GENERALIZATION OF THE DYBVIG INGERSOLL ROSS THEOREM AND ASYMPTOTIC MINIMALITY

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1 Version Ocober 19, 21 To appear in Mahemaical Finance GENERALIZATION OF THE DYBVIG INGERSOLL ROSS THEOREM AND ASYMPTOTIC MINIMALITY VERENA GOLDAMMER AND UWE SCHMOCK Absrac. The long-erm limi of zero-coupon raes wih respec o he mauriy does no always exis. In his case we use he limi superior and prove corresponding versions of he Dybvig Ingersoll Ross heorem, which says ha long-erm spo and forward raes can never fall in an arbirage-free model. Exensions of popular ineres rae models needing his generalizaion are presened. In addiion, we discuss several definiions of arbirage, prove asympoic minimaliy of he limi superior of he spo raes, and illusrae our resuls by several coninuous-ime shor-rae models. 1. Inroducion To price long-erm conracs, like life insurance policies, praciioners model zerocoupon bond prices wih long-erm mauriies o find reasonable discoun facors. Empirical invesigaions of hese prices are difficul, since here are only zero-coupon bonds raded wih mauriy of up o 3 years, and for a life annuiy, for example, discoun facors for up o 1 years are needed, see e. g. Carriere To consruc reasonable models, we need o know how he long-erm zero-coupon raes behave. Dybvig, Ingersoll and Ross 1996 showed ha long-erm zero-coupon raes can never fall in an arbirage-free marke. Therefore, if he raes in a model decrease, i is no arbirage-free. This fundamenal heorem is par of exbooks, see e. g. Cairns 24, and can be used o consrain he parameers of facor models o avoid arbirage. Yao 1999 and El Karoui, Fracho and Geman 1998 discussed he longerm raes for several well-known models and used he heorem in his conex. In he lieraure here are wo approaches o prove he Dybvig Ingersoll Ross heorem. The firs approach consrucs an arbirage sraegy, if long-erm raes fall. Dybvig e al. provide an arbirage sraegy for a general infinie sae space in he appendix of heir paper. In he case of finiely many saes hey consruc a second arbirage sraegy, which was made rigorous by McCulloch 2. Recenly, Schulze 27 showed a furher arbirage sraegy using anoher definiion of arbirage han Dybvig e al. The second approach o prove he Dybvig Ingersoll Ross heorem is o assume he exisence of an equivalen maringale measure. Hubalek, Klein and Teichmann 22 gave a general proof in his seing. 21 Mahemaics Subjec Classificaion. 91G3. Key words and phrases. Dybvig Ingersoll Ross heorem, ineres rae models, long-ime forward rae, long-ime zero-coupon rae, asympoic monooniciy, asympoic minimaliy. This work was financially suppored by he Chrisian Doppler Research Associaion CDG. The auhors graefully acknowledge he fruiful collaboraion and suppor by he Bank Ausria and he Ausrian Federal Financing Agency ÖBFA hrough CDG and he CD-Laboraory for Porfolio Risk Managemen PRisMa Lab hp:// 1

2 2 V. GOLDAMMER AND U. SCHMOCK In his paper we presen a version of he Dybvig Ingersoll Ross heorem, which is more general, because Dybvig e al. as well as Hubalek e al. require he exisence of he long-erm limi of he zero-coupon raes. We show in wo differen ways ha he limi superior of he zero-coupon raes and he forward raes never fall, which is called asympoic monooniciy. For he firs approach, we assume he exisence of an equivalen maringale measure. This proof is inspired by he proof of Hubalek e al. For he second approach, we assume ha here is no arbirage opporuniy in he limi wih vanishing risk, and show again ha asympoic monooniciy holds. Besides he main heorem, Dybvig e al. showed ha he long-erm zero-coupon rae equals is minimum fuure value, if he sae space is finie. Using a sricer definiion of no-arbirage, Schulze exended his resul o infinie sae spaces. Again, he auhors assume he exisence of he long-erm limi. Here we sae condiions for asympoic minimaliy of he limi superior of he zero-coupon raes. Tha means, he limi superior of he long-erm limi of he zero-coupon raes is he larges random variable, which is known a his ime and dominaed by he fuure limi superior of he long-erm limi. The ouline of he paper is he following. In Secion 2 we give he general noaion, sae he main heorem abou asympoic monooniciy, and jusify he use of he limi superior from he invesor s poin of view. Furhermore, we specify condiions for asympoic minimaliy and define wo noions of an arbirage opporuniy in he limi. In Secion 3, we provide several ineres rae models, where he longerm limi of he zero-coupon raes does no exis, o show ha our generalizaion of asympoic monooniciy is useful. Furher examples illusrae he condiions for asympoic minimaliy. In Secion 4 we prove wo auxiliary lemmas. Secion 5 conains he proof for asympoic monooniciy and minimaliy using an equivalen maringale measure. The proofs using arbirage argumens are given in Secion Saemen of he generalized Dybvig Ingersoll Ross heorem and asympoic minimaliy 2.1. Noaion. Le Ω, F, P be a probabiliy space and F = {F } a filraion of F wih a discree-ime parameer N or a coninuous-ime parameer [,. For every mauriy T N or T,, respecively, we assume ha he corresponding zero-coupon bond price process P, T wih {, 1,...,T} or [,T], respecively, is sricly posiive and F-adaped wih normalizaion P T,T=1. Define he zero-coupon rae for mauriy T> in he discree-ime case by R, T :=P, T 1/T 1, {, 1,...,T 1}, 2.1 and in he coninuous-ime case by log P, T R, T :=, [,T. 2.2 T The arbirage-free forward rae F s,, T for a loan over he fuure ime period [, T ], conraced a ime s, is in he discree-ime case defined by F s,, T := 1/T P s, 1, s, {, 1,...,T 1}, s, 2.3 P s, T

3 DYBVIG INGERSOLL ROSS THEOREM AND ASYMPTOTIC MINIMALITY 3 and in he coninuous-ime case by F s,, T := 1 T P s, log, s, [,T, s. 2.4 P s, T For boh ime scales we define he long-erm spo rae process by l := lim sup R, T = lim ess sup R, T, n T>n, 2.5 and he long-erm forward rae process by l F s, := lim sup F s,, T = lim ess sup F s,, T, s, 2.6 n T>n Remark 2.7. For clariy we wan o poin ou ha for each he limi superior l of he zero-coupon raes is he poinwise infimum of {Rn} n N, where each Rn denoes he essenial supremum of {R, T } T>n. The essenial supremum is he smalles F -measurable upper bound. Tha means, RnisanF -measurable random variable, PRn R, T = 1 for all T>n, and every oher random variable X dominaing a. s. hese zero-coupon raes saisfies PX Rn = 1. In paricular, he essenial supremum is uniquely deermined up o a se of P-measure zero. The exisence of he essenial supremum for a collecion of random variables is proved, for example, in [7, Appendix A.5]. Noe ha PRm Rn = 1 for all m n, hence he infimum of {Rn} n N is P-almos surely equal o he almos surely exising poinwise limi. The limi superior of he forward raes is o be undersood in an analogue manner. In comparison o Dybvig e al. and Hubalek e al., we do no assume ha he long-erm limis of he zero-coupon raes or he forward raes exis. In Subsecion 3.1 we presen exensions of popular ineres rae models, which need his generalizaion. From he invesor s poin of view, he limi superior of he zero-coupon raes is he naural definiion, because he/she prefers for he long-erm invesmen hose zero-coupon bonds, which give a high invesmen reurn based on he informaion a ime. The following lemmas proved in Secion 4 show ha he long-erm spo rae l can indeed be approximaed by invesing in a zero-coupon bond wih a suiable mauriy, which is chosen based on he informaion available a ime. Furhermore, l agrees wih he long-erm forward raes, so i suffices o invesigae he behaviour of he long-erm spo raes. Lemma 2.8. Given, here exiss a sequence of F -measurable random mauriies 1 T n :Ω n,, each one aking only a finie number of values, such ha l a.s. = lim R, T n. n Lemma 2.9. The long-erm forward and spo raes are almos surely equal, meaning ha l F s, a.s. = ls for all s. 1 In he discree-ime seing, he random mauriies have o be ineger-valued. This also applies o Remark 2.27, Definiion 2.29, Theorem 2.34 and is corollaries. Since T n aains only a finie number of values, R, T n isf -measurable.

