B H ) iself if and only if C is complee. For a proof of his proposiion we refer o [Pipiras and Taqqu [2]]. There are several ways o come up wih classes of inegrands boh for he ime represenaion and for he specral represenaion of fbm. We will focus on he class of inegrands in he ime domain, because we will use his inegraion approach in chaper 5. So by (ii) of Proposiion 2.15 we sar wih he calculaion of he covariance E[I H (f)i H (g)] = f(u)g(ν)d 2 R H (u, ν), R where R H is he covariance funcion of a fracional Brownian moion. The double inegral is defined o be linear and o saisfy d 2 R H (u, ν) = R H (d, b) R H (d, a) (R H (c, b) R H (c, a)), u, ν R [a,b] [c,d] for any real numbers a < b and c < d. This suggess ha we can define our class of inegrands as Λ H := { f : f(u) f(ν) d 2 R H (u, ν) < }, R R where R H is he oal variaion measure of R H, which is only defined in he case 1/2 < H < 1 as R 13

28 2 Preliminaries R H (E) = sup Π R H (E i ) for all E F, i where Π := i E i is an arbirary pariion of E wih measurable subses E i and F is he σ-algebra of he measure space (Ω, F). By differeniaing he covariance funcion R H as defined in (2.3) wih respec o u and wih respec o ν we ge d 2 R H (u, ν) = H(2H 1) u ν 2H 2 du dν. Hence we can finally define our class of inegrands ha saisfies all condiions of Proposiion 2.15 and accordingly an isomery beween his class and a linear subspace of Sp(B H ) exiss. Definiion Λ H := { f : R R f(u) f(ν) u ν 2H 2 du dν < }, for 1 < H < 1, whereas he inner produc on 2 Λ H can be expressed as (f, g) Λ H = H(2H 1) f(u)g(ν) u ν 2H 2 du dν. R We sae an imporan resul, ha characerizes he funcions in he space Λ H. We provide a large subse of ha space. R Proposiion Le 1 2 < H < 1. Then L 1 (R) L 2 (R) Λ H. Remark We can give an analogon o Iô s isomery in he fracional case wih he help of Proposiion Therefore le I φ α denoe a fracional inegral of order α > of a funcion φ defined by (I α φ)(s) = 1 Γ(α) R φ(u)(s u) α 1 du, s R. Hence, for f E and 1 < H < 1, [Pipiras and Taqqu [2]] find he isomery 2 f(u)db H (u) = d Γ(H + 1/2) (I H C 1 (H) f)(s)db(s). R This gives rise o a class of funcions of he form { ( f : (I H f)(s) ) 2 } ds < R as a sep prior o our Definiion We will specify his isomery in Chaper 5, when we ge o he change of measure. R 14

29 3 Shor-Rae Models In his chaper we wan o give a brief overview abou some popular ineres-rae models and heir evoluion. Ineres-rae modelling in heory was originally based on he assumpions of specific one-dimensional dynamics for he insananeous spo rae process r. For his direc modelling approach all fundamenal quaniies are defined by no arbirage argumens. In paricular, he exisence of a risk-neural measure Q implies ha he arbirage-free price of a zero-coupon bond a ime wih mauriy T is given by he condiional expecaion [ P(, T) = E Q e T r(s)ds F ]. (3.1) This formula calls for some general definiions, which we will assume o hold for all of he hree models covered in his chaper. Le (Ω, F, P) be a complee probabiliy space ha models he uncerainy in our economy, where Ω is he sae space, F is he σ-algebra represening measurable evens and P is he real-world probabiliy measure. Moreover, le (F ) [,T] be a complee, righ-coninuous filraion defined by F := σ(b i (s) : s ) [,T],,...,d, where B i, i = 1,..d, are d independen Brownian moions. We refer o i as he filraion generaed by he Brownian moions, which includes all informaion from he pas. In conras o P, Q is he associaed risk-neural probabiliy measure ha can be aained by a change of measure. We will cover his more precisely in chapers 4 and 5, because in his chaper we will sar modelling righ away wih he dynamics under he risk-neural measure and so we will no need his approach. For he pricing of ineres-rae derivaives i is oally sufficien o direcly model he dynamics under he risk-neural measure. We will cover some of he mos popular early ineres-rae models and give heir sochasic differenial equaions (sde) represening heir dynamics. We will explain he Vasicek model and hen compare i o he Cox-Ingersoll-Ross model and he Hull-Whie model, which inroduce cerain modificaions. All of he hree models belong o he group of affine erm-srucure models where he coninuously compounded zero rae R(, T) is an affine funcion in he shor rae r(), i.e. R(, T) = a(, T) + b(, T)r(), where a and b are deerminisic funcions of ime. Then he model is said o possess an affine erm srucure. This is always saisfied if he zero-coupon bond price can be wrien in he form P(, T) = D(, T)e A(,T)r(). This chaper is based on he one-facor shor-rae models in he book of [Brigo and Mercurio [27]]. 15

