ADMISSIBILITY OF GENERIC MARKET MODELS OF FORWARD SWAP RATES

Transcription

1 ADMISSIBILITY OF GENERIC MARKET MODELS OF FORWARD SWAP RATES Libo Li and Marek Rukowski School of Mahemaics and Saisics Universiy of Sydney NSW 2006, Ausralia April 2, 2010 Absrac The main goal of his work is o re-examine and exend cerain resuls from he papers by Galluccio e al. [5] and Pieersz and van Regenmorel [14]. We will aemp o give he necessary and sufficien condiions for a family of swaps o be suppored by a posiive family of raded bonds. We will also derive he join dynamics of a family of forward swap raes under a single probabiliy measure and we will show ha i can be uniquely deermined by a family of volailiy processes. The research of M. Rukowski was suppored under Ausralian Research Council s Discovery Projecs funding scheme (DP ). 1

2 2 Admissibiliy of Marke Models of Swap Raes 1 Inroducion The goal of his work is o re-examine he admissibiliy problem for an arbirary family of forward swaps, which was previously sudied in Galluccio e al. [5] (see also Pieersz and Regenmorel [14] for he relaed sudy of a special case). Addiionally, we will also examine an exension of his problem o he siuaion when he paymens daes of forward swaps are chosen o be only some of he enor srucure daes beween he sar dae of a swap and is mauriy dae. This generalizaion is moivaed by he case where some forward swaps have quarerly paymens, whereas for some oher swaps he paymens are exchanged according o he semi-annual schedule. Le T = {T 0,..., T n } wih 0 < T 0 < T 1 < < T n 1 < T n be a fixed sequence of daes represening he complee enor srucure for all swaps under consideraion. For every i = 1,..., n, we wrie a i = T i T i 1 o denoe he lengh of he ih accrual period. Le B(, T i ) sand for he price of he zero-coupon bond mauring a T i. Unless saed oherwise, i is assumed ha for every i = 0,..., n he bond price B(, T i ), [0, T i ], follows a posiive process wih he erminal condiion B(T i, T i ) = 1. Le S = {S 1,..., S l } be an arbirary family of l disinc forward swaps associaed wih he enor srucure T, in he sense ha any rese or selemen dae for any swap S j in S belongs o T. The forward swap rae κ j for he (sandard) forward swap S j, which sars a T sj and maures a T mj, is known o be given by he following expression (see, for insance, Secion 13.1 in Musiela and Rukowski [13]) κ j = κ sj,mj = B(, T s j ) B(, T mj ) mj i=s a j+1 ib(, T i ) = P sj,mj, [0, T A sj,mj sj ], (1) where we denoe and The process A s j,m j P sj,mj = B(, T sj ) B(, T mj ), [0, T sj ], A sj,mj = m j i=s j +1 a i B(, T i ), [0, T sj+1]. is called he swap annuiy or he swap numéraire for he forward swap S j. The daes T sj, T sj +1,..., T mj 1 are hen he rese daes for S j, whereas he daes T sj +1, T sj +2,..., T mj are selemen daes. Hence T sj, T sj +1,..., T mj are he rese/selemen daes in he forward swap S j. Of course, hey are he only relevan daes for he (sandard) forward swap S j. Remark 1.1. In fac we will consider a more general se-up, in which he sar dae T sj is he firs rese dae and he mauriy dae T mj is he las selemen dae in he forward swap S j. Oher rese/selemen daes in a forward swap S j can be chosen freely from he se of daes T i T saisfying T sj < T i < T mj. Then all rese/selemen daes in he forward swap S j will be simply referred o as he relevan daes in S j. They can also be called he enor srucure generaed by S j and hus we will use he noaion T (S j ). In he general case, we wrie κ j = κ s j,m j = B(, T s j ) B(, T mj ) i A i ã i B(, T i ) = P s j,m j A s j,m j, [0, T sj ], (2) where A j is he se of all selemens daes in he swap S j and ã i is he adjused accrual period. For insance, if a forward swap S j has one selemen dae, namely, is mauriy, hen A j = {T mj } and equaion (2) becomes κ j = B(, T s j ) B(, T mj ) ã mj B(, T mj ) where ã j = T mj T sj = m j i=s j +1 a i. If all daes from T beween he sar and mauriy dae are selemen daes for a given forward swap, he conrac is referred o as a sandard forward swap.

3 L. Li and M. Rukowski 3 Le us selec he bond mauring a some dae T b T as a bond numéraire and le us denoe by B b he family of he deflaed bond prices B b = { B b (, T i ) = B(, T i )/B(, T b ), i = 0,..., n }. In erms of he deflaed bond prices B b (, T i ), for he sandard forward swap we obain where we wrie and = Bb (, T sj ) B b (, T mj ) mj i=s a j+1 ib b (, T i ) κ sj,mj = P b,sj,mj, [0, T A b,sj,mj sj T b ], (3) P b,s j,m j = B b (, T sj ) B b (, T mj ), [0, T sj T b ], A b,s j,m j = m j i=s j+1 a i B b (, T i ), [0, T sj +1 T b ]. We call he process A b,s j,m j he deflaed swap annuiy or deflaed swap numéraire. As we shall see in wha follows, he (deflaed) swap numéraires are crucial objecs in he specificaion of he Radon-Nikodým densiies for swap measures under which forward swap raes are (local) maringales. Remark 1.2. Noe ha he choice of a bond numéraire is arbirary, in he sense ha one may ake he bond price B(, T b ) for any mauriy dae T b T o play his role. I will be raher clear ha he resuls presened in he sequel do no depend on a paricular choice of a bond numéraire, bu only provided ha all bond prices are eiher assumed or shown o follow posiive processes. We are in a posiion o describe he main problems considered in his work. We are ineresed in deriving he join dynamics of forward swap raes associaed wih a family S = {S 1,..., S l } of forward swaps, as well as in checking wheher hese dynamics are suppored by he exisence of a (unique) family of deflaed bond prices B b = {B b (, T i ), i = 1,..., n} for some (or all) choice of a bond numéraire. We will hus deal wih wo relaed problems, which can be described as follows. Under which assumpions, a given swap rae model can be suppored by he exisence of an arbirage-free erm srucure model consisen wih swap raes, where by a erm srucure model we mean he join dynamics of posiive deflaed bond prices? Under which assumpions, he join dynamics for a given family of forward swaps is uniquely specified under a single probabiliy measure in erms of drifs and volailiies? The answers o he above wo quesions will be examined in Secions 2 and 3, respecively. In boh cases, he echnique is o formulae a suiable inverse problem and o examine is solvabiliy. For he firs quesion, we are able o provide boh necessary condiions and sufficien condiions for he exisence of a unique and posiive soluion o he inverse problem for deflaed bond prices (see Inverse Problem (IP.1)), ha is, for he (weak) T -admissibiliy of a family S (see Definiion 2.3 and Proposiions 2.1, 2.2 and 2.3). I is worh sressing ha our condiions are differen from hese provided in Galluccio e al. [5]. In fac, we show by means of couner-examples ha he main resul in [5] is defecive. For he second quesion, we inroduce he corresponding inverse problem for swap annuiies (see Inverse Problem (IP.2)) and he concep of he A-admissibiliy (see Definiion 3.1). I is clear ha A-admissibiliy implies T -admissibiliy, bu he converse implicaion does no hold. We hen provide a general mehodology for he compuaion of he join dynamics of forward swap raes, which covers all poenially ineresing A-admissible cases.

