Timeinhomogeneous Lévy Processes in CrossCurrency Market Models


 Philip Chase
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1 Timeinhomogeneous Lévy Processes in CrossCurrency Marke Models Disseraion zur Erlangung des Dokorgrades der Mahemaischen Fakulä der AlberLudwigsUniversiä Freiburg i. Brsg. vorgeleg von Naaliya Koval Februar 25
2 Dekan: Prof. Dr. Josef Honerkamp Erser Referen: Prof. Dr. Erns Eberlein Zweier Referen: Prof. Dr. Marek Rukowski Daum der Promoion: 1. Juni 25 Abeilung für Mahemaische Sochasik Fakulä für Mahemaik und Physik AlberLudwigsUniversiä Freiburg Eckersraße 1 D7914 Freiburg im Breisgau
3 Preface For companies operaing in a globalised economy flucuaing foreign exchange and ineres raes represen wo significan sources of risk. For long ime i has been undersood ha hese risk facors are linked by fundamenal economic relaionships, bu up o dae only a few models in he lieraure sysemaically reflec his fac in he valuaion of foreign and crosscurrency derivaives. The majoriy of he marke models, which are used by banks o describe he evoluion of ineres and foreign exchange raes, are based on Brownian moion. However, recen empirical sudies revealed see e.g. Eberlein 1999) and Con and Tankov 24)), ha replacing Brownian moion by an appropriae Lévy process provides a much beer fi of reurn disribuions of financial asses. Furhermore, i is no only he fi of he disribuions ha is improved. Purely disconinuous processes, such as e.g. he generalised hyperbolic Lévy moion, in addiion, give a more realisic picure of price movemens on he level of he microsrucure see e.g. Eberlein and Özkan 23)). In he las en years he Libor Marke Model LMM), which was inroduced by Brace, Gaarek and Musiela 1997), Milersen, Sandmann and Sondermann 1997) and Jamshidian 1997), has gained a wide accepance as a pracical and versaile model of ineres rae dynamics. Based wihin a HJM framework see Heah, Jarrow and Moron 1992)), his model concenraes on he descripion of marke observable forward Libor raes, raher han he mahemaically absrac insananeous forward raes, and has he advanage of pricing ineres rae derivaives caps and floors) by Black s formula see Black 1976)). Recenly, more general models describing Libor dynamics were proposed by Jamshidian 1999) and Eberlein and Özkan 22). In Jamshidian 1999) he Libor rae process is driven by a general semimaringale, bu he quesion of pricing ineres rae derivaives is no considered. The approach of Eberlein and Özkan 22) is based on he represenaion of Libor raes as ordinary exponenials of sochasic inegrals driven by a imeinhomogeneous Lévy process, which allows o derive explici formulas for pricing ineres rae caps and floors. All he menioned Libor models are relaed o one currency of denominaion only. Hence, hey are incapable of handling sysemaically he relaive value of posiions in several fixed income markes, and especially, in foreign exchange and crosscurrency derivaives. Our goal is o consruc a crosscurrency marke model in which he combinaion of foreign exchange and ineres rae risks is specified in a consisen way, imposing resricions II
4 on he coefficiens of forward Libor raes and forward foreign exchange raes, when hese are modeled joinly. In fac, hese resricions provide he link beween ineres rae and currency risk. Several conceps, which we have used in our sudies, are relaed o Schlögl 22), where he Gaussian Libor Marke Model is exended o he mulicurrency seing, bu he problem of model calibraion o he marke prices of heavily raded ineres rae derivaives is no considered. Following he approach of Eberlein and Özkan 22), we focus our research on a racable class of models wih jumps, in which he price of he underlying asse is modeled by he exponenial of a imeinhomogeneous Lévy process. In Chaper 1 we review some noions of sochasic calculus which are sysemaically used in he sequel. Mos of he resuls presened in he firs wo secions, concerning homogeneous and inhomogeneous Lévy processes, are sandard and well known in he lieraure. The las wo secions are concerned wih he Laplace cumulan process and is complex counerpar, he Fourier cumulan process, which have been inroduced in Kallsen and Shiryaev 22). These objecs urned ou o be very imporan in our sudies, since hey allow us o describe some crucial properies of exponenial imeinhomogeneous Lévy processes. Chaper 2 consiss of wo pars. In he firs par we presen he mulicurrency erm srucure model in a semimaringale seing, which is moivaed by he resuls of Björk, Di Masi, Kabanov and Runggaldier 1997) and Björk, Kabanov and Runggaldier 1997). While semimaringales provide a rich framework for heoreical developmens, hey remain oo general for he pracical applicaions. For his reason, in he second par of Chaper 2 we ranslae a few imporan resuls ino he imeinhomogeneous Lévy seing, which allows us o inroduce a mulicurrency Lévy Libor model. In Chaper 3 we consider he problem of pricing foreign ineres rae and crosscurrency derivaives such as foreign caps and floors, foreign exchange rae opions, and differen variaions of crosscurrency swaps. Chaper 4 discusses he problem of model calibraion: rerieving parameers of an opion pricing model from marke prices of heavily raded ineres rae derivaives. We presen an algorihm for solving his problem in he conex of exponenial imeinhomogeneous Lévy model for he Libor raes. The quesions of implemenaion and empirical performance are discussed. I ake his opporuniy o express my graiude o my scienific advisor Prof. Dr. Erns Eberlein for his confidence, encouragemen and suppor. Furhermore, I would like o hank Wolfgang Kluge, Erns Augus von Hammersein, Fehmi Özkan and Jan Liinev for he simulaing discussions, and all he members of Insiu für Mahemaische Sochasik for creaing friendly and moivaing working amosphere. I graefully acknowledge he financial suppor, provided by he Graduae College DFG) Nichlineare Differenialgleichungen: Modellierung, Theorie, Numerik, Visualiesierung. I wan o hank our secreary Monika Haenbach for finding inconsisencies in he layou and bibliography. Las bu no leas, I hank my parens and my husband for heir enormous emoional suppor. III