30 3 Shor-Rae Models 3.1 The Vasicek Model We assume ha he insananeous spo rae under he risk-neural measure evolves as an Ornsein-Uhlenbeck process wih consan coefficiens, drif erm k(θ r()) and diffusion erm σ, i.e. dr() = k(θ r())d + σdb(), r() = r, (3.2) where k, θ, σ and r are posiive real-valued consans. Moreover θ is considered as mean. We can observe ha he drif is posiive whenever he shor rae is below θ and negaive oherwise. I can also be formally shown ha r is mean-revering, ha is E[r() F ] θ for, which is obvious by looking a (3.3) underneah. Hence we can conclude ha r is pushed o be closer o he level θ wih every ime sep. We refer o σ as he volailiy of B, i.e. aking ino accoun he sensiiviy of r wih respec o random shocks, represened by he sandard Brownian moion B. We call k he speed of adjusmen. The simulaions of he Vasicek sde in chaper 6 illusrae he dynamics of his model very well. In order o come up wih he spo-rae process as a soluion of he sochasic differenial equaion (3.2) we use he inegraing facor e k and obain By inegraion we ge which resolves o d(e k r()) = e k dr() + ke k r()d = e k( k(θ r())d + σdb() ) + ke k r()d = e k kθd + e k σdb(). e k r() = r(s)e ks + θk e ku du + σ s r() = r(s)e k( s) + θ(1 e k( s) ) + σ s s e ku db(u), s, e k( u) db(u), s. Due o he driving Brownian moion B(), r() condiional on F s is normally disribued wih mean and variance respecively given by E[r() F s ] = r(s)e k( s) + θ ( 1 e k( s)) and [( ) 2 ] Var(r() F s ) = E σ e k( u) db(u) s [ ] = σ 2 E e 2k( u) du by Iô s Isomery s = σ2 ( ) 1 e 2k( s). 2k (3.3) 16

31 3.1 The Vasicek Model From he expecaion and variance in (3.3) we can easily see ha he rae r() can be negaive wih posiive probabiliy a each ime, which is a major drawback of he Vasicek model, because his is usually no a realisic even, alhough here migh be negaive ineres raes in imes of deflaion. For a suiable choice of he parameers, however, he probabiliy of negaive values can be kep marginally small. On he oher hand, an advanage of he model is is analyical racabiliy implied by a Gaussian densiy, which is hardly achieved when assuming oher disribuions for r. This propery is very helpful for hisorical esimaion. The bond price for he Vasicek model can be derived by saring wih (3.1) and hence compuing he condiiional expecaion under Q. Hence we come up wih he formula P(, T) = D(, T)e A(,T)r(), where and A(, T) = 1 ( ) 1 e k(t ) k ( ) (θ σ 2 )( ) σ 2 D(, T) = exp A(, T) T + A(, 2k 2 T)2. 4k Proof: Wrie X(u) = r(u) θ. So, X(u) is he soluion of he Ornsein-Uhlenbeck equaion dx() = kx()d + σdb() wih X() = r θ. By applying exacly he same procedure as for he sde of he spo-rae process r() before, we obain a soluion o his sde by using he inegraing facor e ku again, i.e. he process u ) X(u) = e (X() ku + σe as db(s). (3.4) Obviously, X(u) is a Gaussian process wih coninuous sample pahs and herefore X(u)du is Gaussian, oo. Hence we have E[X(u)] = X()e ku and hus [ ] E X(u)du = X() ( ) 1 e k. (3.5) k In his chaper all expecaions are aken under he risk-neural measure wihou furher menioning, since we model he dynamics under his measure. Similarly, [ Cov(X(), X(u)) = σ 2 e k(u+) E e ks db(s) u = σ 2 e k(u+) e 2ks ds = σ2 2k e k(u+) (e 2k(u ) 1) u ] e ks db(s) 17