4 4 Admissibiliy of Marke Models of Swap Raes 2 Admissible Families of Forward Swap Raes In his secion, we address he issue wheher a given family of forward swap raes is suppored by he exisence of he implied family of deflaed bond prices. This corresponds o he inverse problem examined by Galluccio e al. [5], which can be informally saed as follows. Give necessary and sufficien condiions, which ensure ha a given family of forward swap raes S = {S 1,..., S l } and he corresponding (diffusion-ype) swap rae processes {κ 1,..., κ l } uniquely specify a family of non-zero deflaed bond prices B b = {B b (, T 1 ),..., B b (, T n )} for any choice of b {0,..., n}. 2.1 Linear Sysems Associaed wih Forward Swaps Le us noe ha for a given family B n = {B n (, T i ), i = 1,..., n} of sochasic processes represening he deflaed bond prices when B(, T n ) is he numéraire bond, we can uniquely deermine he forward swap rae κ s j,m j for any sar dae T sj and mauriy T mj from he enor srucure T. This simple observaion paves he way for he mos commonly used direc approach o modelling of forward swap raes, in which one sars by choosing a erm srucure model and subsequenly derives he join dynamics of a family of forward swap raes of ineres. I is also clear ha each forward swap rae κ s j,m j gives rise o he linear equaion in which deflaed bond prices can be seen as unknowns. Specifically, one obains he following swap equaion, which is associaed wih he forward swap S j and he numéraire bond B(, T b ), B b (, T sj ) + m j 1 i=s j +1 κ sj,mj a i B b (, T i ) + (1 + κ sj,mj )a mj B b (, T mj ) = 0. (4) In his conex, he following inverse problem arises in a naural may: describe all families of forward swaps associaed wih he enor srucure T such ha he corresponding family of swap raes uniquely specifies he associaed family of non-zero (and preferably posiive) deflaed bond prices. To formally address he above-menioned inverse problem, we idenify a forward swap S j wih he corresponding swap equaion, in which a generic value κ j of he forward swap rae κ s j,m j plays he role of a parameer. Le us now inroduce some noaion. For a fixed b, le x i sand for a generic value of he deflaed bond price B b (, T i ) and le κ j be a generic value of he forward swap rae κ j := κ s j,m j. Since x i and κ j are aimed o represen generic values of he corresponding processes in some sochasic model, we have ha (x 0,..., x n ) R n+1 and (κ 1,..., κ l ) R l, in general. Since he bond B(, T b ) is chosen o be he numéraire bond i is clear, by he definiion of he deflaed bond price, ha he variable x b saisfies x b = 1 and hus i is more adequae o consider a generic value (x 0,..., x b 1, x b+1,..., x n ) R n. Using his noaion, we may represen equaion (3) as follows, for every j = 1,..., l, κ j = x s j x mj mj i=s j +1 a ix i. (5) For breviy, we wrie c j,i = κ j a i and c j,mj = (1 + κ j )a mj, so ha equaion (4) becomes x sj + m j 1 i=s j +1 c j,i x i + c j,mj x mj = 0. (6) Definiion 2.1. For a given family S = {S 1,..., S l } of forward swaps, any fixed b {0,..., n} he associaed linear sysem (6) paramerized by a vecor (κ 1,..., κ l ) R l, is denoed as C b (κ 1,..., κ l )x b = e b (κ 1,..., κ l ), where x b = (x 0,..., x b 1, x b+1,..., x n ) is he vecor of unknowns and he vecor e b belongs o R l.

5 L. Li and M. Rukowski 5 The marix C b (κ 1,..., κ l ) and he vecor e b (κ 1,..., κ l ) depend on he vecor (κ 1,..., κ l ) R l. To alleviae noaion, we shall ypically suppress he variables κ 1,..., κ l, however, and we will simply wrie C b and e b. As already menioned in Remark 1.1, we will also consider more general forward swaps for which he paymen daes are only a subse of daes beween he saring and mauriy dae, so ha he sum in he denominaor of (5) over some daes beween T sj +1 and T mj. In ha case, he coefficien a i is adjused o represen he lengh of imes beween he consecuive paymen daes and a suiable version of equaion (6) is derived, so Definiion 2.1 covers he general se-up. For insance, if a forward swap S j seles a mauriy only hen equaion (5) becomes κ j = x s j x mj ã mj x mj where ã j = T mj T sj = m j i=s a j+1 i. The following definiion is also ailored o cover all siuaions considered in his work. Definiion 2.2. A dae T i T is he relevan dae for he swap S j whenever T i = T sj, T i = T mj, or he erm c j,i is non-zero. We wrie T (S j ) o denoe he se of all relevan daes for he swap S j and we se T 0 (S j ) = {T sj, T mj }. The following noaion will be useful: T 0 (S) he se of all sar/mauriy daes for a family S, ha is, T 0 (S) = l T 0 (S j ) = j=1 l {T sj, T mj }, j=1 T (S) he se of all relevan daes for a family S, ha is, T (S) = l T (S j ). j=1 I is clear ha T i is no in T (S) whenever he corresponding column of he marix C b has all enries equal o zero. Manifesly, he non-uniqueness of a soluion o he inverse problem holds in ha case since he deflaed bond price B b (, T i ) canno be rerieved. This is rivial since he variable x i does no appear in any equaion in he linear sysem C b x b = e b. In wha follows, we only consider families S ha generae he enor srucure T in he sense ha T = T (S). 2.2 Inverse Problem for Deflaed Bonds We are in a posiion o sae he inverse problem for he deflaed bond prices and he relaed conceps of admissibiliy of a family S. Inverse Problem (IP.1) A family S = {S 1,..., S l } admis a soluion o he inverse problem (IP.1) if for any b {0,..., n} and for almos every (κ 1,..., κ l ) R l here exiss a soluion x b R n o he linear sysem C b x b = e b associaed wih S. Galluccio e al. [5] prefer o deal direcly wih he sochasic linear sysem C b B b = e b where C b is he marix of sochasic processes obained by replacing he variables (κ 1,..., κ l ) by diffusion-ype forward swap rae processes κ j = κ s j,m j, j = 1,..., l and he unknowns (x 0,..., x b 1, x b+1,..., x n ) R n are replaced by he (unknown) deflaed bond prices B b (, T i ), i = 0,..., b 1, b+1,..., n. Apparenly, i is implicily assumed in [5] ha for any (0, T 0 ] he random variable (κ 1,..., κ l ) has he sricly posiive join probabiliy densiy funcion on (R l, B(R l ), so ha he probabiliy disribuion of (κ 1,..., κ l ) has he suppor R l. In hese circumsances, he inverse problem (IP.1) inroduced above appears o be equivalen o he problem sudied in [5].