5 Conens 1 Some Facs From Sochasic Analysis Lévy processes Timeinhomogeneous Lévy processes Laplace cumulan process Fourier cumulan process CrossCurrency Marke Model Term srucure modelling: semimaringale seing Domesic erm srucure Foreign erm srucure and foreign exchange rae Absence of arbirage in he inernaional bond marke Term srucure modelling: Lévy seing Descripion of he driving process Assumpions on he dynamics of he domesic forward raes Assumpions on he dynamics of he foreign exchange rae and foreign forward raes Domesic forward price process Foreign forward exchange rae Foreign forward price process Discreeenor Lévy domesic Libor rae model Discreeenor Lévy foreign Libor rae model Relaionship beween domesic and foreign markes in discreeenor framework Pricing CrossCurrency Derivaives Domesic forward caps and floors The Libor rae approach The forward rae approach Foreign forward caps and floors The foreign Libor rae approach The foreign forward rae approach Foreign currency opions IV
6 3.4 Crosscurrency swaps Floaingforfloaing i; ; ) and i; l; ) swaps Floaingforfloaing i; l; m) swaps Quano caples Model Calibraion Volailiy srucure Boosrapping of he forward volailiy curve Daa ses descripion Calibraion procedure Calibraion resuls: Daa se I Calibraion resuls: Daa se II Onecurrency seing Twocurrencies seing Threecurrencies seing Conclusions A Daa ses sed for Calibraion of Libor and Forward Rae Models 169 A.1 Daa se I: February 19, A.2 Daa se II: July 2, B Calibraion Resuls 183 B.1 Daa se I: Libor rae model B.2 Daa se I: Forward rae model B.3 Daa se II B.3.1 Onecurrency seing: Libor rae model B.3.2 Onecurrency seing: Forward rae model B.3.3 Twocurrencies seing B.3.4 Threecurrencies seing V
7 Chaper 1 Some Facs From Sochasic Analysis Modern heory of sochasic processes has become a naural language for formulaing quaniaive models of financial markes. I is no surprising, ha Lévy processes play an increasingly imporan role in his conex. Analogously o Brownian moion, Lévy processes and sochasic differenial equaions based on Lévy processes provide a large oolbox for modelling financial phenomena such as jumpype behavior of prices in cerain periods, followed by calm periods wih coninuous pahs. I is worh o noe, ha imporan classes of sochasic processes are obained as generalisaions of he class of Lévy processes, one of hem is he class of semimaringales. This chaper can be divided ino wo pars. In he firs wo secions we give basic definiions concerning homogeneous and inhomogeneous Lévy processes. The disribuions of Lévy processes homogeneous or inhomogeneous) a any ime are infiniely divisible. I is wellknown, [ ha ] for an valued infiniely divisible random variable X and some u holds: E e u X = exp θu)), where θu) denoes he logmomen generaing funcion or cumulan) of X in u if i exiss). This noion can be generalised for semimaringales and leads o he socalled Laplace cumulan process, which has a complex counerpar, called he Fourier cumulan process. The las wo secions of his chaper are concerned wih hese wo processes. They urned ou o be very helpful in descripion of cerain properies of imeinhomogeneous Lévy processes, which are sysemaically used in he sequel. 1.1 Lévy processes Le Ω, F, F ) R, P ) be a filered probabiliy space saisfying he usual condiions, ha is, Ω, F, P) is complee, all he null ses of F are conained in F, and F ) R is righconinuous filraion: F s F F s, R, such ha s and F s = >s F s R. 1
8 We also assume ha F = σ found in Shiryaev 1999). R F. The following definiion of Lévy process can be Definiion An adaped process L = L ) < wih sae space defined on he probabiliy space Ω, F, P) is a ddimensional Lévy process if 1. L = P a.s. 2. L has incremens independen of he pas: ha is L L s is independen of F s, s < <. 3. L has saionary incremens: ha is L L s has he same disribuion as L s, s < <. 4. L is coninuous in probabiliy: for every and ε > holds lim P L L s > ε) =. s 5. for P almos all ω Ω he rajecories L ω)) < belong o he space D ) of càdlàg funcions coninue à droie avec des limies à gauche, ha means: righ coninuous wih lefhand limis) for all >. Processes saisfying he firs hree condiions are called processes wih saionary homogeneous) independen incremens PIIS). If we assume ha process X has only properies 1. 4., hen i can be shown ha here exiss a modificaion L = L ) < of L = L ) < i.e PL L ) = for all ) wih propery 5. This modificaion is again a Lévy process see e.g. Theorem I.3 in Proer 199)). As we will always work wih his modificaion, we incorporaed wihou loss of generaliy) propery 5. in he definiion of Lévy process from he very beginning. Due o he fac ha Lévy process L = L ) belongs o he PIIS class, is disribuion is compleely deermined by he law of L 1, which we will denoe by P L 1. This disribuion is infiniely divisible for each. Le χ z) = E [ exp )] i z L = exp i z x ) P dx), z be he characerisic funcion of L. Then by wellknown Lévy Khinchine formula see e.g. II.2.1 in Jacod and Shiryaev 1987)), we ge χ z) = exp i z B h) 1 2 z C z exp i z x ) 1 i z hx) ) ν dx), 1.1.1) 2
9 where x y := d j=1 x jy j is a Euclidean produc. Wih Corollary II.4.19 in Jacod and Shiryaev 1987), Lévy process are semimaringales and heir characerisics are deerminisic. In paricular, B h), C d is a symmeric posiive semidefinie marix such ha C i,j = L i,c, L j,c, where L c is a coninuous maringale par of L, and ν is a predicable random measure on R such ha ν R {}) = ν {} ) = and x 2 1 ) ν dx) := x 2 1 ) ν < R ) This condiion ensures he inegrabiliy of he funcion x exp i z x ) 1 i z hx) wih respec o measure ν see Remark in Sao 1999)). And finally, h is a runcaion funcion, ha is, any bounded Borel funcion wih compac suppor, which behaves like x near he origin. Tradiionally, hx) = x1 { x 1}, bu generally he choice of runcaion funcion depends on he properies of ν. For insance, if we replace condiion 1.1.2) by sronger inequaliy x 1) ν dx) <, hen we can se hx) =. If L is a special semimaringale, hen x 2 x ) ν dx) < P a.s holds