32 3 Shor-Rae Models and so ( ) ( ) Var X(u)du = Cov X(u)du, X(s)ds [( [ ])( [ = E X(u)du E X(u)du X(s)ds E = = E [( X(u) E[X(u)] )( X(s) E[X(s)] )] du ds Cov ( X(u), X(s) ) du ds = = σ2 2k 3 ( 2k 3 + 4e k e 2k). ])] X(s)ds σ 2 2k e k(u+s)( e 2k(u ) 1 ) du ds (3.6) Since we have X(u) = r(u) θ, we ge [ E ] [ r(u)du = E and so, ogeher wih expression (3.5) we obain [ T ] E r(u)du ] (X(u) + θ)du = r() θ ( ) 1 e k(t ) θ(t ). (3.7) k Moreover, by resul (3.6) we derive ( T ) ( T ) Var r(u)du = Var X(u)du = σ2 2k 3 ( 2k(T ) 3 + 4e k(t ) e 2k(T )). (3.8) From he Iô inegral represenaion of r(), we conclude ha he defining process for he shor rae is Markovian (for a proof see [Karazas and Shreve [1988], p.355]). Hence [ P(, T) = E e ] [ T r(u)du F = E e T r(u)du r() ], where we can wrie r(u) as a funcion of r(), i.e. r(u, r()), so ha [ P(, T) = E e ] [ ] T r(u)du r() := E e T r(u,r())du. In he Gaussian case we know ha { [ T ] P(, T) = exp E r(u, r())du + 12 ( T )} Var r(u, r())du 18

33 3.1 The Vasicek Model and we can conclude he proof by plugging in (3.7) and (3.8): { r() θ ( P(, T) = exp ) 1 e k(t ) θ(t ) k σ 2 } 2k(T )) + 1 ( 2k(T ) 3 + 4e k(t ) e 2 2k 3 { ( ) ( ) 1 e k(t ) 1 e k(t ) = exp r() + θ (T ) k k ( ) ( σ2 1 e k(t ) + σ2 σ2 1 2e k(t ) + e 2k 2 k 2k2(T ) 4k k { 2 = exp A(, T)r() + θa(, T) θ(t ) } σ2 σ2 σ2 2k2A(, T) + 2k2(T ) A(, T)2 4k = D(, T) exp ( A(, T)r() ). 2k(T ) )} 19

34 3 Shor-Rae Models 3.2 The Cox-Ingersoll-Ross Model The Cox-Ingersoll-Ross (CIR) ineres dynamics are formulaed under he risk-neural measure Q wih a modificaion of he diffusion erm in comparison o he Vasicek model, in order o avoid negaive values for r() given ha reasonable values for he parameers are chosen. Therefore a square-roo erm is inroduced ino he diffusion erm. So he risk-neural ineres rae dynamics are assumed o be dr() = k(θ r())d + σ r()db(), r() = r, (3.9) where k, θ, σ and r are posiive consans wih he same inerpreaions as for he Vasicek model. Once again B denoes a sandard Brownian moion. In order o ensure he posiiviy of r() we posulae 2kθ > σ 2. The process r feaures a non-cenral chi-squared disribuion. The zero-coupon bond price is given by he same form as for he Vasicek model, ha is P(, T) = D(, T)e A(,T)r(), bu wih differen specificaions for A(, T) and D(, T), i.e. 2 ( exp((t )h) 1 ) A(, T) = 2h + (k + h) ( exp((t )h) 1 ), ( 2h exp ( (k+h)(t ) ) 2 D(, T) = 2h + (k + h) ( exp((t )h) 1 ) h = k 2 + 2σ 2. )2kθ σ 2, We forego a derivaion in his case, since our focus is no upon he shor-rae models in he firs place and so we refer o [Cox e al. [1985]]. 2