6 6 Admissibiliy of Marke Models of Swap Raes The following definiion reflecs he idea of admissibiliy of a family S, as proposed by Galluccio e al. [5]. I is based on he inverse problem (IP.1), bu i also refers o some addiional feaures of a soluion o his problem, such as uniqueness and posiiviy. Definiion 2.3. We say ha a family S of forward swaps associaed wih T is weakly T -admissible if for any choice of b {0,..., n} he following propery holds: for almos every (κ 1,..., κ l ) R l here exiss a unique non-zero soluion x b R n o he linear sysem C b x b = e b associaed wih S. If, in addiion, he soluion is sricly posiive for almos every (κ 1,..., κ l ) R l + hen we say ha S is T -admissible. We wrie A o denoe he class of all families of forward swaps which are weakly T -admissible. Similarly, A sands for he class of all families of forward swaps which are T -admissible. Remark 2.1. The propery of T -admissibiliy is he sronges form of inveribiliy, which is he mos convenien from he poin of view of furher applicaions. If i holds hen he Radon-Nikodým densiies for various swap measures can be easily compued from he family B b in erms of he underlying forward swap raes. Consequenly, in ha case i will be easy o derive he join dynamics of forward swap raes under a single probabiliy measure. In addiion, hese dynamics will be suppored by an arbirage-free model of posiive deflaed bond prices. Before proceeding, le us commen on he approach presened in he paper by Galluccio e al. [5]. In [5], he auhors formalize he concep of admissibiliy of a family S hrough he following definiion (see Definiion 2.2 in [5]). Definiion 2.4. A family S of forward swaps associaed wih T is admissible if he following condiions are saisfied: (i) he number of forward swaps in S equals n, i.e., l = n, (ii) any dae T i T coincides wih he rese/selemen dae of a leas one forward swap from S, (iii) here are no cycles (also called degenerae subses in [5]) in S. The main heoreical resul in [5] saes ha he admissibiliy of S, in he sense of Definiion 2.4, is necessary and sufficien for he following propery: a se of non-zero deflaed bond prices associaed wih S exiss and is unique, P-a.s., for any choice of he bond numéraire and any generic process (κ 1,..., κ n ). In oher words, he main resul in [5] (see Proposiion 2.1 herein) esablishes he equivalence beween he admissibiliy of S and he exisence of a unique soluion o he inverse problem, P-a.s. As we will argue in wha follows, however, his resul is no rue, in general. Therefore, Definiion 2.4 of admissibiliy of a family S seems o be misleading, since i hinges on he validiy of Proposiion 2.1 in [5]. I is our undersanding ha he mehod of proof of Proposiion 2.1 in [5] relies on he following conjecures. If he number l of forward swaps in S differs from n hen eiher here is no soluion he he linear sysem C b x b = e b or he non-uniqueness of a soluion holds. Hence he necessary requiremen is ha C b should be a square marix, so ha l = n. If l = n, bu a cycle exiss in S (see Secion 2.3 below), hen he rows in he marix C b are linearly dependen, and hus he rank of C b is less han n a leas for some choice of b {0,..., n}. Consequenly, here is no guaranee ha a soluion o he linear sysem C b x b = e b exiss, for almos all realizaions of forward swap raes and for an arbirary choice of he bond B(, T b ) as a numéraire asse. Oherwise, ha is, when l = n and here are no cycles in S hen, for any choice of b {0,..., n}, he rows in he marix C b are linearly independen and hus he marix C b is non-singular, for almos all realizaions of forward swap raes. Consequenly, he unique soluion o C b x b = e b exiss wih probabiliy one for any diffusion-ype dynamics of forward swap raes. Moreover, for almos all realizaions of forward swap raes, he soluion x b is non-zero, meaning ha x i 0 for every i = 0,..., n (recall ha x b = 1).

7 L. Li and M. Rukowski 7 We noe ha he proof of Proposiion 2.1 in [5] suffers from he following shorcomings. The proof of he sufficiency clause is based on he argumen ha if he firs column has only one non-zero enry hen he row vecors are linearly independen. This is, of course, no rue in general (i seems ha he auhors assume ha he coefficiens in he linear combinaion should all be non-zero, whereas in fac i suffices ha no all of hem are zero). A similar argumen is used when here a several non-zero enries. The proof of he necessiy clause is based on he observaion ha he sum of equaions corresponding o he upper and lower pahs in a cycle yields an equaion of a special shape. This fac does no prove, however, ha he row vecors are linearly dependen, as was claimed in [5]. To illusrae his poin, we presen below an example of a family S wih a cycle for which for any choice of b {0,..., n} he corresponding marix C b is non-singular, for almos all (κ 1,..., κ l ) R l. Example 2.1. The following simple couner-example shows ha a family S wih a cycle can be weakly T -admissible. Le n = 3 and le (s j, m j ), j = 1, 2, 3 be given as (0, 2), (2, 3) and (0, 3), respecively. The swaps S 1 and S 2 yield he pah (T 0, T 3 ) and his pah is also given by he swap S 3, so ha a cycle (T 0, T 0 ) exiss. For b = 0, we obain he following linear sysem C 0 x 0 = c 1,1 c 1, c 2,3 c 3,1 c 3,2 c 3,3 x 1 x 2 x 3 = = e 0. One can check by direc compuaions ha, for any choice of b {1, 2, 3}, he following properies hold: (i) for almos all (κ 1, κ 2, κ 3 ) R 3, he marix C b is non-singular and (ii) for almos all (κ 1, κ 2, κ 3 ) R 3, he unique soluion x b has all enries non-zero. We conclude ha he considered family S is no admissible, in he sense of Definiion 2.4, bu i saisfies he Definiion 2.3 of he weak T -admissibiliy. This example makes i clear ha he necessiy clause in Proposiion 2.1 in [5] is incorrec. We will laer commen on he sufficiency clause in Proposiion 2.1 in [5]. 2.3 Graph Theory Terminology Le us firs recall some basic graph heory erminology, which will prove useful in wha follows. For an in-deph inroducion o he graph heory, he ineresed reader may consul he monograph by Bollobás [1]. Definiion 2.5. A graph G is an ordered pair of disjoin ses (V, E) such ha E is a subse of he se of unordered pairs of V. The se V is called he verex se and E is called he edge se. Definiion 2.6. (i) Two verices v 1 and v 2 are said o be adjacen if here exiss an edge e E such ha e is he unordered pair {v 1, v 2 }. (ii) A pah is a finie sequence of verices such ha each of he verex is adjacen o nex verex in he sequence. (iii) A cycle is a pah such ha he sar verex and end verex are he same. (iv) A graph G is said o be conneced if here exiss a pah beween any wo verices. (v) A graph G is acyclic if here exiss no cycles. The nex resul is aken from [1]; i is a combinaion of Theorem 4 on page 7 and Exercise 9 on page 23 herein. Theorem 2.1. The following asserions are equivalen for a graph G wih m verices: (i) G is a minimal conneced graph, ha is, G is conneced and if {x, y} E hen G {x, y} is disconneced, (ii) G is a maximal acyclic graph, ha is, G is acyclic and if x and y any non-adjacen verices of G hen he graph G + {x, y} conain a cycle, (iii) G is conneced and has m 1 edges, (iv) G is acyclic and has m 1 edges.