for any R by Proposiion 2.7 in Kallsen 1998), and one can choose hx) = x. Noe ha C and ν are independen of he runcaion funcion h, while Bh) varies wih h, so ha B h) B h ) = hx) h x)) ν dx). From he saionariy and independence of incremens i follows ha χ s z) = χ z) χ s z), χ z) = exp ψz)), where ψ denoes he cumulan funcion. Noe ha he riple B h), C, ν ) is unambiguously defined by he characerisic funcion in 1.1.1), so ha according o Corollary II.4.19 in Jacod and Shiryaev 1987) we have B h) = bh), C = C, ν [, ] G) = νg) 1.1.3) for G B ), where bh), C is a d d symmeric posiive semidefinie marix, and ν is a Lévy measure, ha is, a posiive measure in such ha ν{}) = and x 2 1) νdx) <. The riple b, C, ν) is called he generaing Lévy Khinchine) riple of L 1, i deermines some fundamenal sample pah properies of he Lévy process L. Figuraively, bh) is he rend componen of L 1, C is he covariance marix of is coninuous Gaussian componen, while he Lévy measure deermines he frequency and magniudes of jumps made by he process. In mahemaical finance equiy prices are commonly modeled by exponenial Lévy processes, for example by he geomeric Brownian moion. Since he price of a coningen 3
10 claim is deermined as he expecaion of is discouned payoff funcion, one is naurally ineresed in he exisence and form of he exponenial momen of L. Le us define a se C := c Rd : 1 { x >1} exp c x ) νdx) <. Wih respec o Theorem in Sao 1999) we have 1. E [ exp c L )] < if and only if c C. 2. If w C d is such ha Re w) C, hen E [ exp w L ) ] <, and E [ exp w L )] = exp θw)), where θw) is a logmomen generaing funcion of L 1 in w given by θw) = bh) w 1 2 w Cw exp w x ) 1 w hx) ) νdx). 1.2 Timeinhomogeneous Lévy processes Dropping he saionariy of incremens requiremen for Lévy processes, we obain he class of imeinhomogeneous Lévy processes, which are also called addiive processes or processes wih independen incremens and absoluely coninuous characerisics, shorly PIIAC see Eberlein, Jacod and Raible 23)). Definiion Timeinhomogeneous Lévy process X ) is a sochasic process saisfying properies 1, 2, 4 and 5 in Definiion A process saisfying only properies 1, 2 and 4 is called an addiive process in law. As in he case of Lévy processes here always exiss a modificaion of an addiive process in law, which also saisfies propery 5 see Theorem in Sao 1999)). Timeinhomogeneous Lévy processes are described by sysems of infiniely divisible disribuions. In paricular, if X ) { is a imeinhomogeneous Lévy process on, hen i deermines he sysem of riples B h), C, λ ) } :, where each B h), C, λ ) is he generaing riple of he disribuion of X. Evidenly, he characerisic funcion of X, which is given by χ z) := E [ exp iz X )] has a Lévy Khinchine represenaion χ z) = exp iz B h) 1 2 z C z exp iz x ) 1 z hx) ) λ dx). 4
11 Furhermore, we have B h) = bs h) ds, C = C s ds, λ G) = ν [, ] G), G B ), where ν is a unique measure on [, ) such ha for all holds ν {} ) = and x 2 1 ) νds, x)) <. Noe ha in case of Lévy processes he measure ν is he produc of he Lebesgue measure on [, ) and he Lévy measure ν see equaion 1.1.3)). According o Theorem i) in Sao 1999) he generaing riple of a imeinhomogeneous Lévy process in law X ) saisfies he following condiions: 1. B h) =, C = λ =. 2. Posiiveness: if s <, hen z Cs z z C z for z and λ s G) λ G) for G B ) ) 3. Coninuiy: as s in [, ), B s h) B h), z Cs z z C h)z, λs G) λ G) 1.2.5) for z, and for all G B ) such ha G {x : x > ɛ}, ɛ >. Le ψ, u) = E [ ] e iux denoe he characerisic funcion of X and ψs,, u) = E [ e ] iux X s ) denoe he characerisic funcion of X X s. Due o he independence of he incremens for 1 < 2 < 3 < holds ψ 1, 2, u) ψ 2, 3, u) = ψ 1, 3, u). For more deails on he properies of imeinhomogeneous Lévy processes see Skorohod 1986) and Sao 1999). In he following Lemma we inroduce a imeinhomogeneous Lévy process X, which will be ofen used in he sequel. Lemma Se T o be a fixed ime horizon and T := T, T 1,..., T N ) o be a finie pariion of he inerval [, T ] such ha = T < T 1 < < T N = T. 5
12 Le L 1, L 2,..., L N be N Lévy processes wih characerisic riples B k h) = b k h), C k = C k, ν k = ν k), k = 1,..., N. The process X ) [,T ] wih X =, which is given by [ N k 1 ) ] X := 1 Tk 1,T k ]) L k L m Tm L m1 T m for, T ] 1.2.6) k=1 m=1 is a imeinhomogeneous Lévy process. Is generaing riple B h), C, λ ) saisfies B h) = bs h) ds, C = C s ds, λ G) = ν s G) ds for, T ], 1.2.7) where b h) = N b k h)1 Tk 1,T k ]), C = k=1 N C k 1 Tk 1,T k ]), ν G) = k=1 N ν k G)1 Tk 1,T k ]) ) Furhermore, for = holds B h) =, C =, ν =. Proof. One can easily check ha generaing riple B h), C, λ ) saisfies posiiveness and coninuiy condiions 1.2.4) and 1.2.5). Hence, according o Theorem 9.8 ii) in Sao 1999) here exiss a imeinhomogeneous Lévy process X such ha is generaing riple is given by B h), C, λ ). Plugging 1.2.8) ino 1.2.7), we obain [ N B h) = 1 Tk 1,T k ]) b k C = λ G) = k=1 [ N 1 Tk 1,T k ]) C k k=1 k 1 m=1 k 1 m=1 [ N 1 Tk 1,T k ]) ν k G) k=1 T m b m b m1)], T m C m C m1)], k 1 m=1 k=1 T m ν m G) ν m1 G) )]. Represenaion of X given by equaion 1.2.6) follows direcly from he above form of characerisics B h), C, λ ). In he following secions we are going o inroduce Laplace and Fourier cumulan processes, which allow us o describe cerain properies of imeinhomogeneous Lévy processes and will be ofen used in he sequel. 6