4 4 V. GOLDAMMER AND U. SCHMOCK 2.2. Resuls using he exisence of a forward risk neural probabiliy measure. Par of our main resuls, namely asympoic monooniciy in Theorem 2.17 and asympoic minimaliy in Theorem 2.21 are based on he following wo condiions: Condiion 2.1. We say ha his condiion holds for imes s and wih s<, if here exis a probabiliy measure Q s, on Ω, F, which is equivalen o P F, and a T >such ha, for all T T, P s, T P s, E Qs, [P, T F s ] a. s This condiion is sufficien for asympoic monooniciy. For asympoic minimaliy in Theorem 2.21 we need he sronger condiion: Condiion 2.12 Exisence of forward ime s risk neural probabiliy measure. We say ha his condiion holds for imes s and wih s<, if here exiss a probabiliy measure Q s, on Ω, F, which is equivalen o P F such ha, for all T>, P s, T a.s. = P s, E Qs, [P, T F s ] We call Q s, he forward ime s risk neural probabiliy measure for mauriy. Condiion 2.12 says ha, simulaneously for all mauriies T>, he arbiragefree forward price P s, T /P s,, conraced a ime s for he T -mauriy zerocoupon bond a ime, can be expressed as he F s -condiional expecaion of he price P, T a ime wih respec o he measure Q s,. Remark Suppose a money marke accoun B wih N or [, is given by a sricly posiive and F-adaped process. Then he following consrucion yields a model, where a forward risk neural probabiliy measure Q s, exiss simulaneously for all imes s and wih s<. If Q is a probabiliy measure such ha B /B T is Q-inegrable for every T>, hen we can define zero-coupon bond prices by [ B ] F P, T =E Q = B [ B ] F E Q, [,T], 2.15 B T B B T and he forward ime s risk neural probabiliy measure Q s, on Ω, F by for every s [,. Since dq s, dq = B s P s, B [ B ] s Fs a.s. E Q =1, P s, B i follows by using Bayes formula and he ower propery, ha [ E Qs, [P, T F s ] a.s. B [ s B ] ] F a.s. = E Q E Q F s = P s, T P s, B B T P s,, hence Condiion 2.12 holds for all imes s and wih s<. We will use his consrucion for he examples in Secion 3.

5 DYBVIG INGERSOLL ROSS THEOREM AND ASYMPTOTIC MINIMALITY 5 Example In he discree-ime case, le {r } N be an ineres rae process, which is F-adaped and 1, -valued. We define he money marke accoun by B = B 1 + r i, N, i=1 where B is sricly posiive and F -measurable. For a probabiliy measure Q such ha B /B T is Q-inegrable for every T N, we define he corresponding zerocoupon bond prices by 2.15, which means [ T ] 1 P, T =E Q 1+r i=+1 i F, {, 1,...,T}. By Remark 2.14, a forward risk neural probabiliy measure exiss in his model simulaneously for all imes s, N wih s<. The following resul, which we prove in Secion 5, saes ha he long-erm spo and forward raes, given by 2.5 and 2.6, respecively, almos surely never fall. This is also called asympoic monooniciy. Under he assumpion, ha he long-erm limis of he spo and forward raes exis, his is he so-called Dybvig Ingersoll Ross heorem. Economically, from ime s o a laer ime, he available informaion increases, so a more informed decision concerning he bes zero-coupon bonds for long-erm invesmens can be made. However, o ake advanage of his addiional informaion, he gains during [s, ] on zero-coupon bonds wih a large mauriy T should be negligible compared o he oal gains unil T, a leas in he limi, see Example 2.18 for a counerexample. Therefore, in a reasonable economic environmen as specified by Condiion 2.1, he long-erm spo rae a ime should be greaer han he long-erm spo rae a ime s. Theorem If Condiion 2.1 holds for imes s and wih s<, hen a ls l a. s. and b l F s, s l F, a. s. for all s s and. Examples 3.16, 3.2 and 3.22 show ha he inequaliies can be sric everywhere on Ω. Example Given s <, he deerminisic, coninuous-ime example wih P s, T =e T s for all T s and P, T = 1 for all T shows, ha ls =1>l = can happen, if here is arbirage by invesing in he zero-coupon bonds wih mauriy T>. To exploi he arbirage in his example, sell a ime s one -mauriy bond and buy e T zero-coupon bonds of mauriy T wih T>. Asympoic monooniciy raises he quesion, wheher ls is he larges F s - measurable random variable, which is almos surely dominaed by l. To discuss his quesion, we need he following definiion. Definiion Le Ω, F, P be a probabiliy space and G a sub-σ-algebra of F. For an R-valued random variable X, we define he upper G-measurable envelope X G as he essenial infimum of all R-valued, G-measurable random variables Z wih Z X a. s. Similarly, we define he lower G-measurable envelope X G as he essenial supremum of all R-valued, G-measurable Z wih Z X a. s. Observe ha X G X X G a. s., and asympoic monooniciy implies ls l Fs a. s. Noe ha even in case of convergence of he zero-coupon raes R, T