35 3.3 The Hull-Whie Model 3.3 The Hull-Whie Model Hull and Whie ried o improve he poor fiing of he iniial erm srucure of ineres raes implied by he Vasicek model. Therefore hey inroduced ime-varying parameers ino he Vasicek model, i.e. deerminisic funcions insead of he consans in he shorrae dynamics. They assume ha he risk-neural shor rae evolves according o dr() = (ν() a()r())d + σ()db(), r() = r, (3.1) where a, σ and ν are deerminisic and posiive funcions of ime ha are chosen so as o exacly fi he erm srucure of ineres raes being currenly observed in he marke. However, we choose a and σ as posiive consans because quoes of marke volailiies are no always significan due o liquidiy issues in some markes and so perfec fiing can be dangerous. Thus we can speak of a Vasicek model wih a ime-dependen reversion level of he form dr() = (ν() ar())d + σdb(), r() = r. In he Hull-Whie model r() condiional on F s is normally disribued and again he zero-coupon bond price akes on he form where his ime A(, T) = 1 a( 1 e a(t ) ), D(, T) = P(, T) P(, ) exp P(, T) = D(, T)e A(,T)r(), (A(, T)f(, ) σ2 4a (1 e 2a(T ) ) 3 2a wih P(, ) denoing he iniial bond price given by he marke. ), 21

36 3 Shor-Rae Models 3.4 Conclusion The Vasicek model and he CIR model are called endogenous erm-srucure models, which refers o he fac ha he curren erm srucure of ineres raes is an oupu and no an inpu of he models. We can illusrae ha by seing he evaluaion ime =, which yields he iniial ineres-rae curve as an oupu. In pracice one has o find suiable parameers ha force he iniial bond price curve o be as close as possible o he marke curve, bu usually hree parameers are no enough o reproduce he marke erm srucure saisfacorily. This is conrased by he so-called no-arbirage models like he Hull-Whie model or he more general Heah-Jarrow-Moron approach, on which we will focus in he following chapers. They are called exogenous and oday s erm srucure of ineres raes is an inpu. A clear drawback of he Vasicek model is ha ineres raes can assume negaive values wih posiive probabiliy, a problem ha is ackled by he CIR model. On he oher hand is lineariy enables an explici soluion which makes he model aracive from an analyical poin of view. As a consequence several expressions and disribuions of useful quaniies relaed o he ineres-rae world, for insance volailiy, can be easily obained. This is no given for every model, e.g. he CIR model wih is non-cenrally chi-squared disribued ineres rae is less analyically racable. As we have already seen by he differen pros and cons one has o weigh up wha is more imporan when choosing a cerain model. I is a maer of posiive ineres raes, disribuion of he process and explicily compuable bond prices or opion prices. Moreover, quesions abou mean reversion, implied volailiy srucures, compuaion and esimaion echniques arise - in a nushell: a very difficul choice for which many facors have o be considered. 22

37 4 The Heah-Jarrow-Moron Model The Heah-Jarrow-Moron (HJM) model is a more general approach o an ineres-rae model han wha we have already seen for he shor-rae models. We will cover his model more precisely because undersanding of his classical case is essenial for a deailed analysis of he fracional Heah-Jarrow-Moron model in chaper 5, he main focus of his hesis. The HJM approach sars off by modelling he forward rae insead of he shor rae which allows o capure he evoluion of he enire forward rae curve. This will faciliae a more precise calibraion of he model o he inial forward rae curve. As i will urn ou in he arbirage-free framework, he forward-rae dynamics will be fully specified by heir insananeous volailiy srucures. This is a major difference o he one-facor shor-rae models covered in he previous chaper, where also he drif has o be specified in order o characerize he relevan ineres-rae model. We will ge o he advanages and disadvanages of his approach in our conclusion laer on. This chaper is based on he original aricle of [Heah e al. [1992]] and adds o i. 4.1 The Se-Up Le (Ω, F, P) be a complee probabiliy space ha characerizes he uncerainy in our economy and le T > so ha [, T ] is he rading inerval in our coninuous rading economy. Le (F ) [,T ] be a complee, righ-coninuous and augmened filraion defined by F := σ(b i (s) : s ) [,T],,...,d, where B i, i = 1,..., d, d N, are d independen sandard Brownian moions. There is a coninuum of defaul-free discoun bonds, each bond rading wih a differen mauriy, one for each dae T [, T ]. Hence for he price of he T-mauriy bond P(, T) we require: (i) P(T, T) = 1 for all T [, T ]; (ii) P(, T) > for all T [, T ], [, T]; (iii) T log P(, T) exiss for all T [, T ], [, T]. The firs equaion implies ha his is a defaul-free marke, because he bond payoff equals 1% a mauriy. The second equaion exludes rivial arbirage opporuniies. The las equaion ensures ha he forward raes are well-defined. 23