8 8 Admissibiliy of Marke Models of Swap Raes As previously inroduced, he se T 0 (S) is he se of all sar or mauriy daes of swaps in S. To ranslae ino graph heory erminology, we regard he se T 0 (S) as a verex se. A swap S j S which sar a T sj and maures a T mj is characerized by he pair of elemens {T sj, T mj }. This characerizaion allows us o regard S as he edge se. Noe ha we only consider graphs ha have finie number of verices. Definiion 2.7. The graph of S is he verex and edge pair given as (T 0 (S), S). I is clear ha he graph of S has l edges and a mos n + 1 verices. The exisence of a cycle in he graph of S does no seem o be of primary ineres in he sudy of T -admissibiliy of a family S (see Secion 2.4, Remark 2.2). However, in order o re-examine Proposiion 2.1 in [5], we will firs presen some basic properies of he graph of S and relaed o he exisence of a cycle in he graph of S. The following lemma is a sraighforward consequence of Theorem 2.1. Lemma 2.1. Le S = {S 1,..., S l } be any family of swaps wih T (S) = T. If l > n hen here exiss a cycle in he graph of S. The nex auxiliary resul deals wih he acyclic case. Lemma 2.2. Le S = {S 1,..., S n } be a family of n swaps wih T (S) = T. If here are no cycles in he graph of S hen he equaliy T 0 (S) = T holds. Proof. Le us assume ha here exiss a dae T k T which does no belong o T 0 (S). Then here exis a family S of n swaps wih T 0 ( S) = {T 0 < < T k 1 < T k+1 < < T n }. To obain S from S, i suffices o delee he dae T k from all swap equaions. Hence, by Lemma 2.1 a cycle necessarily exiss in S and hus also in S. Lemma 2.3. Assume ha l = n and he graph S is conneced. Then T 0 (S) = T and hus here are no cycles in S. 2.4 Weak Admissibiliy of Forward Swaps We posulae hroughou ha he family S is such ha he equaliy T (S) = T is valid. Equivalenly, for an arbirary choice of b {0,..., n} here is no null column in he marix C b associaed wih S. Before we proceed, le us quickly noe Lemma 2.4. The following properies are equivalen: (i) he sysem C b x b = e b has a non-zero soluion for some b {0,... n}, (ii) he sysem C b x b = e b has a non-zero soluion for all b {0,... n}. Proof. (i) (ii) If a non-zero soluion exiss for a paricular b, hen for any b {0,... n} he soluion o he sysem C bx b = e b can be obained by aking he raios of he soluion o he sysem C b x b = e b. This argumen corresponds o he simple observaion ha o obain he deflaed bond prices for any oher choice of he numéraire bond, one may use he equaliy The implicaion (ii) (i) is obvious. B b(, T i ) = Bb (, T i ) B b (, T b), b b (T, b)-inadmissibiliy The following concep of (T, b)-inadmissibiliy will prove useful in he sequel.

9 L. Li and M. Rukowski 9 Definiion 2.8. A subse S of a family S of swaps is said o be (T, b)-inadmissible if he number of swaps in S is sricly greaer han he number of daes in T ( S) \ {T b }. We denoe by M 0 he class of all families of swaps ha do no conain a (T, b)-inadmissible subse for every b {0,..., n}. Noe ha he number of daes in T ( S) \ {T b } is equal o he number of all variables from he se x 0,..., x b 1, x b+1,..., x n ha are associaed wih a given subse S. The following properies are obvious: if a subse S is (T, b)-inadmissible for some b such ha T b T ( S) hen i is also (T, b)-inadmissible for any b such ha T b T ( S). if a subse S is (T, b)-inadmissible for some b such ha T b / T ( S) hen i is also (T, b)-inadmissible for any b {0,..., n}. Example 2.2. The family S inroduced in Example 2.1 represens a cycle S c, which is no (T, b)- inadmissible for any choice of b {0,..., 4}. Indeed, we have here S c = S and T (S c ) = T (S) = T. Hence, for any b {0,..., 4}, he number of daes in he se T (S c ) \ {T b } is equal o 3 and hus i always coincides wih he number of swaps in S c. Remark 2.2. The exisence a (T, b)-inadmissible cycle S c for some b {0,..., n} implies he exisence of a (T, b)-inadmissible subse S for some b {0,..., n} (simply ake b = b). Noe, however, ha, in general, he exisence of a (T, b)-inadmissible subse S for some b {0,..., n} does no imply he exisence of a (T, b)-inadmissible cycle S c, as he following couner-example shows. Example 2.3. The forward swaps ha we will consider here are sandard forward swaps, so ha all daes beween he saring dae and mauriy dae are selemen daes. Le S be S 1 c S 2 c, where S 1 c = {S 0 = {T 0, T 3 }, S 1 = {T 0, T 4 }, S 2 = {T 3, T 4 }} and S 2 c = {S 3 = {T 1, T 2 }, S 4 = {T 2, T 5 }, S 5 = {T 1, T 5 }}. I is easy o see ha S is a T b -inadmissible subse for every b = {0,..., 5}. However, he only cycles S 1 c and S 2 c are clearly no (T, b)-inadmissible cycles for any b = {0,..., 5}. We denoe by C he class of all families of forward swaps ha do no conain a cycle. Lemma 2.5. The exisence of a (T, b)-inadmissible subse implies he exisence of a cycle, so ha C M 0. Proof. By definiion, here exiss a subse of swaps S such ha he number of relevan daes (variables) is sricly less han he number of swaps in he S. Now, le us consider he graph (T 0 ( S), S). Since S is a (T, b)-inadmissible subse, he number of edges is sricly greaer o he number of verices. By Theorem 2.1, here mus exis a cycle (any oher relevan dae ha is no in T ( S) bu falls beween he smalles and larges dae in T ( S) simply sreches he cycle). In view of Lemma 2.5, he condiion of he non-exisence of a cycle in S is sronger han he assumpion of he non-exisence of a (T, b)-inadmissible subse, which will be shown o be a necessary condiion for he weak T -admissibiliy (see Lemma 2.7). Recall ha Example 2.1 describes a family wih a cycle which is weakly T -admissible. We hus see ha he non-exisence of a cycle is no a necessary condiion for he weak T -admissibiliy, ha is, he class A is no included in C. The quesion wheher he non-exisence of a cycle is a sufficien condiion for he weak T - admissibiliy, ha is, wheher C A (i is clear ha C A) remains open (see Conjecure 2.1 below) Necessary Condiions for he Weak T -Admissibiliy By a verical block, we mean he collecion of rows in which he number of non-null columns is less han he number of rows. In he language of he linear algebra, a verical block corresponds o