6 6 V. GOLDAMMER AND U. SCHMOCK o l as, he exisence of a forward risk neural probabiliy measure does no imply asympoic minimaliy in he sense ha ls a.s. = l Fs, as Example 3.22 shows. In his example of a sochasic ineres rae model, he long-erm spo rae l jumps up from o 1 a ime = 1 wih probabiliy 1. The following, purely analyical condiion is a convenien addiional assumpion for proving asympoic minimaliy in Theorem 2.21 below. Definiion 2.2. Le Ω, F, P be a probabiliy space and G a sub-σ-algebra of F. An R-valued random variable X is said o dominae he random variables {X } > in he G, P-superexponenial sense along a G-measurable subsequence, if 2 lim inf 1 log E[ max{x X, } G ] a.s. =. Theorem 2.21 Asympoic minimaliy. Le Condiion 2.12 be saisfied for imes s and wih s<. Assume in addiion ha he upper F s -measurable envelope V F s of he limiing annual discoun facor a ime given by 3 { 1/l+1 in he discree-ime case, V = 2.22 exp l in he coninuous-ime case, dominaes {P, + u 1/u } u> in he F s, Q-superexponenial sense along an F s - measurable subsequence, which means ha [ lim inf EQ max{p, T V F s T, } ] 1/T a.s. Fs = Then ls a.s. = l Fs and l F s, s =l F, Fs a. s. for all s s and. Remark For asympoic minimaliy we canno weaken he requiremens for he probabiliy measure, because we use he probabiliy measure in Condiion 2.1 o show asympoic monooniciy, bu we need also he reversed inequaliy P s, T P s, E Qs, [P, T F s ] a. s., for he esimae in 5.15 in he proof of Theorem a.s. = exp l Fs, respec- Remark Noe ha V F s ively, and since a.s. V = { lim 1 inf R,T +1 lim inf exp R, T a.s. = 1/l Fs + 1 and V F s in he discree-ime case, in he coninuous-ime case, we obain a.s. V = lim inf P, T 1/T Remark If here exiss a sequence {T n } n N of F s -measurable random imes, aking a mos counable many values in, and ending o infiniy as n, such ha for every ε> here exiss n ε N saisfying P, T n 1/Tn V F s + ε a. s for all n n ε, hen 2.23 holds. Due o 2.26 and V V F s, his uniformiy cerainly holds for all s [,] simulaneously when F is finie and he limi 2 We use here he convenion log =. Analogously o 2.5 and 2.6, he limi inferior is he limi as n of he essenial infima over all >n. 3 We use here he convenions 1/ =, 1/ =, exp =, and exp =.

7 DYBVIG INGERSOLL ROSS THEOREM AND ASYMPTOTIC MINIMALITY 7 inferior in 2.26 is aained along a deerminisic sequence {T n } n N. The laer condiion in urn is saisfied when l = lim n R, T n a.s Resuls using differen noions for absence of arbirage. In he las secion we assumed he exisence of a forward risk neural probabiliy measure, resp. ha Condiion 2.1 holds. The second approach uses no-arbirage argumens o show asympoic monooniciy and minimaliy. The nex definiion gives wo differen noions of arbirage and applies o discree as well as coninuous ime. I is inspired by he definiion of arbirage in he limi, which is used by Schulze, and he definiion of arbirage used in Dybvig e al. Definiion Given imes s<, he zero-coupon bonds wih mauriy T provide an arbirage opporuniy in he limi for imes s and, if here exis a sequence {ϕ n,ψ n } n N of F s -measurable, R 2 -valued porfolio composiions and a sequence {T n } n N of F s -measurable random mauriies T n :Ω n,, each one aking only a finie number of values, such ha a V n s :=ϕ n P s, T n +ψ n P s, a.s. = for all n N, b Plim inf n V n > >, where V n :=ϕ n P, T n +ψ n, and c lim inf n V n a.s. We say ha he zero-coupon bonds provide an arbirage opporuniy in he limi wih vanishing risk for imes s and, if c is replaced by d for every ε> here exiss n ε N such ha V n ε a. s. for all n n ε. Remark 2.3. Par a in Definiion 2.29 always holds if ψ n := ϕ n P s, T n /P s, for all n N. Remark Since d implies c, he assumpion of no arbirage opporuniy in he limi is sronger han no arbirage opporuniy in he limi wih vanishing risk. If F is finie, hen poinwise implies uniform convergence, hence c implies d and boh noions of arbirage are equivalen. Example 3.26 below shows ha even he sronger assumpion of no arbirage opporuniy in he limi does no imply he exisence of a forward risk neural probabiliy measure in Condiion Even he weaker Condiion 2.1 does no hold in his example. Lemma 2.32 below shows ha Condiion 2.12 implies he weaker no-arbirage condiion, which by Theorem 2.33 is sufficien for asympoic monooniciy. Acually, he no-arbirage condiion can be furher weakened by excluding only arbirage due o a posiive invesmen in he long-erm zero-coupon bonds. The lemma and he following heorems are proved in Secion 6. Lemma If here exiss a forward ime s risk neural probabiliy measure for mauriy as in Condiion 2.12 wih s<, hen here is no arbirage opporuniy in he limi wih vanishing risk for imes s and. Theorem Consider imes s<. Assume ha here is no arbirage opporuniy in he limi wih vanishing risk for imes s and in he sense of Definiion 2.29 by invesing in he long-erm zero-coupon bonds wih ϕ n for all n N. Then ls l a. s. and l F s, s l F, a. s. for all s s and. For he remaining resuls, we need he sronger assumpion of no arbirage opporuniy in he limi, however, for Theorem 2.34 below we only have o exclude his limiing arbirage by shor-selling of he long-erm zero-coupon bonds. The

8 8 V. GOLDAMMER AND U. SCHMOCK heurisic jusificaion of he following heorem is as follows: If, wih sricly posiive probabiliy, he wors long-erm spo rae, which we will incur by placing our invesmen orders for ime already a an earlier ime s based on he informaion available a s, is sricly larger han he bes long-erm spo rae we can earn by invesing already a ime s, hen he prices of he long-erm zero-coupon bonds mus fall subsanially during [s, ], offering an arbirage possibiliy in he limi by shor-selling hese bonds. Theorem Consider imes s<. Assume ha here is no arbirage opporuniy in he limi for imes s and in he sense of Definiion 2.29 by shorselling he long-erm zero-coupon bonds wih ϕ n for all n N. Then for every sequence {T n } n N of F s -measurable random mauriies T n :Ω n,, each one aking only finiely many values, lim inf R, T n ls a. s n F s Examples 3.3 and 3.2 show ha, for cerain sequences of random or even deerminisic mauriies, he inequaliy in 2.35 can be sric everywhere on Ω. In hese models, he limi of R, T n asn does no exis. Noe ha he deerminisic model of Example 2.18, which admis an arbirage possibiliy by invesing in he zero-coupon bonds wih mauriy T>, saisfies he assumpions of Theorem Using he definiion of he long-erm zero-coupon rae l from 2.5, he F s - measurabiliy of ls and he definiion of he lower F s -measurable envelope in Definiion 2.19, we obain from Theorems 2.33 and 2.34: Corollary 2.36 Asympoic minimaliy. Consider s <. If here is no arbirage opporuniy in he limi for s and in he sense of Definiion 2.29, hen lim inf R, T ls lim sup R, T a. s F s F s In paricular, if lim R, T exiss a. s., hen ls a.s. = l Fs. If he limi of R, T asdoes no exis a. s., we migh sill ge asympoic minimaliy. Using asympoic monooniciy and Theorem 2.34, each sequence {T n } n N of F s -measurable random mauriies, each one aking only finiely many values, saisfies lim inf R, T n ls l Fs, a. s., n F s if here is no arbirage opporuniy in he limi. If a special sequence of mauriies saisfies addiionally he reversed inequaliy, we have asympoic minimaliy. Noe ha he sequence from Lemma 2.8 canno be used in general, because hese mauriies are only F -measurable. Corollary 2.38 Asympoic minimaliy. Consider s <. If here is no arbirage opporuniy in he limi for imes s and in he sense of Definiion 2.29 and if here exiss a sequence {T n } n N of F s -measurable random mauriies T n : Ω n,, each one aking only finiely many values, such ha hen ls a.s. = l Fs. l Fs lim inf n R, T n a. s., 2.39