38 4 The Heah-Jarrow-Moron Model Recall he definiion of he insananeous forward rae f(, T) a ime R + for mauriy T > from (2.1), ha is f(, T) = T log P(, T) for all T [, T ], [, T]. Conversely, his yields for he bond price ( T ) P(, T) = exp f(, s)ds for all T [, T ], [, T]. We observe he major difference beween he Heah-Jarrow-Moron model and he shorrae models, ha is he HJM model is based on forward raes and no on shor raes. As a consequence we will see ha in he HJM framework we can derive arbirage-free condiions for he sochasic evoluions of he enire yield curve. Condiion 1 - A family of forward-rae processes We will sar off wih an assumpion on he family of sochasic processes for he forward rae movemens, ha is f(, T) f(, T) = α(ν, T)dν + σ i (ν, T)dB i (ν) for all T, (4.1) where T [, T ] is fixed, bu arbirary and B i, i = 1,..d are d independen Brownian moions ha model he sochasic flucuaion of he enire forward rae curve saring from a fixed, non-random iniial forward rae curve {f(, T) : T [, T ]} given by he marke. Furhermore we impose he following condiions: (i) f(, ) : ([, T ], B[, T ]) (R, B) is measurable wih B[, T ] denoing he Borelσ-algebra resriced o [, T ]. (ii) α : {(, s) : s T } Ω R is a B{(, s) : s T } F-joinly measurable and adaped sochasic process iself. Moreover α saisfies α(, T) d < P-a.s. T (iii) The volailies σ i : {(, s) : s T } Ω R are B{(, s) : s T } F-joinly measurable and adaped sochasic processes. They saisfy T σ2 i (, T)d < P-a.s. for i = 1,..., d. The differing volailiy coefficiens reflec he sensiiviy of a paricular mauriy forward rae s change o each Brownian moion. The d independen Brownian moions imply ha he resricions for our economy are he coninuous sample pahs of he forward-rae processes and a finie number of random shocks across he enire forward rae curve, i.e. for all mauriies. This is a major difference o shor-rae models where differen mauriies are no aken ino accoun when modelling he spo rae. 24

39 4.1 The Se-Up Now, given he forward-rae process in (4.1) we can easily compue he spo-rae process as r() = f(, ) = f(, ) + α(ν, )dν + σ i (ν, )db i (ν) for all [, T]. (4.2) Condiion 2 - Regulariy of he money marke accoun Noaion: For convenience we define ( B () = exp ) r(y)dy for all [, T ] (4.3) as our money marke accoun or numeraire. We need o make sure ha he value of he money marke accoun is finie. Therefore we posulae for he money marke accoun o hold and T ( T f(, ν) dν < ) α(ν, ) dν d < Q-a.s. The dynamics of he bond price process Condiion 3 - Regulariy of he bond price process In order o ensure ha he bond price process is well-behaved, i.e. ha inegrals are well-defined, we impose hree condiions on he bond price process: (i) ( 2 σ ν i(ν, y)dy) dν P-a.s. for all [, T ], i = 1,..., d; (ii) ( 2 T σ i (ν, y)dy) dν P-a.s. for all [, T], i = 1,..., d; (iii) T ( ) σ i(ν, y)db i (ν) dy is coninuous P-a.s. for all T [, T ], i = 1,..., d. 25

40 4 The Heah-Jarrow-Moron Model Theorem 4.1. Le Condiions 2 and 3 hold. Then he dynamics of he bond price process are log P(, T) = log P(, T) where a i (, T) = and b(, T) = T T (r(ν) + b(ν, T))dν a i (ν, T) 2 dν + σ i (, ν)dν α(, ν)dν for i = 1,..., d a i (, T) 2. a i (ν, T)dB i (ν) P-a.s., (4.4) Proof: In order o prove Theorem 4.1 we will need he following lemma: Lemma 4.2 (Generalized form of Fubini s heorem for sochasic inegrals). Le (Ω, F, P) be a probabiliy space and le (F ) [,T ] be a filraion saisfying he usual condiions (i.e. complee and righ-coninuous), which is generaed by he Brownian moion B. Le {Φ(, a, ω) : (, a, ω) [, T ] [, T ] Ω} be a family of real random variables such ha: (i) ((, ω), a) {([, T ] Ω) [, T ]} Φ(, a) is L B[, T ]-measurable where L is he predicable σ-field; (ii) Φ2 (s, a)ds < a.s. for all [, T ]; (iii) ( 2 T Φ(s, a)da) ds < a.s. for all [, T ]. If T ( ( T Φ(s, a)db(s)) da is coninuous a.s., hen: ) T ( Φ(s, a)da db(s) = ) Φ(s, a)db(s) da for all [, T ]. Moreover we will apply he following corollary ha we will sae wihou proof since i is jus one sraighforward rearrangemen. Corollary 4.3. Le he hypoheses of Lemma 4.2 hold. Define { σ(s, a)1{s a} if (s, a) [, ] [, ], Φ(s, a) := if (s, a) / [, ] [, ]. Then y ( s ) σ(s, a)da db(s) = ( s y ) σ(s, a)db(s) da for all y [, ]. 26