13 1.3 Laplace cumulan process Le X be a realvalued semimaringale, defined on a complee probabiliy space Ω, F, F ) R, P ), whose characerisics B, C, ν) relaive o some runcaion funcion h : are given in he form B h) = b s h) ds, C = C s ds, ν X [, ], dx) = F s dx) ds, where b is a predicable valued process, C is a predicable valued process, whose values are posiive semidefinie, symmeric marices, F is a measure on, which inegraes x 2 1) and saisfies F {}) = for all, and finally hx) = x1 { x 1} x) is a runcaion funcion. Wih Theorem II.2.34 in Jacod and Shiryaev 1987) he canonical decomposiion of X is given by X = X Bh) X c hx) µ X ν X ) x hx)) µ X. For he purpose of lucidiy we use he noaion for inegrals wih respec o random measures given by II.1.5 in Jacod and Shiryaev 1987). Noe ha wih Proposiion II.2.1 i) in Jacod and Shiryaev 1987) X is a quasilefconinuous process. Hence, ν X {}, ) =. Wih respec o Definiion I.4.21 in Jacod and Shiryaev 1987) a realvalued semimaringale is called special, if i admis a decomposiion X = X) M V, where M is a local maringale, and V is a predicable process of finie variaion, boh saring a. The unique process V is called compensaor or drif process of X. I can be expressed expliciely in erms of semimaringale characerisics B, C, ν X ). In paricular, V = Bh) x hx)) ν X x). Definiion A semimaringale X is called exponenially special if exp X X ) is a special semimaringale. 2. A predicable process of finie variaion V is called an exponenial compensaor of a semimaringale X if he process exp X X) V ) is a local maringale. Wih Lemma 2.15 in Kallsen and Shiryaev 22) a realvalued semimaringale X has an exponenial compensaor if and only if i is exponenially special. Moreover, in his case exponenial compensaor is up o indisinguishabiliy unique. Definiion Le φ ) be a process inegrable wih respec o X, i.e. φ LX) in he sense of Definiion III.6.17 of Jacod and Shiryaev 23), and such ha φ X := is exponenially special. 7 φ s dx s
14 1. The Laplace cumulan process K X φ) of X in φ is defined as he compensaor of he special semimaringale L exp φ X )), where L ) denoes sochasic logarihm The modified Laplace cumulan process K X φ) of X in φ is he logarihmic ransform of K X φ), i.e. K X φ) = ln E K ) X φ)), where E ) denoes sochasic exponenial. The expressions for Laplace cumulan process and is modificaion are given in Theorem 2.18 in Kallsen and Shiryaev 22) in he mos general case. Since in our seing X is a quasilef coninuous semimaringale, KX ) and K X ) coincide and are given by K X φ) = K X φ) = κ s φ) ds, where κ φ) = φ b h) 1 2 φ C φ exp φ x ) 1 φ hx) ) F dx). In paricular, since K X φ) is a process of finie variaion wih K X φ) =, we obain exp K X φ) ) = E K X φ)). Remark If X is a imeinhomogeneous Lévy process on, he Laplace cumulan process K X φ) is deerminisic if and only if φ : R is a deerminisic process see Remarks III.7.15 in Jacod and Shiryaev 23)). According o Theorem 2.19 in Kallsen and Shiryaev 22) he modified Laplace cumulan process K X φ) is he exponenial compensaor of φ X. More specifically, Z := exp φ X K X φ) ) = exp φ X ) E K M loc. X φ)) Lemma Le X be an valued imeinhomogeneous Lévy process, and φ : R be a deerminisic funcion inegrable wih respec o X such ha U exponenially special, where z R. Then E [exp U )] = exp K X zφ) ) = := z ) zφ s b s h) ds z2 2 φ s C s φ s ds exp zφ s x ) 1 zφ s hx) ) F s dx) ds. φ s dx s is 1 The definiion of sochasic logarihm can be found in Theorem II.8.3 in Jacod and Shiryaev 23). See also Definiion in he nex secion. 8
15 Proof. This is obvious due o he fac ha K X ) is a deerminisic funcion. Remark Noe ha if X has saionary incremens, i.e if i is a Lévy process, hen E exp z φ s dx s = exp K X zφ) ) = exp θzφ s ) ds, where θ is a logmomen generaing funcion of X 1. Lemma Le X be an addiive process and φ be defined as in Lemma above. Then for u < and z R holds E exp z φ s dx s = exp θ zφ s ) ds. u u Proof. Observe ha Y := z of he incremens we ge φ s dx s is an addiive process. Hence, by he independence E [exp Y )] = E [exp Y u Y ) Y Y u ))] = E [exp Y u )] E [exp Y Y u )] = exp u θzφ s ) ds E exp z φ s dx s = exp θzφ s ) ds u and he saemen of he lemma follows. Proposiion Le L 1, L 2,..., L N be N Lévy processes wih characerisic riples B k h) = b k h), C k = C k, ν k = ν k), k = 1,..., N and T := T, T 1,..., T N ) be a finie pariion of he inerval [, T ], where T = and T N = T. Furhermore, le X be a imeinhomogeneous Lévy process given by equaion 1.2.6) in Lemma and φ : R be a deerminisic funcion inegrable wih respec o X. Then for [, T ] holds N φ k 1 T m s dx s = 1 Tk 1,T k ]) φ s dl m s. k=1 T k 1 φ s dl k s m=1 T m 1 Moreover, if E exp z T k φ s dl k s < for every k = 1,..., N and some z R, T k 1 9
16 hen E exp z φ s dx s = exp θzφ s ) ds, where θz) = N 1 Tk 1,T k ])θ k z) and θ k ) is he logmomen generaing funcion of L k 1. k=1 Proof. Observe ha using he addiiviy propery of he sochasic inegral can wrie φ s dx s = N 1 Tk 1,T k ]) k=1 From equaion 1.2.6) i follows ha dx = T k 1 φ s dx s k 1 T m m=1 T m 1 φ s dx s, we φ s dx s ) N 1 Tk 1,T k ]) dl k ) k=1 Plugging 1.3.1) ino 1.3.9), we obain he firs saemen of he proposiion. To simplify he noaion, le us pu Y k incremens of E exp z = = k=1 φ s dx s we obain φ s dx s [ N 1 Tk 1,T k ])E exp Y k N k=1 := z [ 1 Tk 1,T k ])E exp Y k Y k Applying Lemma 1.3.6, we ge E exp z φ s dx s = ) k 1 YT k k 1 φ s dl k s. Due o he independence of he m=1 T k 1 ))] k 1 m=1 Y m Tm Y m T m 1 ) )] E [ exp Y m T m Y m T m 1 ))]. = N 1 Tk 1,T k ]) exp k=1 T k 1 θ k zφ s ) ds k 1 m=1 exp T m T m 1 θ m zφ s ) ds 1
17 N = 1 Tk 1,T k ]) exp θ k zφ s ) ds k=1 T k 1 N = exp 1 Tk 1,T k ]) θ k zφ s ) ds k=1 T k 1 = exp which complees he proof. θzφ s ) ds, k 1 T m m=1 T m 1 k 1 T m m=1 T m 1 θ m zφ s ) ds θ m zφ s ) ds The following resuls, describing he properies of exponenially special semimaringales, are essenial for he exposiion in he nex chaper. Lemma Le X be a realvalued exponenially special semimaringale. The process exp X) is a local maringale if and only if he exponenial compensaor of X is equal o zero up o indisinguishabiliy. Proof. ) Suppose ha X is exponenially special. I means ha exp X X ) is a special semimaringale X is finie and F measurable). By Lemma 2.15 in Kallsen and Shiryaev 22), here exiss a predicable process of finie variaion V such ha exp X X V ) is a local maringale. This process is up o indisinguishabiliy unique and is called he exponenial compensaor of X. Obviously, if V is equal o zero, e X X, and hence, e X, is a local maringale. ) Le exp X) be a local maringale, hen i is a special semimaringale and X is by definiion exponenially special. From he firs par we know ha here is an exponenial compensaor V of X such ha exp X X V ) is a local maringale. By uniqueness, we ge ha V = up o indisinguishabiliy. The following sae Lemma Consider realvalued semimaringales X and X. mens are equivalen: 1. X and X are exponenially special { x >1} x) exp x ) ν X ds, dx) < P a.s. 11