9 DYBVIG INGERSOLL ROSS THEOREM AND ASYMPTOTIC MINIMALITY 9 Remark 2.4. In Corollaries 2.36 and 2.38, i is acually sufficien o assume ha here is no arbirage opporuniy in he limi wih vanishing risk for imes s and by invesing in he long-erm zero-coupon bonds wih ϕ n for all n N and ha here is no arbirage opporuniy in he limi by shor-selling he long-erm zero-coupon bonds wih ϕ n for all n N. Remark Using he almos sure equivalence of he long-erm spo and forward raes in Lemma 2.9, we can also ransfer Theorem 2.34 and is corollaries o he long-erm forward raes. We refrain from spelling ou he deails. Remark We could relax Definiion 2.29d o e here exiss n N, such ha he negaive pars Vn :=max{, V n } for all n n are uniformly inegrable and lim inf n V n a.s. and ge more limiing arbirage opporuniies in his way. This would srenghen he no-arbirage assumpion. However, using a more general version of Faou s lemma for condiional expecaions 4, he proof of Lemma 2.32 carries over, where he exisence of a forward risk neural probabiliy measure for imes s and in Condiion 2.12 implies no arbirage in he limi wih vanishing risk. Hence his Condiion is sill sronger. Therefore, his sronger no-arbirage assumpion would no be srong enough o imply asympoic minimaliy, as Example 3.22 illusraes. In paricular, he limiing arbirage sraegies given here canno saisfy e. Remark The proofs of he above heorems, corollaries and lemmas do no use pah properies of he processes {P, T } T like being càdlàg or a semimaringale, and we also do no need a bank accoun process or addiional assumpions on he filraion F like conaining all null ses of F or being righ-coninuous. Furhermore, we allow for PP, T > 1 >, which can happen for models wih negaive ineres raes like he Vasiček model or he Heah Jarrow Moron model. 3. Examples In his secion we show wih illusraive examples firs, ha he long-erm zerocoupon raes do no always exis and our generalizaion of he Dybvig Ingersoll Ross heorem is herefore useful. In hese examples asympoic monooniciy holds for he limi superior of he zero-coupon raes, resp. he forward raes. Four furher examples illusrae he asympoic minimaliy condiions as explained in Secion 2. In Example 3.2 we describe a very simple sochasic ineres rae model wih Ω ={, 1}. Alhough his model provides no arbirage opporuniy in he limi and a forward risk neural probabiliy measure exiss, asympoic minimaliy does no hold. There does no exis a deerminisic sequence {T n } n N wih T n such ha 2.39 holds. Therefore, he absence of arbirage opporuniies in he limi for imes s and wih s<or he exisence of a forward risk neural probabiliy measure is no sufficien for asympoic minimaliy in he sense of ls a.s. = l Fs. These condiions are no even necessary, see Example Example 3.24 shows ha asympoic minimaliy is no an inerval propery, meaning ha for imes < s<<uhe propery ls a.s. = lu Fs does no imply l a.s. = lu F. Furhermore, he example shows ha even if here is no arbirage opporuniy in he limi for imes s and u, i is possible o have an arbirage opporuniy for imes and u. 4 See Faou s lemma a en.wikipedia.org/wiki/, version of Ocober 11, 28.

10 1 V. GOLDAMMER AND U. SCHMOCK All hese examples are coninuous-ime shor-rae models, and here exiss a forward risk neural probabiliy measure for all imes defined in Condiion 2.12 by consrucion as poined ou in Remark Hence by Lemma 2.32, hese models do no provide an arbirage opporuniy in he limi wih vanishing risk. For a model no saisfying Condiion 2.12, see Example The general se-up of hese models wih he excepion of he las one is given as follows. For a given F-progressive ineres rae inensiy process {r } wih locally inegrable pahs, we define he money marke accoun by B = exp r u du, [,. Assume ha 1/B is Q-inegrable for every >. Using 2.15, he zero-coupon bond prices are given by P, T =E Q [exp ] r u du F, T. 3.1 Therefore, he definiion of R, T in 2.2 implies R, T = 1 [ ] T log E Q exp r u du F, <T Models where he limis of he zero-coupon raes do no exis. In he following examples we discuss models, where he limis of he zero-coupon raes R, T for do no exis. The idea is o vary he behaviour of he shor rae on longer and longer ime periods o ge oscillaing means. We illusrae his wih a simple deerminisic model and hen wih wo shor-rae models having an exponenially affine erm srucure. More specifically, we consider a varian of he familiar Vasiček model wih ime-dependen coefficiens, which was proposed by Vasiček 1977 and Hull and Whie 199. Secondly, we sudy he behaviour of he long-erm spo rae in he model of Cox, Ingersoll and Ross 1985 wih ime-dependen coefficiens bu consan dimension. In boh examples he mean level or he volailiy of he shor rae changes cyclically bu deceleraes over ime. An economical jusificaion for his behaviour can be he dependence on he business cycles, which become longer and longer. So, if he lenghs of he business cycles increase exponenially, hen he limis of he zero-coupon raes migh no exis, as our examples show. In our las example, we use he well-known Heah Jarrow Moron framework, proposed in [9], and choose an oscillaing bu decaying volailiy funcion for he forward raes such ha he limis of he zero-coupon raes do no exis, see Example 3.16 below. Since we specialize o a deerminisic volailiy funcion in produc form, his example is relaed o he exended Vasiček model, cf. [14, Secion 1.2]. Example 3.3 Deerminisic model. Define he se A = [ 1 3, 1 [ 2 2k+1, 2 2k+2, 3.4 k= he càdlàg ineres rae inensiy r =1 A for, and he coninuous funcion R A, T = 1 T 1 A u du = λa [, T ], <T, 3.5 T

11 DYBVIG INGERSOLL ROSS THEOREM AND ASYMPTOTIC MINIMALITY 11 where λ denoes he Lebesgue measure. Since {r } is deerminisic, 3.2 implies R, T =R A, T for all <T. Noe ha T 1 is a local minimum of R A, if and only if here exiss n N wih T =2 2n+1. Since λa [, 2 2n+1 ] = 2 n k+1 = n 3 k= = 22n+1, n N, 3 we have R A, 2 2n+1 =1/3. Furhermore, T 2 is a local maximum of R A, if and only if here exiss n N wih T = 2 2n+2. Since λa [, 2 2n+2 ] = λa [, 2 2n+1 ]+2 2n+1 =2 2n+3 /3, we ge R A, 2 2n+2 =2/3. Hence, we have R A,T [1/3, 2/3] for all T 1, and he inerval [1/3, 2/3] is also he se of all accumulaion poins of {R A,T} T>. Since R A, T R A,T 2/T for <T, he laer is also rue for {R A, T } T>, in paricular he limi of R, T asdoes no exis. Since l = lim sup R A, T =2/3 for all [,, asympoic minimaliy holds. This can also be shown by verifying he assumpions of Corollary For [, and T n := 2 2n+2 wih n N such ha T n >, R A, T n l = R A, T n R A,T n 2 n 2 2n+2, hence 2.39 is saisfied. Since he model is deerminisic, he σ-algebra F is finie. Remark 2.31 implies ha no arbirage opporuniy in he limi is equivalen o no arbirage opporuniy in he limi wih vanishing risk. Furhermore, he example illusraes ha he inequaliy 2.35 in Theorem 2.34 can be sric, because l=2/3 bu lim inf n R,T n =1/3 for T n := 2 2n+1 wih n N. Noe ha his example can be generalized o an ineres inensiy process r = a + b1 A for, where a, b R and b. A broad class of ineres rae models have an exponenially affine erm srucure, i. e., he price process of a zero-coupon bond wih mauriy T> admis he represenaion P, T = exp A, T +B, T r, [,T, wih deerminisic real-valued funcions A and B, cf. [2, Chaper 22.3]. Hence, he zero-coupon rae process for T> is given by R, T = A, T +B, T r, [,T. 3.6 T Therefore, if for he shor rae r is no deerminisic, hen he limi of {R, T } T> exiss a. s. if and only if he limis of A, T /T and B, T /T for exis. In he following we consider generalizaions of he familiar Vasiček and Cox Ingersoll Ross models, which boh belong o he exponenially affine erm srucure models. In hese generalized models we show ha, wih appropriae choices of ime-dependen coefficiens, he limi of A, T /T as does no exis. Example 3.7 Vasiček model wih ime-dependen coefficiens. Le α> be a parameer for he mean revering srengh. Suppose he mean level μ: [, R is a locally inegrable funcion and he volailiy σ: [, R is a locally squareinegrable funcion. Le {W } be a sandard Brownian moion under Q, and