10 10 Admissibiliy of Marke Models of Swap Raes an over-deermined sub-sysem. However, for breviy, we will also frequenly use he erm verical block. By a permuaion of he marix C b we mean a finie number of eiher row or column swaps (or boh) performed on he marix C b. Lemma 2.6. Le us fix some b {0,..., n}. The following properies are equivalen: (i) here exiss a (T, b)-inadmissible subse S, (ii) here exiss a verical block in he marix C b afer some permuaion. Proof. I is clear ha any (T, b)-inadmissible subse S corresponds, afer a suiable permuaion, o a verical block. The converse implicaion follows immediaely from he definiion of a (T, b)- inadmissible subse. The following lemma shows ha he exisence of a (T, b)-inadmissible subse in S implies he non-exisence of a soluion o C b x b = e b a leas for some b {0,..., n}. Lemma 2.7. If a family S conains a (T, b)-inadmissible subse S for some b {0,..., n} hen S is no weakly T -admissible. Consequenly, he inclusion A M 0 is valid. Proof. I suffices o focus on he variables corresponding o he daes from he se T ( S) \ T b, where b is such ha he subse S is (T, b)-inadmissible. If T b / T ( S) hen hese variables need o saisfy an over-deermined homogeneous linear sysem. We hen deal wih he following wo subcases: (i) if he deerminan of a square sub-sysem is a non-zero raional funcion hen he null soluion is he unique soluion o he paricular subse of unknowns, (ii) he deerminan of a square sub-sysem is zero and hus he soluion eiher does no exiss or is no unique. Therefore, we see ha in boh cases he family S is no weakly T -admissible. If, on he conrary, T b T ( S) hen hese variables need o saisfy a non-homogeneous linear sysem in which he number of equaions is larger han he number of variables. Then by changing he dae T b o some dae T b / T ( S), we obain a once again a homogeneous linear sysem, in which he number of equaions is a leas equal o he number of variables. Therefore, we are back in eiher case (i) or case (ii) (if all daes are already in T ( S), ha is, when T = T ( S), we can add one more dae o T in order o perform his sep). Remark 2.3. I can be shown by means of an example ha he inclusion A M 0 is in fac sric, i.e. A M 0, in general. Specifically, for any n 2 here exiss a family S of forward swaps such ha S A c M 0 (e.g., i is easy o produce a family wih a cycle, which is no weakly T -admissible). In oher words, he propery ha S belongs o M 0 is a necessary, bu no sufficien, condiion for he weak T -admissibiliy of a family S. The nex sep oward a sufficien condiion for he weak T -admissibiliy of a family S is he following simple resul, which provides a sronger necessary condiion. Lemma 2.8. If for some b {0,..., n}, he linear sysem C b x b = e b associaed wih S has a verical block afer a finie number of row operaions hen he family of forward swaps S is no weakly T -admissible. Proof. Suppose afer finie number of row operaion, here exiss a verical block and eiher (i) he verical block forms a homogeneous over-deermined sub-sysem, or (ii) he verical block forms a non-homogeneous over-deermined sub-sysem. In case (i), if any square sub-sysem has non-zero deerminan hen he zero soluion is he unique soluion o he subsysem. On he oher hand, if any square sub-sysem has zero deerminan hen he original marix is no inverible. In case (ii), one can adjus he choice of he numéraire bond. By picking B(, T b) such ha x b is no in he over-deermined sub-sysem, he non-homogeneous over-deermined sub-sysem becomes a homogeneous over-deermined sub-sysem and we are back in case (i) (similar argumen applies if he sub-sysem is square). Then, by Lemma 2.4, he family S is no weakly T -admissible.

11 L. Li and M. Rukowski 11 Definiion 2.9. For a fixed b {0,..., n}, we say ha he marix C b is diagonalizable if here exiss a permuaion of C b such ha all enries on he firs diagonal of C b are non-zero. An l n marix (l n) is diagonalizable if here exiss a permuaion such ha he diagonal enries of a l l sub-marix are non-zero. The proof of he following lemma is deferred o he appendix. Lemma 2.9. Le us fix b {0,..., n}. The following properies are equivalen: (i) he marix C b is diagonalizable, (ii) here is no (T, b)-inadmissible subse S in S. Recall ha, by Lemma 2.6, he exisence of a verical block is equivalen o he exisence of a (T, b)-inadmissible subse S. Suppose ha for some b here exiss a verical block in C b afer some permuaion π b of variables. I may sill happen ha he marix is diagonalizable for some oher b. Example 2.4. One can provide examples of marices, which are diagonalizable for every b {0,..., n}, bu fail o have a non-zero deerminan. The following marix is diagonalizable for every b {0, 1, 2, 3, 4, 5, 6}, bu is deerminan equals zero C b = 1 a 1 κ 1 a 2 κ 1 a 3 κ a 4 κ a 1 κ 2 a 2 κ 2 a 3 κ 2 a 4 κ a 5 κ 2 1 a 1 κ 3 a 2 κ 3 a 3 κ 3 a 4 κ 3 a 5 κ a 5 κ a 2 κ a 3 κ e b = a 6 κ a 6 κ 5 0 Noe ha he fourh column can be wrien as linear combinaions of second and hird columns. In wha follows, by an inernal dae of a swap we mean a relevan dae, which is neiher he sar nor he mauriy dae of a given swap. Definiion A diagonalizable family of forward swaps is said o have a firs order (T, b)- inadmissible subse if i has he following characerisics: here exiss k daes for some k 2 such ha: (i) hey are inernal daes o a leas wo swaps and (ii) hey are relevan o a mos k 2 oher swaps. We denoe by M 1 he class of all families of forward swaps ha does no conain a firs order (T, T b )-inadmissible subse for all b {0,..., n}. Lemma The non-exisence of firs order (T, b)-inadmissible subse is necessary for a family of swaps o be weakly T -admissible, so ha A M 1. Proof. Suppose ha here exiss a firs order (T, b)-inadmissible subse. I is sufficien o concenrae on hose k inernal daes (columns). Wihou loss of generaliy, we may pick one column as he pivo column and perform column operaions o eliminaed oher columns. By he end, he sub-sysem wihou he pivo column consiss of k 1 columns and a mos k 2 non-zero rows. This gives linear dependence of he columns. The following example will make his argumen more explici, he reader should ake some ime o convince hemselves ha he argumen is general. Le us focus on columns number wo, hree and four in he marix in Example 2.4 a 1 κ 1 a 2 κ 1 a 3 κ 1 a 1 κ 2 a 2 κ 2 a 3 κ 2 a 1 κ 3 a 2 κ 3 a 3 κ a 2 κ a 3 κ 5 c 3 = a 3 a 1 c 1 c 3 c 2= a 2 a 1 c 1 c 2 I is clear ha he columns are linearly dependen. a 1 κ a 1 κ a 1 κ a 2 κ 5 a 2 a a 3 κ 5 a 3 a 1..