18 3. 1 {x } x) exp x) 1 hx)) ν X ds, dx) < and 1 {x<} x) exp x) 1 h x)) ν X ds, dx) < P a.s. Proof. Obviously, X) = X). 1 2: Lemma and 3) in Kallsen and Shiryaev 22). 1 3: Lemma and 2) in Kallsen and Shiryaev 22). 3 1: Noice ha e x 1 h x) e x 1 hx), x e x 1 hx) < e x 1 h x), x <. Hence, if condiion 3 holds, hen e x 1 hx)) ν X V and e x 1 h x)) ν X V, which is equivalen o he fac ha X and X are exponenially special semimaringales. 1.4 Fourier cumulan process In his secion we are going o inroduce a complex valued counerpar of he Laplace cumulan process. Since here is a igh connecion beween he noions of cumulan process, sochasic exponenial, and sochasic logarihm, i is firs necessary o define he las wo erms for he class of complexvalued semimaringales. In wha follows he sochasic basis Ω, F, F ) R, P ) remains fixed. Definiion Le Y be a complexvalued semimaringale given by Y := Y iy, where Y and Y are realvalued semimaringales. The sochasic exponenial EX) is defined as he up o indisinguishabiliy unique) soluion Z o he sochasic differenial equaion Z = 1 Z Y or dz = Z dy wih Z = 1, ) which is equivalen o and Z = Z iz. Z = 1 Z Y Z Y, Z = Z Y Z Y, ) 12
19 According o Theorem I.4.61 in Jacod and Shiryaev 1987) he soluion of equaion ) is a semimaringale, which is denoed by EY ), and is given by EY ) = exp Y Y 1 2 Y, Y c 1 ) 2 Y, Y c i Y, Y c s 1 Y s ) exp Y s ). The mapping Y EY ) can be invered. In analogy o real calculus he converse is called sochasic logarihm of Y see Choulli, Krawczyk and Sricker 1998), Kallsen and Shiryaev 21) and Kallsen and Shiryaev 22)). Definiion Le Z be a complexvalued semimaringale such ha Z and Z are C \ {}) valued. Then here exiss an up o indisinguishabiliy unique complexvalued semimaringale Y such ha Y = and Z = Z EY ). I is called sochasic logarihm of Z, and is given by Y := LZ) = 1 Z Z. Lemma A complexvalued semimaringale Z = Z iz one of he following condiions holds: is exponenially special if 1. Z is exponenially special. 2. exp Z Z ) is a locally bounded semimaringale. Proof. By definiion, Z is exponenially special if e Z Z) = e Z Z ) e iz Z ) is a special semimaringale. Noe ha e iz Z ) is bounded and, hence, a complexvalued special semimaringale. The saemen of he lemma follows from he fac ha a produc of wo semimaringales XY is a special semimaringale if X is special and Y is locally bounded see VII.32 in Dellacherie and Meyer 198)). Lemma Le Z = Z iz be a complexvalued semimaringale. The process L exp Z)) is a special semimaringale if Z is exponenially special. Proof. Observe ha Lexp Z)) = exp Z ) exp Z ) = exp Z ) exp iz )) exp Z ). By definiion if Z is exponenially special, hen exp Z ) is a special semimaringale. Evidenly, Z and Z are boh càg lefconinuous) and adaped, hence, hey are predicable and locally bounded. Consequenly, exp Z ) is locally bounded and predicable process, and he asserion follows from Proposiion 2.51 in Jacod 1979). Definiion For any complexvalued semimaringale Z wih Z =, we call Z := L exp Z)) he exponenial ransform of Z. 13
20 Lemma Le Z = Z iz exponenial ransform is given by be a complexvalued semimaringale wih Z =. Is Z = Z 1 2 Z, Z c e x 1 x) µ Z, ) where f µ Z sands for he inegral of f wih respec o he jump measure µ Z of Z. Proof. By definiion, Z = Lexp Z)). Applying Iô s formula, we obain he following expression for he exponen of Z exp Z ) = exp Z iz ) = 1 exp Z ) Z exp Z ) iz 1 2 exp Z ) Z, Z c 1 2 exp Z ) Z, Z c i exp Z ) Z, Z c s [exp Z s ) exp Z s ) exp Z s ) Z s i Z s )] = 1 exp Z ) Z 1 2 exp Z ) Z, Z c s exp Z s ) [exp Z s ) 1 Z s ]. Noe ha exp Z ) saisfies he equaion: exp Z ) = 1exp Z ) Z. Hence, we can wrie exp Z ) in he form of a sochasic exponenial. More specifically, exp Z ) = E Z) = E Z 1 ) 2 Z, Z c ex 1 x) µ Z ) By Lemma 2.4 5) in Kallsen and Shiryaev 22), L EX)) = X X if X 1 ouside some evanescen se. Observe ha Z = exp Z) 1) 1 and Z =, since we assumed ha Z =. Thus, Lexp Z)) = Z and he asserion follows. Le H = H ih be a predicable complexvalued process in C d such ha is real and imaginary pars are boh inegrable wih respec o X, i.e H, H LX) for he deailed descripion of he se LX) see Jacod 198) and Chaper 2 in Lipcer and Shiryaev 1986)). Wih respec o Proposiion 5.2 in Goll and Kallsen 2), he semimaringale H X admis he following represenaion: H X = H X c H x1 cx) µ X ν X ) H x1 x) µ X H B, ) 14
21 where B = B x1 cx) hx)) ν X is a predicable process of finie variaion and he se is defined by := { ω,, x) Ω R : x > 1 or H ω)x > 1 } P B d. Definiion Le H LX) such ha H X is a complexvalued exponenially special semimaringale. The Fourier cumulan process K X H) ) of X in H is defined as he compensaor of he special semimaringale L exp H X )), which means ha L exp H X )) K X H) is a local maringale. Proposiion Le H LX) such ha H X is exponenially special. Then he Fourier cumulan process of X in H is given by K X H) = H s b s ds 1 2 Hs C s H s ds exp H s x ) 1 H s hx) ) F s dx) ds. Proof. Wih Lemma 1.4.6, we have L exp H X )) = H X 1 H X, H X c 2 exp H s x ) 1 H s x ) µ X ds, dx) ) Plugging ) ino ), we obain: L exp H X )) = H B H X ) c 1 H X, H X c H x1 x) µ X 2 H x1 cx) µ X ν X) exp H x ) 1 H x ) µ X ν X) exp H x ) 1 H hx) H x H x1 cx) ) ν X. Since we assumed ha H X is exponenially special, wih Lemma 1.4.4, i follows ha L exp H X )) is a special semimaringale. Hence, H x1 x) µ X A loc in he above decomposiion, and correspondingly H x1 x) µ X A loc. By Proposiion II.1.28 in Jacod and Shiryaev 1987), we have H x1 x) µ X = H x1 x) µ X ν X) H x1 x) ν X. 15