12 12 V. GOLDAMMER AND U. SCHMOCK le he iniial value r be normally disribued possibly wih zero variance and independen of he Brownian moion. Define he ineres rae inensiy process by r = e α r + α e αs μ s ds + e αs σ s dw s,. 3.8 Using Iô s formula, i follows ha {r } is a srong soluion of he sochasic differenial equaion dr = αμ r d + σ dw,, wih iniial value r. Noe ha {r } is a Gaussian process wih coninuous pahs, see e. g. [1, Chaper 8]. I follows from 3.8 ha r u = e αu r + α u e αu s μ s ds + u e αu s σ s dw s, u, hence he condiional disribuion of he inegral I,T = r u du given r is a normal one. In paricular, he process {r } is Markovian and 3.2 simplifies o R, T = 1 T log E [ ] Q exp I,T r, <T. 3.9 Using he sochasic Fubini heorem, see e. g. Proer 24, we obain I,T r e αu du α μ s e αu s du ds = s σ s e αu s du dw s, T. s }{{} =1 e αt s /α Since he sochasic inegral on he righ-hand side is independen of r wih zero expecaion, i follows ha [ ] αt E Q I,T r 1 e T = r + 1 e αt s μ s ds, T, α and, using he Iô isomery, Var Q I,T r = 1 1 e αt s 2 σ 2 α 2 s ds, T. If X has a normal disribuion, hen log E [ e X] = E[X]+ 1 2 VarX. Applying hese resuls o 3.9 leads o αt 1 e R, T =r e αt s μ s ds αt T 1 T 1 e αt s 2 σ 2 2α 2 T s ds, <T. 3.1 Given, he limi of he zero-coupon raes {R, T } T> exiss in R if and only if he limi of he difference of he las wo erms in 3.1 exiss in R as. I remains o choose suiable ime-dependen funcions for he mean level μ or he volailiy σ such ha his is no he case. Le us discuss hree specific choices.

13 DYBVIG INGERSOLL ROSS THEOREM AND ASYMPTOTIC MINIMALITY 13 If μ is bounded and lim s σs 2 =, hen, for every n N, here exiss T n such ha σs 2 n for all s T n. Since 1 e αt s 1/2 for s T 1/α, we ge for all T T n +1/α, 4 T 1 e αt s 2 σ 2 s ds 1 T 1/α σs 2 ds n T T n 1/α T n T n. Hence l = lim sup R, T = by 3.1 and, in paricular, asympoic minimaliy holds. Noe, ha 2.39 is saisfied. A similar argumenaion shows ha l =± if σ is bounded and lim s μ s = ±, respecively. We now discuss cases where he mean level μ and he volailiy σ remain bounded. Noe ha, for every a> and bounded measurable funcion g: [, R, e at s gs ds 1 e at g g a a, T. Therefore, if follows from 3.1 ha, for every, R, T = 1 T μ s ds 1 2α 2 T 1 σs 2 ds + O T 3.11 as T>ends o infiniy. We firs consider a consan volailiy σ R and a ime-dependen mean level μ s := a + b1 A s for s wih A given by 3.4, a R and b>. Using 3.11 we obain, for every, R, T =a + br A, T σ2 1 2α 2 + O as, T wih R A, T given by 3.5. I is shown in Example 3.3 ha he limi of R A, T as does no exis, hence he limi of {R, T } T> does no exis eiher. Since lim sup R A, T =2/3 by he resuls from Example 3.3, we see ha l = lim sup R, T =a + 2b 3 σ2 2α 2,, hence asympoic minimaliy holds for all s <. A similar resul can be obained, if we choose a consan mean level μ R and a ime-dependen volailiy σ s := a + b1 A s wih a, b R saisfying 2ab + b 2. To illusrae explicily ha a bounded, coninuously varying volailiy funcion σ can also lead o oscillaing zero-coupon raes, we consider a consan mean level μ R and a volailiy funcion of he form σ = a + b sinlog b coslog + 1,, wih a, b, saisfying a 2 b. Then σ 2a for all. Furhermore, for all T>and [,T, 1 T σ 2 s ds = a + b T + 1 sinlogt sinlog + 1 T Togeher wih 3.11 we obain for he long-erm spo rae process l = lim sup R, T =μ a b, 2α2, 3.13 bu for he limes inferior of he zero coupon raes lim inf R, T =μ a + b 2α 2,.

14 14 V. GOLDAMMER AND U. SCHMOCK Hence, he limi of {R, T } T> as does no exis. Since he long-erm spo-rae process given by 3.13 is a deerminisic consan, asympoic minimaliy holds for all imes s<. Example 3.14 Cox Ingersoll Ross model wih ime-dependen coefficiens. Le α: [,, and β: [,, be wo coninuously differeniable funcions and le {W } be a sandard Brownian moion under Q. Analogously o [14, Secions and 1.3.3], by considering a squared Bessel process of dimension δ, wih respec o some probabiliy measure P, applying a suiable measure change o Q using Girsanov s heorem, and rescaling he sae space by he funcion α, we can consruc an ineres rae inensiy process {r } which solves he sochasic differenial equaion dr = δα+ 2β+ α α r d +2 αr dw,, wih deerminisic iniial value r. If for given <T here is a soluion F T :[, T ] R o he Riccai equaion FT 2 u+f T u =2αu+β 2 u+β u, u [, T ], wih he erminal condiion F T T =βt, hen i follows as in [14, Secions and 1.3.4] ha he corresponding zero-coupon rae is given by 1 FT β T R, T = r + δ FT u βu du, 2T α which corresponds o 3.6 resuling from an exponenially affine erm srucure. We now make specific choices for α and β. For b> and a> 2b define he funcion β = a + b sinlog b coslog + 1,. Noe ha β is coninuously differeniable and ha a 2b β < for all. Furhermore, for c> wih c 2 > a + 2b 2 + 2b, define he funcion α = 1 2 c 2 β 2 β,. Since β 2 a + 2b 2 and β 2b, i follows ha α > for all. For hese funcions α and β, he Riccai equaion simplifies, for each T>, o FT 2 u+f T u =c 2, u [,T]. The soluion for he erminal condiion F T T =βtis F T u =c anhcu + g T, u [,T], where g T := aranhβt /c ct. Since βt a + 2b< c for all T>, he area angens hyperbolicus of βt /c is well-defined and bounded wih respec o T. Noe ha d dx logcoshcx + g T = c anhcx + g T for all x R. Therefore, cosh aranh βt F T u du = log cosh aranh βt c Using cosh x =e x + e x /2 for x R and he boundedness of aranhβt /c, i follows ha lim c ct, T. 1 T F T u du = c,. T