12 12 Admissibiliy of Marke Models of Swap Raes Lemma The exisence of a firs order (T, b)-inadmissible subse implies he exisence of a cycle, ha is, C M 1. Proof. By definiion, here exis k inernal daes, such ha a mos k 2 swaps sar or maure on hose k inernal daes. Therefore, here mus be a leas n k + 2 swaps saring and mauring ouside he k inernal daes. If one consider he graph of S wihou hose k inernal daes, i is a graph wih n k verices and a leas n k + 2 edges, and hus, by Theorem 2.1, here mus be a cycle Sufficien Condiions for he Weak T -Admissibiliy In his secion, we se l = n and we consider an arbirary family S of forward swaps. Our nex goal is o provide a sufficien condiion for he weak T -admissibiliy of a family S. Le M sand for he class of all families of forward swaps which, for any choice of b {0,..., n}, he diagonalizaion propery is preserved under elemenary row operaions. Lemma Assume ha he marix C b associaed wih a family S is diagonalizable and he diagonalizaion propery is preserved under elemenary row operaions for every b {0,..., n}, ha is, C b M for every b. Then a family S is weakly T -admissible. This means ha A = M M 0. Proof. Le us firs prove sufficiency. By assumpion, he marix is diagonalizaion for every b {0,..., n}. Therefore for a fixed b {0,..., n} he marix C b can be diagonalized, ha is, Nex, we ake he firs row as he pivo row and we eliminae he enries below he firs diagonal and, by assumpion, he sub-marix is again diagonalizable. The procedure hen coninues wih he second row becoming he pivo row, so ha he enries below he second diagonal are eliminaed. A each sage, he assumpion ha diagonalizaion propery is preserved is used o diagonalize he marix afer each row operaion. By he end of he procedure, we obain a marix in row echelon form wih he diagonals consising of non-zero raional funcions in κ 1,..., κ n. I is hen easy o see ha he deerminan is non-zero, for almos all generic values of (κ 1,... κ n ) R n. The necessiy clause will be proved by conradicion. Le us assume ha he diagonalizaion propery is no preserved under elemenary row operaions for every b {0,... n} and he sysem has a unique non-zero soluion. This implies ha for some b {0,... n}, he marix C b is no longer diagonalizable (i.e., here exiss a verical block ) afer for finie number of elemenary row operaions. By Lemma 2.8, he family S is no weakly T -admissible. However on he oher hand, he soluion is invarian under elemenary row operaion and herefore we have a conradicion. Inaddiion, i is imporan o assume ha he marix is diagonalizable for every b {0,..., n}. Hence here does no exis a square block in some C b such ha he corresponding variables in e b are all zero. Indeed, if such a siuaion occurs hen by choosing he dae T b from inside he square block, we would obain a C b wih a verical block and by Lemma 2.7 family S is no weakly T -admissible. Le us sress once again ha i is imporan o assume ha for all b {0,..., n}, he marix C b belongs o M, ha is, he diagonalizaion propery is invarian under elemenary row operaions (cf. Example 2.4). In order o summarize he resuls regarding he weak T -admissibiliy of a family S, le us firs recall he noaion: A - he class of all families of forward swaps which are weakly T -admissible for all b {0,..., n}, C - he class of all families of forward swaps ha do no conain a cycle, M 0 - he class of all families of forward swaps ha do no conain a (T, T b )-inadmissible subse for all b {0,..., n}, M 1 - he class of all families of forward swaps ha do no conain a firs order (T, T b )-inadmissible subse for all b {0,..., n}. M - he class of all families of forward swaps for which he diagonalizaion propery is preserved under elemenary row operaion for any b {0,..., n}.

13 L. Li and M. Rukowski 13 Proposiion 2.1. Le S be a family of forward swaps such ha l = n. Then: (i) A M 1 M 0, (ii) C M 1 M 0, (iii) A C c, (iv) A = M M 0. Proof. I suffices o use he auxiliary resuls esablished hus far. We noe ha: (i) by definiion M 1 M 0 and, by Lemma 2.7, he inclusion A M 1 holds, (ii) by Lemma 2.11, we have ha C M 1, (iii) Example 2.1 describes a family belonging o A C c, (iv) by Lemma 2.12, we have ha A = M M 0. Recall ha we have argued ha he (non)-exisence of an cycle is no of primary ineres, as opposed o he (non)-exisence of a (T, T b )-admissible subse. For he necessiy clause, Galluccio e al. [5] failed o observe ha no every cycle is a (T, T b )-inadmissible subse. Unforunaely, he sufficiency clause in he main resul in Galluccio e al. [5] is sill unclear, since i is no known wheher C A. Le us hen formulae wo perinen conjecures. Conjecure 2.1. The non-exisence of cycles is a sufficien condiion for a family of forward swaps o be weakly T -admissible, ha is, C A. Conjecure 2.2. The non-exisence of a firs order (T, T b )-inadmissible subse for all b {0,..., n} is sufficien for a family of swaps o be weakly T -admissible, ha is, M 1 A. In view of par (ii) in Proposiion 2.1, we see ha if one can prove ha Conjecure 2.2 is rue, hen he validiy of Conjecure 2.1 would follow auomaically. 2.5 T -Admissibiliy of Forward Swaps Recall ha, by Definiion 2.3, a family of forward swaps is T -admissible, if i is weakly T -admissible and he soluion o he linear sysem associaed wih ha family is sricly posiive for all generic values of (κ 1,..., κ n ) R n +. In his secion, we will give condiions for which a weakly T -admissible family is T -admissible. The proof of he following lemma follows raher closely he proof of sufficiency in Theorem 1 of [14]. I is imporan o poin ou, however, ha Theorem 1 in [14] gives he sufficien and necessary condiions for exisence of deflaed bonds, when given a family of swaps ha has he propery ha no wo swaps sar on he same dae. However, heir se-up is differen han ours, and he posiiviy of he deflaed bond prices was simply a by-produc of exisence of deflaed bond prices up o he bonds mauriy daes. In fac, he assumpion of posiiviy was no used a all in he proof of he necessiy clause in Theorem 1 in [14]. For his reason, we need o re-examine he necessiy clause. For he reader s convenience, we firs give he proof of sufficiency (cf. [14]). Proposiion 2.2. Assume ha a family S of forward swaps is weakly T -admissible. If no wo swaps sar on he same dae hen S is T -admissible. Proof. I is sufficien o show ha a posiive soluion exiss for b = n. Assuming no wo swaps sar on he same dae, one can arrange he row of he marix C n such ha i is an upper riangular marix wih non-zero enries on he diagonal. I is hus clear ha he family S is weakly T -admissible. We will show posiiviy of soluion by inducion and back subsiuion. By picking b = n, we have B n (, T n ) = 1, which is sricly posiive and greaer or equal o 1. Le us assume ha B(, T j ) 1 for all j > i, we would like o show B n (, T i ) 1 is posiive for all generic values of (κ 1,... κ n ) R n +. By he back subsiuion algorihm B n (, T i ) = κ i m i j=i+1 a j B n (, T j ) + B n (, T mi ), a j 0, j = {i,..., m j }. (7)