22 Consequenly, L exp H X )) = H B H X ) c 1 H X, H X c 2 exp H x ) 1 H hx) ) ν X exp H x ) 1 ) µ X ν X). Thus, he compensaor of L exp H X )), which is he predicable process of finie variaion in he canonical decomposiion of he special semimaringale L exp H X )), is given by K X H) = H B 1 H X, H X c exp H x ) 1 H hx) ) ν X. 2 By Theorem in Lipcer and Shiryaev 1986) H X ) c = H X c. Hence, K X H) = H s b s ds 1 2 Hs C s H s ds exp H s x ) 1 H s hx) ) ν X ds, dx), which complees he proof. Remark Recall ha H = H ih. One can easily check, ha K X H) = H s ih s ) b s ds i 1 2 H s C s H s ds H s C s H s ds H s C s H s ds ) ) exp H s ih s ) x 1 H s ih s ) hx) ν X ds, dx). In a special case, when H =, he Fourier cumulan process of X in H coincides wih he Laplace cumulan process of X in H inroduced in Kallsen and Shiryaev 22). Proposiion Le H LX) such ha H X is exponenially special. Then K X H) is he exponenial compensaor of H X. More specifically, he process U, defined by U := exp H X K X H) ) = exp H X ) E K X H) ) is a local maringale. = E H X c exp H x ) 1 ) µ X ν X)), )
23 Proof. Observe ha, by definiion, he Fourier cumulan process K X H) is a predicable process of finie variaion. Is jumps are given by K X H) = H B exp H x ) 1 H hx) ) ν X {} dx) = exp H x ) 1 ) ν X {} dx) =, because X is a quasilefconinuous process. Hence, E K X H)) = exp ) K X H). Noe ha wih equaion ), we obain exp H X ) = E H X 1 H X, H X c exp H x ) 1 H x ) ) µ X. 2 Applicaion of he Yor s formula EX)EY ) = EX Y [X, Y ]) yields exp H X K X H) ) = E H X 1 H X, H X c H B 2 exp H x ) 1 H x ) µ X 1 2 exp H x ) 1 H hx) ) ) ν X. H X, H X c The canonical represenaion of he semimaringale H X saisfies: H X = H B H X c H hx) µ X ν X) H x hx)) µ X. Furhermore, he process exp H x ) 1 H hx) ) ν X belongs o A loc, herefore exp H x ) 1 H hx) ν X A loc, and i follows wih Proposiion II.1.28 in Jacod and Shiryaev 1987) ha exp H x ) 1 H hx) ) ν X = exp H x ) 1 H hx) ) µ X ν X) Hence, sraighforward calculaions yield exp H x ) 1 H hx) ) µ X. exp H X K X H) ) = E H X c exp H x ) 1 ) µ X ν X ) ) and he saemen of he proposiion follows. 17
24 Remark By Lemma 2.15 in Kallsen and Shiryaev 22), a realvalued semimaringale, which is exponenially special has a realvalued exponenial compensaor, which is up o indisinguishabiliy unique. However, he complexvalued exponenial compensaor V = V iv of he complexvalued semimaringale Z is generally no unique. Indeed, exp Z Z V ) = exp Z Z V ) exp 2kπi) = exp Z Z V i V 2kπ))), where k Z. Lemma Le X be an valued imeinhomogeneous Lévy process and H : R C d be a deerminisic funcion inegrable wih respec o X and such ha exponenially special complexvalued semimaringale. Then E exp H s dx s = exp K X H) ). H s dx s is an Proof. Since X is a imeinhomogeneous Lévy process, is characerisics are deerminisic, and i follows ha K X H) : R C d is a deerminisic funcion. Wih respec o Proposiion we have exp H X K X H) ) M loc. Hence, E exp H s dx s K X H) = E exp H s dx s exp K X H) ) F = 1. Noe ha exp K X H) ) is F measurable, herefore we can ake his erm ou of he condiional expecaion, and he saemen of he Lemma follows. Remark Noe ha, if he funcion H = H ih in Lemma is such ha Re H ) = H = for all, hen H s dx s = i H s dx s is auomaically exponenially special, because exp i H s dx s has bounded jumps, and herefore, is a special ) semimaringale. Thus, we obain E exp i H = exp K X ih ) ) s dx s = exp i H s b s ds 1 H s C s H s ds 2 exp ) ih s x 1 ih s hx) 18 ) ν X ds, dx).
25 Corollary Le X be a Lévy process and H : R be a deerminisic funcion inegrable wih respec o X. Then he characerisic funcion ψ, z) of he following represenaion: ψ, z) := E exp iz H s dx s = exp where θ ) sands for he logmomen generaing funcion of X 1. θ izh s ) ds, H s dx s admis Corollary Le L 1, L 2,..., L N be N Lévy processes wih characerisic riples B k h) = b k h), C k = C k, ν k = ν k), k = 1,..., N and T := T, T 1,..., T N ) be a finie pariioning of he inerval [, T ], where T = and T N = T. Furhermore, le X be a imeinhomogeneous Lévy process given by equaion 1.2.6) in Lemma and φ : R be a deerminisic funcion inegrable wih respec o X. Then for all [, T ] and k = 1,..., N holds E exp iz φ s dx s = = N 1 Tk 1,T k ]) exp k=1 T k 1 θ k izφ s ) ds k 1 T m m=1 T m 1 θ m izφ s ) ds = exp θizφ s ) ds, where z R, θz) = funcion of L k 1. N 1 Tk 1,T k ])θ k z) and θ k ) denoes he logmomen generaing k=1 19
26
27 Chaper 2 CrossCurrency Marke Model I is naural o expec ha flucuaions of ineres raes and exchange raes are highly correlaed. This fac should be refleced in he valuaion and hedging of he whole range of foreign and crosscurrency derivaive securiies, which are raded on he marke. The heory of ineresrae modelling was originally based on he specificaion of a cerain dynamics for he insananeous shor rae process. On he firs glance, such a modelling approach appears o be very convenien, because expressions for mos widely used insrumens on he fixed income marke, such as bonds for example, can be easily obained by noarbirage argumens as expecaions of some funcionals of he shor rae process. However, imehomogeneous) shor rae models 1 possess a few clear drawbacks. One of hem is heir inabiliy o reproduce exacly a given iniial erm srucure. Furhermore, from an economic poin of view i makes no sense o assume, ha he enire money marke is governed by only one explanaory variable. One of he firs alernaives o shorrae models has been proposed by Ho and Lee 1986), who modeled he evoluion of he enire yield curve in a binomialree seing. This idea was hen employed in coninuous ime seing by Heah, Jarrow and Moron 1992), who developed a quie general framework for he modelling of ineres rae dynamics. More specifically, heir approach is based on he exogenous specificaion of he dynamics of insananeous, coninuously compounded forward raes. Amin and Jarrow 1991) were probably he firs who exended he HJM mehodology by incorporaing foreign markes. Their model combines a fully developed sochasic heory of he erm srucure of ineres raes wih models for he valuaion of exchange rae and sock opions. Following heir approach, Frey and Sommer 1996) deal wih he valuaion and hedging of non pahdependen European opions on several underlyings in a mulicurrency seing. However, hey use bond prices insead of forward raes as primiives for he modelling of ineres rae risk. A special case of he HJM model wih sochasic volailiies is considered in Fracho 1995), where he bond price and he foreign exchange rae are assumed o be deerminisic funcions of a single sae variable. 1 A good overview of shorrae models accompanied by he discussions of heir advanages and drawbacks can be found in Brigo and Mercurio 21), Musiela and Rukowski 1997b) and Björk 1998). 21