15 DYBVIG INGERSOLL ROSS THEOREM AND ASYMPTOTIC MINIMALITY 15 Finally, inegraion of β, cf. 3.12, yields l = lim sup R, T = δ c a + b, bu lim inf R, T = δ c a b,. 2 Since l in 3.15 is a deerminisic consan, asympoic minimaliy holds. Our nex example is he well-known Gaussian Heah Jarrow Moron model, cf. [14, Chaper 11], wih deerminisic bu ime-dependen volailiy of he forward raes. For his volailiy we choose a non-negaive funcion, which flucuaes over ime bu converges o zero when he mauriies end o infiniy. Again, we assume, ha he volailiy varies wih he business cycles of exponenially increasing lenghs. Example 3.16 Gaussian Heah Jarrow Moron model. Le σ 1,σ 2 :[, R denoe bounded measurable funcions. Define he volailiy σ: [, 2 R of he forward raes by σu, v =σ 1 uσ 2 v for all u, v. Suppose ha he deerminisic forward rae curve f, : [, R a ime zero is locally inegrable. We se up he model direcly using he spo maringale measure Q, under which zero-coupon bond prices, discouned by he bank accoun process, are maringales. Therefore, le {W } be a sandard Brownian moion under Q and le he forward raes saisfy f, T =f,t+ σu, T σ u, T du + σu, T dw u, T, wih inegraed volailiy σ u, T := σu, v dv so ha hey obey he Heah u Jarrow Moron drif condiion. The shor-erm ineres rae inensiy process is given by r = f, for. Then, for each mauriy T>and ime [,T, he zero-coupon rae is given by R, T = 1 f,u du ru 1 T 2 σ u, T 2 du + σ u, T dw u, see e. g. [14, Chaper 11, pp ]. Given, he limi of he zero-coupon raes {R, T } T> as migh no exis, if he averages of he iniial forward raes {f,u} u [,T ] do no converge, see Examples 3.3 and 3.7 for such funcions. In he following, we herefore assume he exisence of he limi of hese averages so ha we can define f 1 T = lim f,u du. T To furher discuss he limiing behaviour of he zero-coupon raes, we firs consider heir sochasic componen. Subsiuing he sochasic inegral from he shor-rae r v ino he formula for R, T and using he sochasic Fubini heorem, we obain 1 T σ u, T dw u = 1 T σ 2 v dv v σu, v dw u dv σ 1 u dw u, <T, 3.17

16 16 V. GOLDAMMER AND U. SCHMOCK which converges o zero as whenever he averages of {σ 2 v} v [,T ] do. To obain he oscillaing behaviour of he zero-coupon raes, define 1 σ 2 v = 2 a + sinb logv +1+2bcosb logv + 1, v, v +1 wih parameers 5 a, b R. Then lim v σ 2 v =, hence he sochasic par given in 3.17 ends o zero as. For all T>and u [,T], u σ 2 v dv = T +1 a + sinb logt + 1 u +1 a + sinb logu + 1. Hence, using he above expression for R, T, he long-erm spo rae is given by l = lim sup R, T = f lim sup σ 2 2T 1u σ 2 v dv du u = f lim sup a + sinb logt + 1 σ 2 1u 2 du = f a +1 b 2 σ1u 2 du,, where 1 b equals 1 if b and oherwise. In paricular, if a and b are no boh zero, hen, for all imes s<wih s σ2 1u du >, asympoic monooniciy in he sense ls =l Fs does no hold. The same reasoning as above yields lim inf R, T =f + 1 max{ a 1b, } 2 σ 2 1u 2 du,, 3.19 hence, for, he limi of he zero-coupon raes does no exis if σ2 1u du > and b. Furhermore, if a > 1 b and σ 1 is no he zero funcion, hen his model provides arbirage opporuniies in he limi for all imes s<saisfying s a +1 b 2 σ1u 2 du < a 1 b 2 σ1u 2 du by shor-selling he long-erm zero-coupon bonds: For every deerminisic sequence {T n } n N ending o infiniy, we have ls < lim inf n R, T n by 3.18 and 3.19, hence he assumpions of Theorem 2.34 have o be violaed Models violaing he asympoic minimaliy. We now presen four shorrae models in coninuous ime, which illusrae he link beween asympoic minimaliy and he exisence of a forward risk neural probabiliy measure in Condiion 2.12, no arbirage in he limi and convergence of he spo rae. Example 3.2. On Ω = {, 1} consider Xω =ω for ω Ω, le Q denoe he uniform disribuion, F = {, Ω} for [, 1/3 and F equal o he power se of Ω for 1/3. Wih A given by 3.4, define he ineres rae inensiy process by r = X1 A +1 X1 Ac [1/3,, [,. 5 If we choose a 1 b +4b 2, hen σ 2 v for all v.

17 DYBVIG INGERSOLL ROSS THEOREM AND ASYMPTOTIC MINIMALITY 17 Noe ha {r } is adaped and càdlàg. Using 3.2 and Jensen s inequaliy, we ge for all [, 1/3 and T>1/3, R, T = 1 T log exp λa [, T ] + exp λac [1/3,T] 2 λa [, T ] + λac [1/3,T] = T 1/3 2T 2T 1 2, hence l 1/2. For 1/3, X is F -measurable and we ge from 3.2 R, T = XλA [, T ]+1 XλAc [, T ] T = XR A, T +1 X1 R A, T, T >, wih R A, T given by 3.5. Therefore, l = lim sup R, T =2/3 for all 1/3, because he poins in [1/3, 2/3] are he accumulaion poins of R A, T as, see Example 3.3. In his example asympoic minimaliy fails for all imes s [, 1/3 and [1/3,T]. By consrucion here exiss a forward risk neural probabiliy measure, which implies wih Lemma 2.32, ha here is no arbirage opporuniy in he limi wih vanishing risk. Since F is finie for each, he model provides also no arbirage opporuniy in he limi wih Remark Therefore Condiion 2.12, resp. he weaker Condiion 2.1, and he wo differen noions of no-arbirage are no sufficien for asympoic minimaliy. Indeed, he inequaliy 2.39 fails, which is sufficien for asympoic minimaliy in combinaion wih no arbirage opporuniy in he limi. Consider an arbirary deerminisic sequence {T n } n N ending o infiniy. Then lim inf n R, T n=x lim inf n R A, T n +1 X 1 lim sup n R A, T n Assume 2.39 holds for ω = 1, hen lim inf n R A, T n l Fs =2/3. Therefore, lim sup n R A, T n 2/3. Wih 3.21 follows ha he inequaliy 2.39 fails for ω =. Finally, suppose T n := 2 n for n N. We have seen in Example 3.3, ha lim inf n R A, T n =1/3 and lim sup n R A, T n =2/3 for all. Hence, for 1/3 s he inequaliy 2.35 is sric on Ω, i. e. lim inf R, T n=1/3 < 2/3 =ls. n Even, if he limi of he zero-coupon bonds and a forward risk neural probabiliy measure exis, his is no sufficien for asympoic minimaliy, which is shown by he following example. Example Consider Ω =, 1] wih Lebesgue measure Q, define F = {, Ω} for [, 1 and le F denoe he Borel σ-algebra of, 1] for 1. Le τω =1/ω for ω Ω denoe he random ime, when he ineres rae inensiy jumps o 1, i. e., we define he ineres rae inensiy process by r =1 [τ, for. Then τ is F 1 -measurable and 3.2 implies for T>1 R1,T= 1 T 1 1 r u du = T T τ T 1 1