14 14 Admissibiliy of Marke Models of Swap Raes Nex, by he inducion hypohesis and he assumpion ha (κ 1,... κ n ) R n + he following holds κ i m i j=i+1 a j B n (, T j ) 0, B n (, T mi ) 1. Thus i follows easily from (7) ha B n (, T i ) 1 for almos all generic values of (κ 1,..., κ n ) R n +. Noe i was imporan o assume he marix was upper riangular wih non-zero diagonal or else he back subsiuion algorihm would no apply. In he following, we will give a parial converse of Lemma 2.2. Assumpion 2.1. Le S i and S l be wo forward swaps such ha T si = T sl and, wihou loss of generaliy, T ml < T mi. Then T (S l ) T (S i ). The proof of he nex resul is given in he appendix. Proposiion 2.3. If a family S is T -admissible and Assumpion 2.1 is saisfied hen here are no wo swaps in S saring on he same dae. Noe he assumpion ha T (S l ) T (S i ) is crucial for he proof. Of course, his assumpion is saisfied in he case of sandard swaps. The following is an example of a family wih wo forward swaps saring a he same dae, for which he soluion is sricly posiive. Example 2.5. The associaed linear sysem is given by C 0 x 0 = a 2κ a 3 κ 2 x 1 x 2 = 1 0 = e a 3 κ 3 x 3 1 By solving his sysem, we obain x 0 = 1+a 3 κ 2 1+a 3 κ 3 (1 + a 2 κ 1 ) 1 (1 + a 3 κ 3 ) 1. From he form of he soluion, we see ha he deflaed bonds prices are sricly posiive for all generic values of (κ 1, κ 2, κ 3 ) R Admissible Marke Models of Forward Swap Raes In his secion, we will examine he second issue menioned in he inroducion, ha is, he problem wheher he join dynamics of a given family of forward swap raes is uniquely deermined. To his end, we will focus on he Radon-Nikodým densiies of he corresponding family of forward swap measures. The crucial objec in he sudy of forward swap measures is in urn he swap numéraire, ha is, he denominaor appearing in he definiion of he swap rae. For he deailed analysis of some special cases of models considered in his secion, such as: he LIBOR marke model and he marke model for co-erminal swaps, we refer o Brace e al. [2], Davis and Maaix-Pasor [4], Jamshidian [7, 8, 9], Musiela and Rukowski [12], Rukowski [15] (see also, Chapers 12 and 13 in [13] and he references herein). Due o he limied space, we will no deal here neiher wih any paricular examples of marke models of forward swap raes, nor wih oher relaed quesions such as he posiiviy of raes whose dynamics are no direcly posulaed, bu are implicily given by a specificaion of a marke model of some family S of forward swaps.

15 L. Li and M. Rukowski Inverse Problem for Swap Annuiies By an absrac forward swap we mean he sar dae T sj, he mauriy dae T mj as well a pair P s j,m j, A s j,m j of processes, where A s j,m j is a posiive process. A family of absrac swaps S is simply a collecion of absrac swaps. The forward swap rae κ j in an absrac forward swap saring a T sj and mauring a T mj is defined by he formula κ j = κ s j,m j Recall from Secion 1 ha he deflaed swap annuiy is given by = A b,s k,m k = P sj,m j A s j,m j, [0, T sj ]. (8) i A k ã k B b (, T i ), (9) where A k represens he se of all selemen daes in S k. Le us define, for a fixed d {1,..., l}, he annuiy deflaed swap numéraire and le us denoe Then (8) becomes à d,j = As j,m j A s d,m d κ j = κ s j,m j = Ab,s j,m j A b,s d,m d = P d,j d,j P à d,j = P sj,m j, [0, T sj T sd ], (10) A s d,m d In view of he second equaliy in (11), i is also clear ha he process Ãd,j raio of deflaed swap annuiies, for any choice of b.., [0, T sj T sd ]. (11) can be represened as he One should ake noe here ha he annuiy deflaed swap numéraire process Ãd,j is really he process of ineres, as i would laer on ac as he Radon-Nikodým densiy process in he specificaion of he join dynamics of he processes κ 1,..., κ l under a single probabiliy measure. I is hus naural o consider he following inverse problem for swap numéraires. Inverse Problem (IP.2): A family S = {S 1,..., S l } admis a soluion o Inverse Problem (IP.2) if, for any choice of d {1,..., l}, every annuiy deflaed swap numéraire Ãd,j can be expressed as a funcion of variables (κ 1,..., κ l ). The following definiion describes he class of families of forward swaps possessing he required properies. Definiion 3.1. We say ha S is A-admissible if, for almos every (κ 1,..., κ l ) R l and any choice of he swap deflaor A s d,m d, here exiss a unique non-zero soluion Ãd,j o he sysem of annuiy deflaed swap equaions (11) and his soluion is given in erms of (κ 1,..., κ l ) only. Le us firs consider a family of forward swaps S = {S 1,..., S l }, which admis a (possibly nonunique, bu non-zero) soluion o he inverse problem (IP.1). This means ha, a.s. wih respec o he Lebesgue measure in R l, B b (, T k ) = g b,k (κ 1,..., κ l ) + i j=1 g b,k j (κ 1,..., κ l )B b (, T nj ) T k T, (12) where {B b (, T n1 ),..., B b (, T ni )} is he sub-family of deflaed bonds which parameerizes he soluion o he linear sysem associaed wih S. By subsiuing equaion (12) ino equaion (9), we