28 Le us summarise he conen of his chaper. The firs secion is relaed o Björk, Di Masi, Kabanov and Runggaldier 1997), where a general semimaringale approach is used for modelling of he domesic bond marke. Having inroduced he basic assumpions, we specify he models for domesic and foreign bond prices hrough he dynamics of domesic and foreign insananeous forward raes. We derive expressions for he foreign spo and forward exchange raes and consider he problems of exisence of foreign maringale and foreign forward maringale measures respecively. As a byproduc we obain HJMype condiions on he coefficiens for he foreign marke. Finally, we invesigae he quesion of absence of arbirage on he inernaional bond marke. The second secion is moivaed by Eberlein and Özkan 22), where Lévy Libor model based on a imeinhomogeneous Lévy processes has been inroduced. Afer presenaion of several echnical resuls concerning he properies of he driving imeinhomogeneous Lévy process, we ranslae a few models from he semimaringale seing in he firs secion o he curren Lévy seing. In paricular, we specify he models for domesic and foreign insananeous forward raes, bond prices and foreign spo and forward exchange raes. This allows us o proceed wih he specificaion of he dynamics for domesic and foreign forward processes, followed by he models for domesic and foreign forward Libor raes. Finally, we consider he relaionship beween domesic and foreign fixed income markes in he discreeenor framework. 2.1 Term srucure modelling: semimaringale seing Le T be a fixed ime horizon for all marke aciviies. In our inernaional economy we consider M 1 markes currencies) indexed by i {,..., M}, where sands for he domesic marke. When working wih only wo markes, domesic and a foreign one, here will be no superscrip for he domesic marke quaniies. The choice of he domesic counry is arbirary and depends on he paricular pricing problem under consideraion. We rade domesic and foreign zerocoupon bonds of all mauriies wihin he ime inerval [, T ] and oher domesic and foreign asses or indices of asse values. There are no axes, ransacion coss, and invesors can rade he menioned domesic and foreign securiies wihou fricion a each rading day. A zerocoupon bond wih mauriy dae T T is a financial securiy paying o is owner one currency uni a a prespecified dae T in he fuure. In his seing we consider only defaulfree bonds; ha is, he possibiliy of defaul by he bond s issuer is excluded. The price of a zerocoupon bond of mauriy T a any insan T on he ih marke will be denoed by B i, T ). I is obvious ha B i T, T ) = 1 for any mauriy dae T T and for each currency i =,..., M. The funcion T B i, T ) for T is ofen referred o as erm srucure of discoun facors. I sars from B i, ) = 1 and is decreasing in T due o he posiiviy of ineres raes. Mos radiional sochasic ineres rae models 22
29 Figure 2.1.1: Term srucure of discoun facors for EUR, USD and GBP on July 2, 23. are based on he exogenous specificaion of a shorerm rae of ineres. We wrie r i ) o denoe shorerm ineres rae for riskfree borrowing or lending prevailing a ime over he infiniesimal ime inerval [, d] in he ih economy. We assume hroughou ha r i is a sochasic process wih almos all sample pahs inegrable on [, T ] wih respec o Lebesgue measure. Given shorerm ineres raes we define riskfree savings accoun by B i ) := exp r i s) ds, B i ) = 1, i =,..., M. Le f i, T ) be he forward ineres rae a dae T for insananeous riskfree borrowing or lending a dae T on he ih marke. I can be inuiively inerpreed as ineres rae over he infiniesimal ime inerval [T, T dt ] as seen from ime. For his reason f i, T ) is ofen referred o as insananeous forward rae. Assuming ha here exiss a measurable version of he process f i, ), [, T ], we obain he following equaion for he savings accoun: B i ) = exp f i s, s) ds, [, T ]. A widely acceped approach o he bond price modelling, due o Heah, Jarrow and Moron, is based on he exogenous specificaion of a family f i, T ), T T of forward raes. Given such a family f i, T ), he bond prices are hen defined by B i, T ) = exp T f i, u) du, [, T ]. 23
30 2.1.1 Domesic erm srucure In his secion we mainly sae he resuls obained in Eberlein and Özkan 22). We also show how he condiions, which guaranee he absence of arbirage in he domesic erm srucure model, presened in Proposiion 5.3 of Björk, Di Masi, Kabanov and Runggaldier 1997), can be reformulaed using exponenially special semimaringales inroduced in Kallsen and Shiryaev 22). For furher noaion we refer o Eberlein and Özkan 22) as well as Jacod and Shiryaev 1987). Le Ω, F T, F ) T, P) be complee filered probabiliy space, where he filraion saisfies he usual condiions. The predicable and opional σ fields on Ω [, T ] are denoed by P and O respecively. Furhermore, sands for he Eucledian vecor norm. Assumpion We assume ha i) The dynamics of he domesic insananeous forward raes for T [, T ] are given by df, T ) = α, T ) d ς, T ) dw δ, x, T )µ ν)d, dx), 2.1.1) where W is a sandard Brownian moion in, µ is he random measure of jumps of a semimaringale wih coninuous compensaor ν, i.e. here exiss λ : R B ) R such ha νd, dx) = λ, dx) d and ν, {}) =. ii) The filraion F) is he naural filraion generaed by W and µ, ha is F = σ {W s, µ[, s] G), B; s [, ], G B ), B N }, where N is he collecion of P null ses from F. iii) The sochasic basis Ω, F T, F ) T, P) has he predicable represenaion propery: any local maringale M is of he form M = M fs) dw s gs, x)µ ν)ds, dx), where f is a predicable process and g is a P B ) measurable funcion such ha for all finie fs) 2 ds <, gs, x) νds, dx) < P a.s. 24