18 18 V. GOLDAMMER AND U. SCHMOCK everywhere on Ω, hence l1 = 1. For [, 1 and T 1, 3.2 implies ha R, T = 1 [ ] T log E Q exp r u du 1 1 T log 1, T }{{} 1 {τ T } hence l = due o non-negaive ineres raes. Therefore, asympoic minimaliy does no hold. This does no conradic Corollary 2.36, because his model provides an arbirage opporuniy in he limi for he imes s [, 1 and =1by shor-selling long-erm zero-coupon bonds. Choose a 1, -valued deerminisic sequence {T n } n N ending o infiniy, define ϕ n = expt n 1/2 for each n N and fix {ψ n } n N according o Remark 2.3 so ha Definiion 2.29a holds. Then 1 V n 1 = exp T n 1 2 R1,T n + exp 1 2 T n 1 Rs, T n T n s, P s, 1 hence lim inf n V n 1 a.s. = and pars b and c of Definiion 2.29 hold. The exisence of a forward risk neural probabiliy measure or he absence of arbirage opporuniies in he limi, is no even necessary for asympoic minimaliy. Example Consider Ω =, 1] wih Lebesgue measure Q, define F = {, Ω} for [, 1 and le F denoe he Borel σ-algebra of, 1] for 1. Le τω = 1/ω for ω Ω be a random ime. Define he ineres rae inensiy process 6 by r =1 1 1 [1,τ for. Wih 3.1 he zero-coupon bond price for mauriy T 1 is given by P, T =e T E Q [τ T ]=e T 1 + log T, [, 1. Using he definiion of he zero-coupon raes in 2.2, we obain for every [, 1 and T 1 log1 + log T R, T =1 1. T For = 1 he zero-coupon prices for T 1 are given by P 1,T=e T 1 τ T and herefore he zero-coupon raes equal R1,T=1 logτ T T 1 1. Hence asympoic minimaliy holds. On he oher hand, we can consruc an arbirage opporuniy in he limi for he imes s = and = 1 according o Definiion 2.29 by shor-selling long-erm zero-coupon bonds. For his define T n = n +1, 1 ϕ n = ep,t n = etn 1, 1 + log T n and ψ n = 1 for all n N. Then V n = for all n N and V n 1 = τ T n +1 n 1 on Ω. 1 + log T n 6 This example can be slighly simplified if we omi he 1 and allow negaive ineres raes.

19 DYBVIG INGERSOLL ROSS THEOREM AND ASYMPTOTIC MINIMALITY 19 Example Consider Ω = N, define he filraion {, Ω} for [, 1, F = {, {1}, Ω \{1}, Ω} for [1, 2, PΩ for [2,, where PΩ denoes he power se, and he probabiliy measure Q on Ω, PΩ by Q{ω} =1/ω 1/ω +1 for all ω Ω. Le τω =ω for ω Ω denoe he random ime, when he ineres rae inensiy jumps o 1 1/ω, i. e., we define he ineres rae inensiy process by r =1 1/τ1 [τ, for. Then τ is F 2 -measurable and 3.2 implies for T>2 R2,T= 1 T 2 2 r u du = 1 1 T T τ 2 τ T τ 3.25 everywhere on Ω, hence l2 = 1 1/τ. Therefore l2 F = and l2 F1 = Ω\{1}. For T>, we always have ha R,T and 3.2 implies ha R,T= 1 [ ] T log E Q exp r u du 1 T log 1, T }{{} 1 {τ T } hence l = and asympoic minimaliy holds for imes and 2. For T > 1, 3.2 implies as in 3.25 ha l1 = on {1} and ha on he complemen Ω \{1} [ ] exp r u du τ 2, R1,T= 1 T 1 log E Q 1 } {{ } 1 {τ T } 1 T 1 log 2 T hence l1 = on Ω and asympoic minimaliy does no hold for imes 1 and 2. To consruc an arbirage opporuniy in he limi for imes 1 and 2 according o Definiion 2.29 by shor-selling he long-erm zero-coupon bonds, define for each n N he deerminisic mauriy T n = n + 2 and he sraegy by ϕ n = 1 Ω\{1} exp T n 1R1,T n R1, 2 and ψ n =1 Ω\{1}. Then ϕ n,ψ n isf 1 -measurable and V n 1 = for all n N. Furhermore, we obain for all n N ϕ n P 2,T n = 1 Ω\{1} exp T n 2R1,T n R2,T n + R1,T n R1, 2. Since l1 = lim n R1,T n = on Ω as well as l2 = lim n R2,T n = 1 1/τ 1/2 onω\{1} by 3.25, we ge lim inf n ϕ n P 2,T n =. Therefore, lim inf n V n 2 = ψ n and wih probabiliy QΩ\{1} =1/2 he limes inferior is sricly greaer han zero A model wihou forward risk neural probabiliy measure and wihou limiing arbirage opporuniies. The following example is inspired by he infinie-horizon model considered in Example 7.2 in Pliska I shows ha in general for a model, which does no provide an arbirage opporuniy in he limi, here mus no exis a forward risk neural probabiliy measure. The oher implicaion is also no rue, see Example 3.22.