41 4.1 The Se-Up Now we can sar wih our proof by using Condiions 1-3: T log P(, T) = = T f(, s)ds f(, y)dy T = f(, y)dy T ( ( T ) α(ν, y)dν dy ) α(ν, y)dy dν T ( ( T ) σ i (ν, y)db i (ν) dy ) σ i (ν, y)dy db i (ν), where all inegrals are well-defined due o Condiions 1 and 2 and where we used Lemma 4.2. We are now spliing up he inegrals by changing heir limis and hen rearrange and apply Corollary 4.3. Moreover recall expression (4.2) where r() = f(, )+ α(ν, y)dν + d σ i(ν, y)db i (ν). So we can rearrange log P(, T) = + = log P(, T) + = log P(, T) + T f(, y)dy ( T ν ) α(ν, y)dy dν ( ) f(, y)dy + α(ν, y)dy dν + ν }{{} r(y)dy = ν α(ν,y)dνdy ( T α(ν, y)dy ν }{{} (r(y) + b(y, T))dy 1 2 = b(ν,t)+ 1 2 d a i(ν,t) 2 = a i (ν,t) ( T ν ) σ i (ν, y)dy db i (ν) ( ) σ i (ν, y)dy db i (ν) } ν {{ } ) dν ( T ) σ i (ν, y)dy db i (ν) ν }{{} a i (ν, T) 2 dν + = ν σ i (ν,y)dydb i (ν) a i (ν, T)dB i (ν). In order o come up wih he sochasic differenial equaion (sde) ha represens he bond price dynamics and of which he bond price process P(, T) is a srong soluion, we apply Iô s lemma. Firs recall: Lemma 4.4 (Iô s lemma). Le X() = X()+ b(s)ds+ σ(s)db(s) be an Iô process wih drif b and diffusion σ, R + and B a Brownian moion. Le f C 2 (R, R). Then (f(x())) R+ is an Iô process of he form: f(x()) = f(x()) + f (X(s))dX(s) f (X(s))d[X, X] s, 27

42 4 The Heah-Jarrow-Moron Model where {[X, X] } R+ wih [X, X] := σ2 (s)ds is called he quadraic variaion of X. [Kallsen [27], Theorem 5.4.8] So, in our case we define X() = (r(ν) + b(ν, T))dν 1 2 a i (ν, T) 2 dν + a i (ν, T)dB i (ν). Then Iô s lemma wih f(x) = e x yields: dp(, T) = df(x()) = + e X() dx() + e X()1 2 a i (, T) 2 d = e X() ( (r() + b(, T))d e X()1 2 a i (, T) 2 d = e X() ((r() + b(, T))d + = (r() + b(, T))P(, T)d + a i (, T) 2 d + ) a i (, T)dB i () a i (, T)P(, T)dB i (). ) a i (, T)dB i () (4.5) Our bond price process is in general non-markov because is drif erm (r() + b(, T)) and is volailiy coefficiens a i (, T) can depend on he hisory of he Brownian moions B i, i = 1,..., d. The relaive bond price process We can easily deermine he relaive (or discouned) bond price process for a T-mauriy bond. Le Z (T) = P(,T) B denoe he relaive bond price for a T-mauriy bond a ime () for T [, T ] and [, T], where he numeraire B () was defined earlier in expression (4.3). Z (T) is he bond s value expressed in unis of he accumulaion facor and no in dollars, which is paricularly useful for analysis and no-arbirage heory which we will see laer on. Analogously o Theorem 4.1 we ge he relaive bond price process as log Z (T) = log Z (T) + + a i (ν, T)dB i (ν) b(ν, T)dν 1 2 wih a i (, T) and b(, T) defined as in Theorem 4.1. P-a.s. a i (ν, T) 2 dν (4.6) 28