16 16 Admissibiliy of Marke Models of Swap Raes obain he following sysem of equaions A b,sj,mj = g b,j (κ 1,..., κ l ) + i k=1 g b,j k (κ1,..., κ l )B b (, T nk ). (13) I is obvious ha, independenly of he choice of he bond numéraire B(, T b ), he annuiy deflaed swap numeráire can now be expressed as follows à d,j /B(, T b ) /B(, T b ) = gb,j (κ 1,..., κ l ) + g b,d (κ 1,..., κ l ) + i = Asj,m j A s d,m d i k=1 gb,j k (κ1,..., κ l )B b (, T nk ) k=1 gb,d k (κ1,..., κ l )B b (, T nk ). (14) The family S is also A-admissible, provided ha equaion (14) can be simplified so ha à d,j = G d,j (κ 1,..., κ l ) for some non-zero raional funcions G d,j (κ 1,..., κ l ). Clearly, if he family S is weakly T -admissible hen i A-admissible as well. If, however, he parameerizing family of bond prices can no be compleely eliminaed in (14) for some annuiy deflaed swap numéraire in A d = {Ãd,j, j = 1,..., d 1, d + 1,..., l} hen he family of annuiy deflaed swap numéraires A d is no uniquely deermined by forward swap raes, so ha he family S is no A-admissible. Example 3.1. I is worh sressing ha he weak T -admissibiliy is no a necessary condiion for he A-admissibiliy of S. Le us give an example of an family of swaps, which is A-admissible, bu is no weakly T -admissible. Consider he sandard forward swaps S 1 = {T 0, T 4 }, S 2 = {T 0, T 3 } and S 3 = {T 2, T 3 }. The linear sysem associaed wih S for b = 0 is given as follows a 1κ 1 a 2 κ 1 a 3 κ a 4 κ 1 a 1 κ 2 a 2 κ a 3 κ a 4 κ 3 x 1 x 2 x 3 x 4 = A soluion o he corresponding inverse problem (IP.1) is given by x 1 x 0 = x 2 x 3 = x 4 ( a 2κ 1 γ a 2 κ 1 a 4 κ 3 γ+a 2 κ 2 γ+a 2 κ 1 κ 2 a 4 γ)+a 4 κ 3 a 3 κ 1 +a 3 κ 2 a 3 κ 1 a 4 κ 3 +a 3 a 4 κ 3 κ 2 a 4 κ 1 ( κ 1 κ 1a 4κ 3+κ 2+κ 2a 4κ 1)a 1 γ (1+a 4 κ 3 )(κ 1 κ 2 ) κ 1 κ 1a 4κ 3+κ 2+κ 2a 4κ 1 κ 1 κ 2 κ 1 κ 1a 4κ 3+κ 2+κ 2a 4κ 1, so ha is parameerized by a generic value γ of B 0 (, T 2 ). I is hus clear ha he family is no weakly T -admissible. We will show ha he family S is A-admissible. For his purpose, we subsiue a soluion ino he deflaed swap annuiy equaion (or generally (9)) and we obain A 0,s1,m1 = A 0,s2,m2 = A 0,s 3,m 3 a 4 ( κ 3 + κ 2 ) κ 1 κ 1 a 4 κ 3 + κ 2 + κ 2 a 4 κ 1, a 4 ( κ 3 + κ 1 ) κ 1 κ 1 a 4 κ 3 + κ 2 + κ 2 a 4 κ 1, κ 1 κ 2 =. κ 1 κ 1 a 4 κ 3 + κ 2 + κ 2 a 4 κ 1 We conclude ha he deflaed swap annuiies can be uniquely expressed as funcions of forward swaps raes (κ 1, κ 2, κ 3 ) R 3, and hus he considered family of forward swaps is indeed A-admissible.

17 L. Li and M. Rukowski Dynamics of Forward Swap Raes Consider a family of swap rae processes denoed by S κ = {κ 1,..., κ l }. We aim o show under some weak condiions imposed on a family S of swap rae processes heir join dynamics are uniquely specified by a family of volailiy process wih respec o some spanning maringale. To his end, we need o recall some resuls from sochasic calculus, which will be used in wha follows. In his secion, he mulidimensional Iô inegral should be inerpreed as he vecor sochasic inegral (see, for insance, Shiryaev and Cherny [16]). Le us firs quoe a version of he Girsanov heorem (see, for insance, Brémaud and Yor [3] or Theorem in Jeanblanc e al. [10]). Proposiion 3.1. Le P and P be equivalen probabiliy measures on (Ω, F T ) wih he Radon- Nikodým densiy process Z = d P, [0, T ]. (15) dp F Suppose ha M is a (P, F)-local maringale. Then he process 1 M = M d[z, M] s Z s is a ( P, F)-local maringale. (0,] Le M loc (P) (M(P), resp.) sand for he class of all (P, F)-local maringales ((P, F)-maringales, resp.). Assume ha Z is a posiive (P, F)-local maringale such ha Z 0 = 1. Also le an equivalen probabiliy measure P be given by (15). Then he linear map Ψ Z : M loc (P) M loc ( P) given by he formula Ψ Z (M) = M (0,] 1 Z s d[z, M] s, M M loc (P), (16) is called he Girsanov ransform associaed wih he Radon-Nikodým densiy Z. By he symmery of he problem, he process Z 1 is he Radon-Nikodým densiy of P wih respec o P. The corresponding Girsanov ransform Ψ Z 1 : M loc ( P) M loc (P) associaed wih Z 1 is hus given by he formula Ψ Z 1( M) = M Z s d[z 1, M] s, M M loc ( P). (0,] Proposiion 3.2. Le P be a probabiliy measure equivalen o P on (Ω, F T ) wih he Radon-Nikodým densiy process Z. Then for any ( P, F)-local maringale Ñ here exiss a (P, F)-local maringale N such ha Ñ = N 1 d[z, Z N] s. (17) s The process N is given by he formula N = Ñ0 + where L = ÑZ. (0,] (0,] 1 L s dl s Z s (0,] Zs 2 dz s (18) From Proposiion 3.2, i follows ha he process N 1 given by (18) belongs o he se (Ψ Z ) (Ñ). In fac, we have ha N 1 = (Ψ Z ) (Ñ), as he following resul shows. Lemma 3.1. Le Ñ be any ( P, F)-local maringale and le he process N be given by formula (18) wih L = ÑZ. Then he process N is also given by he following expression N = Ñ Z s d[z 1, Ñ] s. (19) (0,] The linear map Ψ Z : M loc (P) M loc ( P) is bijecive and he inverse map (Ψ Z ) 1 : M loc ( P) M loc (P) saisfies (Ψ Z ) 1 = Ψ Z 1.