20 2 V. GOLDAMMER AND U. SCHMOCK Example Define Ω = N, F = {, N} and F 1 = PN, he se of all subses of N. Le P be any probabiliy measure on F 1 wih P{ω} > for all ω Ω. Define zero-coupon bond prices by P,n = 1 for all n N and 1 if ω n 2, P 1,nω = n 2 +1/2 if ω = n 1, 1/2 if ω n, for all ω Ω and inegers n 2. By 2.1 and 2.5, i follows ha l = l1 =, hence asympoic monooniciy and minimaliy hold for imes s = and =1. To verify ha here is no arbirage opporuniy in he limi for imes s = and = 1, consider an F -measurable, hence deerminisic sequence {T n } n N of mauriies wih T n >nfor all n N and deerminisic porfolios ϕ n,ψ n wih ϕ n = ψ n for all n N so ha Definiion 2.29a is saisfied. Then V n 1ω = for all ω {1, 2,...,n 2}, hence lim inf n V n 1 = on Ω. Therefore, par c of Definiion 2.29 is saisfied, bu par b does no hold. To show ha Condiion 2.1 for imes s = and = 1 is no saisfied and here does no exis a forward risk neural measure for imes s = and =1 like in Condiion 2.12, we argue by conradicion. Assume ha here exiss an equivalen probabiliy measure Q = Q,1 such ha P,n E Q [P 1,n] for all inegers n n 2. This implies Q{n 1,n,...} n2 +1 Q{n 1}+ 1 Q{n, n +1,...}, 2 2 hence n 2 Q{n 1} Q{n 1,n,...} for all inegers n n. Define he consan c =n 1 Q{n 1,n,...}/n >. Then we ge by inducion Q{n 1,n,...} cn 3.27 n 1 for all inegers n n, because Q{n, n +1,...} =Q{n 1,n,...} Q{n 1} 1 1 n 2 Q{n 1,n,...} 1 1 cn n 2 n 1 = cn +1 n for all inegers n n + 1. However, 3.27 for n implies Q =c>, which is impossible for a probabiliy measure. 4. Proofs of auxiliary resuls Proof of Lemma 2.8. Consider a finie non-empy se I n, of zero-coupon bond mauriies, which is required o be also a subse of N in he discree-ime case. Le M I := max u I R, u denoe he maximal available zero-coupon rae. Define he random mauriy T I :Ω I as he firs one realizing his maximal rae, i. e. T I = u I u 1 {R,u=MI,R,v<M I for all v I,v<u}. Noe ha T I is F -measurable and ha R, T I =M I. By [7, Theorem A.32b], here exiss, for every n N, an increasing sequence {I k,n } k N of finie subses of n,, which are also subses of N in he discree-ime case, such ha S n := ess sup R, T = lim R, T I k,n T>n k a.s.

21 DYBVIG INGERSOLL ROSS THEOREM AND ASYMPTOTIC MINIMALITY 21 Hence, for every n N, here exiss k n N such ha he essenial supremum is nearly reached wih high probabiliy, e. g. wih he abbreviaion T n := T Ik n,n, P min{s n,n} 2 n R, T n S n 1 2 n. The a. s. limi of {S n } n N exiss due o he monooniciy of he essenial suprema. Hence, using he firs Borel Canelli lemma, he a. s. limi of {R, T n } n N exiss and agrees wih he one of {S n } n N. Proof of Lemma 2.9. Fix s. In he coninuous-ime case, using he definiion of he arbirage-free forward rae in 2.4 and he zero-coupon rae in 2.2, log P s, F s,, T = + T s Rs, T, T,. T T Since he firs summand ends o zero almos surely as, i follows ha T s l F s, = lim sup F s,, T = lim sup Rs, T =ls T a. s., by he definiion of he long-erm forward rae in 2.6 and he long-erm zerocoupon rae in 2.5. In he discree-ime case, using he definiion 2.3 of he arbirage-free forward rae and he definiion 2.1 of he zero-coupon rae, we see ha i is enough o prove 1/T P s, a.s. lim sup log = lim sup log P s, T 1/T s. P s, T However, ha is wha we jus verified for he coninuous-ime case. 5. Proofs for asympoic monooniciy and minimaliy assuming he exisence of a forward risk neural probabiliy measure The key observaion for our generalizaion is he following lemma, which uses noaion inroduced in Definiion Lemma 5.1. Le Ω, F, P be a probabiliy space and G a sub-σ-algebra of F. a For every non-negaive random variable X on Ω, F, P, he funcion, E[X G] 1/ is non-decreasing a. s. and X lim E [ X G ] 1/ = X G a. s. 5.2 b Le {X } > be a collecion of non-negaive random variables on Ω, F, P. For each n N le Y n denoe he essenial infimum of {X } >n. Then X := lim inf X lim n E[Y n n G] 1/n = X G lim inf E[X G] 1/ a. s. 5.3 c If in b he random variable X G dominaes {X } > in he G, P-superexponenial sense along a subsequence according o Definiion 2.2, hen he las inequaliy in 5.3 is an a. s. equaliy. Remark 5.4. For he rivial case G = {, Ω}, Lemma 5.1a implies he well-known resul lim p X L p = X L. Remark 5.5. For a non-negaive random variable Z wih E[Z] =, we define E[Z G] = sup n N E[min{Z, n} G]. For a σ-inegrable random variable wih respec o G, he generalizaion of he condiional expecaion is given in [8, Chaper 4].

22 22 V. GOLDAMMER AND U. SCHMOCK Remark 5.6. Par b of Lemma 5.1 was inroduced by Hubalek e al. 22 wih he addiional assumpion ha he sequence {X } > converges. A furher applicaion of he lemma is o prove ha he long volailiies, implied by he Black Scholes formula, canno fall, which was done by Rogers and Tehranchi 26. Example 5.7. Noe ha X < X G is possible in 5.2, even for a bounded X. As an example, consider Ω =, 1 wih Lebesgue measure and Borel σ-algebra, G = {, Ω} and Xω =ω for ω Ω. Then E[X n G] 1/n =n+1 1/n and X G =1. Example 5.8. Noe ha he las inequaliy in 5.3 can be sric for a bounded sequence {X n } n N which converges everywhere in a monoone way. In he seing of Example 5.7, consider X n =1,1/n for n N wih poinwise limi X =, hence X G =. However, E[X n n G] 1/n = n 1/n 1asn. Proof of Lemma 5.1. a Consider <s<<. Jensen s inequaliy for condiional expecaions, applied o he convex funcion ϕx =x /s implies E[X s G] 1/s = ϕe[x s G] 1/ E[ϕX s G] 1/ = E [ X G ] 1/ a. s. Due o his monooniciy he almos sure limi C := lim n E[X n G] 1/ n exiss along every sequence n and every oher sequence gives a. s. he same limi. Noe ha C is G-measurable. If Z is a G-measurable random variable saisfying PX Z = 1, hen E[X G] 1/ E[Z G] 1/ = Z a. s. for all >. I remains o show ha X C a. s., which we do by conradicion. We assume for he se A := {X >C} ha PA >. Since A {C< } here exis k N wih PA {C k} >. Furhermore, here exiss l N such ha PB > for B := {X C +1/l, C k}. We obain E[X1 B ] E[C1 B ]+PB/l > E[C1 B ], 5.9 because PB > and E[C1 B ] k PB <. In he remaining par of he proof, we use he convenion = for producs. Using he condiional Hölder inequaliy 7 and he fac ha E[X n G] 1/n C a. s., i follows for all n N ha E[X1 B G] E[X n G] 1/n E[1 B G] 1 1/n C E[1 B G] 1 1/n a. s. Passing o he limi n and using he G-measurabiliy of C, E[X1 B G] C E[1 B G]=E[C1 B G] a.s. Taking expecaions gives E[X1 B ] E[C1 B ], which is a conradicion o 5.9. b Since Y m Y n sup k N Y k = X for all m, n N wih m n, we obain E[Ym n G] 1/n E[Yn n G] 1/n E[X n G] 1/n X G a. s., using par a for he las inequaliy. Hence, by par a, for every m N, Y m Ym G = lim E[Y m n G] 1/n sup E[Yn n G] 1/n n Therefore, n N lim n E[Xn G] 1/n = X G a. s. X = sup Y m sup Ym G lim E[Y n n G] 1/n X G a. s. m N m N n 7 For a proof, cf. Hölder s inequaliy a en.wikipedia.org/wiki/, version of April 6, 28.