31 iv) The coefficiens of he insananeous forward rae are coninuous in he second variable, where α : Ω [, T ] [, T ] R and ς : Ω [, T ] [, T ] are P B[, T ]) measurable and δ : Ω [, T ] [, T ] R is P B ) B[, T ]) measurable. The coefficiens saisfy he following condiions: Denoe := {s, u) R R s u T } hen for all, T ) / holds α, T ) = δx,, T ) =, ς, T ) =,..., ), whereas for all, T ) T T αs, u) du ds < P a.s., T T ςs, u) 2 du ds < P a.s., 2.1.2) T T δs, x, u) 2 du νds, dx) < P a.s. We pu T A, T ) := and D, x, T ) := T α, u) du, δ, x, u) du. T S, T ) := ς, u) du, Wih respec o Proposiion 5.3 in Björk, Di Masi, Kabanov and Runggaldier 1997), P is he domesic local maringale measure if and only if for any, T ) holds exp Ds, x, T )) 1 Ds, x, T )) νds, dx) < P a.s ) and A, T ) 1 2 S, T ) 2 exp D, x, T )) 1 D, x, T )) λ, dx) =, dp d a.e. One can easily check ha equaion 2.1.1) has he following soluion: f, T ) = f, T ) αs, T ) ds ςs, T ) dw s δs, x, T )µ ν)ds, dx). 25
32 The iniial values f, T ) are deerminisic and bounded in T. Noe ha due o assumpions 2.1.2) we made ono he coefficiens, he process f, T )) T T is a special semimaringale wih respec o he sochasic basis Ω, F T, P, F) T ) and is canonical decomposiion is given by he equaion above. The dynamics of he shor rae r) can be easily obained from r) = f, ), more precisely r) = f, ) αs, ) ds ςs, ) dw s δs, x, )µ ν)ds, dx). The price a ime of a domesic zerocoupon bond wih mauriy T is hen given by B, T ) = B, T ) exp rs) As, T )) ds Ss, T ) dw s Ds, x, T )µ ν)ds, dx) ) Noe ha, seing = T in he las equaion, we obain he following represenaion of he domesic savings accoun: 1 B) = B, ) exp As, ) ds Ss, ) dw s Ds, x, )µ ν)ds, dx). Le us denoe he discouned domesic bond price a ime by Equaion 2.1.4) yields Z, T ) = Z, T ) exp As, T ) ds Z, T ) := B, T ) B). Ss, T ) dw s Ds, x, T )µ ν)ds, dx). ) 2.1.5) Consider he process ZT := ln Z, T ))) T T, which is a logarihm of he T T discouned domesic bond price process. Lemma For each T [, T ] process Z T is a special semimaringale, and is semimaringale characerisics wih respec o he domesic local maringale measure P are given by: B e Z = As, T ) ds, 26
33 Z C e = Ss, T ) 2 ds, 2.1.6) ν ez [, ] G) = 1 {[,] G} s, Ds, x, T ))νds, dx). where G B R \ {})). Proof. Observe ha B ez is a predicable process of finie variaion. By Lemma I.3.1 in Jacod and Shiryaev 1987), i is a process wih locally inegrable variaion, i.e. B ez A loc, and he firs saemen follows. The jumps of B ez Z saisfy B e = D, x, T )µ{}, dx). Thus, γs, v)µ B e Zds, dv) = γs, Ds, x, T ))µds, dx) R for any measurable funcion γ on R R. Evidenly, for he P compensaor of µ B Z e have a similar propery. Noe ha for u R he process we Au) := iub e Z u2 2 C Z e R exp iu v) 1 iu v) ν ez ds, dv) is a predicable process of finie variaion. One can easily check ha for all [, T ] holds Au) c = and Au) =. Hence, wih respec o Theorem I.4.62 in Jacod and Shiryaev 1987), he Doléans Dade exponenial of Au) is given by EAu)) = exp Au) ). Consider he process Gu) := exp iu Z ) T Au) = exp iu Z T iuss, T ) dw s u 2 iuds, x, T )µ ν)ds, dx) Ss, T ) 2 ds 2 exp iuds, x, T )) 1 iuds, x, T )) νds, dx). 27
34 By Lemma 2.6 in Kallsen and Shiryaev 22), is sochasic logarihm LGu)) is given by LGu)) = iu Z T iuss, T ) dw s iuds, x, T )µ ν)ds, dx) exp iuds, x, T )) 1 iuds, x, T )) µ ν)ds, dx). Evidenly, LGu)) is a local maringale, and i follows from Lemma 2.4 6) in Kallsen and Shiryaev 22) combined wih Theorem I.4.62 in Jacod and Shiryaev 1987) ha Gu) is also a local maringale for each u R. According o Corollary II.2.48 in Jacod and Shiryaev 1987) his is equivalen o he fac ha semimaringale characerisics of Z are given by equaion 2.1.6). Lemma Condiion 2.1.3) is saisfied if and only if he process Z T is exponenially special. The exponenial compensaor of Z is given by is Laplace cumulan process K Z e wih Z K e = As, T ) 1 2 Ss, T ) 2 exp Ds, x, T )) 1 Ds, x, T )) λs, dx) ds ) Proof. Boh saemens follow from he resuls presened in Kallsen and Shiryaev 22). The firs one is saed in Lemma and 2). Then, wih respec o Lemma 2.15, Z has an exponenial compensaor ) V, which is a predicable process of finie variaion such ha exp Z Z V is a local maringale. Moreover, V is up o indisinguishabiliy unique. Noe ha Z is a quasilefconinuous process see I.2.25 in Jacod and Shiryaev 1987) for he definiion of quasilef coninuiy). Hence, by Theorems 2.19 and 2.18, he Laplace cumulan process of Z given by equaion 2.1.7) is is exponenial compensaor. Summing up, we can reformulae he Proposiion 5.3 in Björk, Di Masi, Kabanov and Runggaldier 1997) in he following way Proposiion The domesic erm srucure model is arbiragefree if and only if he logarihm of he discouned bond price process is an exponenially special semimaringale and is exponenial compensaor is equal o zero up o indisinguishabiliy. Observe ha for any fixed T [, T ] he process Z, T )) T saisfies he following sochasic inegrodifferenial equaion dz, T ) Z, T ) = S, T ) dw exp D, x, T )) 1) µ ν)d, dx) 28
35 wih he soluion given by Z, T ) = Z, T ) exp Ss, T ) dw s 1 Ss, T ) 2 ds 2 R r Ds, x, T )µ ν)ds, dx) exp Ds, x, T )) 1 Ds, x, T )) νds, dx) ) Le us consider wo mauriies T and U such ha T U. We inroduce he domesic forward process by F B, T, U) := B, T ) B, U). Applying Iô s formula for semimaringales see e.g. Proer 199)), we obain F B, T, U) = B, T ) B, U) exp As, T ) As, U)) ds Ss, T ) Ss, U)) dw s Ds, x, T ) Ds, x, U)) µ ν)ds, dx). Definiion Le δ > wih δ T T, hen he δforward)libor rae is defined by L, T ) := 1 δ F B, T, T δ) 1) = 1 ) B, T ) δ B, T δ) 1. The forward Libor rae L, T ) represens he addon rae prevailing a ime over he fuure ime inerval [T, T δ] wih he accrual period 2 δ >. Thus, a pary enering ino a conrac a ime o borrow one uni of he domesic currency over he inerval [T, T δ] will receive his one uni a ime T and will pay he lender 1 δl, T )) unis of he domesic currency a ime T δ. Wih respec o Theorem 2.1 in Eberlein and Özkan 22) he P dynamics of L, T ) saisfy δ dl, T ) 1 δl, T ) = S, T ) S, T δ)) dw S, T δ) d) exp D, x, T ) D, x, T δ)) 1) µ ν)d, dx) 2 Accrual period is expressed as a fracion of a year, for insance hreemonhs rae corresponds o δ = 1/4. 29
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