Models of Forward LIBOR and Swap RatesMAREK RUTKOWSKI �Instytut Matematyki, Politechnika Warszawska, 00-661 Warszawa, PolandAbstractThe backward induction approach is systematically used to produce various models of for-ward market rates. These include the lognormal model of forward LIBOR rates examined inMiltersen et al. (1997) and Brace et al. (1997), as well as the lognormal model of (�xed-maturity)forward swap rates proposed by Jamshidian (1996, 1997). The valuation formulae for Europeancaps and swaptions are given. In the last section, the Eurodollar futures contracts and optionsare examined within the framework of the lognormal model of forward LIBOR rates.Keywords: zero-coupon bond, LIBOR rate, swap rate, cap, swaption, Eurodollar futures1 IntroductionThe aim of this note is to present in a systematic way the backward induction approach to themodelling of forward market rates (a generic term market rate is used here to describe any kind ofmarket interest rate, such as LIBOR rate or swap rate). In the context of market rate modelling,such an approach was initiated by Musiela and Rutkowski (1997), who employed this methodologyto construct the lognormal model of forward LIBOR rates (with �xed accrual period). The ideaof term structure modelling with the direct use of forward measures was previously exploited byHansen (1994), who used the forward induction to produce an arbitrage-free di�usion-type1 modelfor prices of a �nite family of zero-coupon bonds with di�erent maturities. Let us focus on the mostimportant features of the backward induction approach (also termed the forward measure approach).First, in contrast to a considerable number of published papers on term structure modelling, weare not interested (at least, not primarily) in producing a multi- (or in�nite-) dimensional processwhich would describe the bond prices (not to mention the process of the short-term interest rate),but rather we focus directly on a �nite collection of speci�c market rates. The basic inputs of eachmodel are thus the quantities (typically, volatilities) associated with these rates, as well as the initialterm structure, re ected here by prices of a �nite family of bonds with prespeci�ed maturities.Second, the arbitrage-free features of a model are embedded in assumptions which are imposed(in an implicit way) on the dynamics of relative bond price under the (generalized) forward measure.In this sense, the methodology presented in this note is also closely related to the change of numerairetechnique which is now commonly standard in arbitrage pricing (see, e.g., Geman et al. (1995)).The �rst feature leads to the following natural and important question: is it possible to �nd anarbitrage-free family of (either some or all) bond prices with a particular model of forward marketrates? Due to second feature of the backward induction procedure employed in the modelling ofmarket rates, the answer to this question is positive, in general. It is worthwhile to stress that theunderlying bond prices are not uniquely determined, however. Therefore the problem of the rightchoice of the underlying family of bond prices appears here in a natural way (for a practical solutionto this problem, we refer to Sawa (1997)).�Present address: School of Mathematics, University of New South Wales, 2052 Sydney, NSW, Australia.1Hansen's work is indeed much closer in the spirit to the paper Ball and Torous (1983), who modelled bond pricesthrough Brownian bridges. In contrast to their rather crude approach, a clever choice of the drift and di�usioncoe�cients guarantees the arbitrage-free property of Hansen's model.1

2 Models of Forward LIBOR and Swap Rates2 PreliminariesLet T � > 0 be a horizon date for our model of economy. In this preliminary section, we focus onLIBOR rates which correspond to a �xed time period, denoted by �: Let B(t; T ) stand for the priceat time t of a unit zero-coupon bond that matures at time T: By de�nition, the forward �-LIBORrate L(t; T ) = L(t; T; �) prevailing at time t for the future date T � T ��� (or rather, over the futuretime period [T; T + �]) is de�ned through the formula1 + �L(t; T ) = B(t; T )B(t; T + �) ; 8 t 2 [0; T ]: (1)Put more explicitly, we haveL(t; T ) = B(t; T )�B(t; T + �)�B(t; T + �) ; 8 t 2 [0; T ]: (2)In particular, the initial term structure of forward LIBOR rates and the initial term structure ofbond prices are related to each other through the following relationshipL(0; T ) = B(0; T )�B(0; T + �)�B(0; T + �) ; 8T 2 [0; T � � �]:Equivalently, in terms of the initial yield curve Y (0; �) we haveL(0; T ) = ��1 �e(T+�)Y (0;T+�)�TY (0;T ) � 1�since Y (0; �) satis�es B(0; T ) = e�TY (0;T ) for any maturity T:To introduce the uncertainty in our model, we assume throughout that we are given a d-dimensional standard Brownian motion W; de�ned on an underlying �ltered probability space(;F;P): The �ltration F = (Ft)t2[0;T�] coincides with the P-completion of the natural �ltrationof the underlying noise process W:3 Model of Forward LIBOR RatesIn this section, we describe the backward induction approach to the modelling of forward LIBORrates. The construction presented below is a slight modi�cation and extension of that given byMusiela and Rutkowski (1997a). Let us start by introducing some notation. We are given a pre-speci�ed collection of reset/settlement dates T0 = 0 < T1 < � � � < TM = T �; referred to as the tenorstructure. Let us denote �j = Tj � Tj�1 for j = 1; : : : ;M: Then obviously Tj = Pji=1 �i for everyj = 0; : : : ;M: We �nd it convenient to denoteT �m = T � � MXj=M�m+1 �j = TM�m; 8m = 0; : : : ;M:For any j = 1; : : : ;M � 1; we de�ne the forward LIBOR rate L(�; Tj) by settingL(t; Tj) = L(t; Tj ; �j+1) = B(t; Tj)�B(t; Tj+1)�j+1B(t; Tj+1) ; 8 t 2 [0; Tj ]:De�nition 3.1 Let us �x j = 1; : : : ;M: A probability measure PTj on (;FTj ); equivalent to P; issaid to be the forward LIBOR measure for the date Tj if, for every k = 1; : : : ;M; the relative bondprice UM�j+1(t; Tk) def= B(t; Tk)�jB(t; Tj) ; 8 t 2 [0; Tk ^ Tj ];follows a local martingale under PTj :

Marek Rutkowski 3It is clear that the notion of forward LIBOR measure is in fact identical with that of a forwardprobability measure for a given date (see, e.g., Geman et al. (1995)). The slight modi�cation of thestandard terminology emphasizes our intention to make a clear distinction between various kinds offorward probabilities, which we are going to study in the sequel. Also, it is trivial to observe thatthe forward LIBOR rate L(�; Tj) necessarily follows a local martingale under the forward LIBORmeasure for the date Tj+1: If, in addition, it is a strictly positive process, the existence of theassociated volatility process can be easily justi�ed.In our further development, we shall go the other way around; that is, we will assume that for anydate Tj ; the volatility �(�; Tj) of the forward LIBOR rate L(�; Tj) is exogenously given. Basically, itcan be a deterministic Rd-valued function of time, a Rd-valued function of the underlying forwardLIBOR rates, or a d-dimensional adapted stochastic process. For simplicity, we assume throughoutthat the volatilities of forward LIBOR rates are bounded.Our aim is to construct a family L(�; Tj); j = 1; : : : ;M � 1 of forward LIBOR rates, a collectionof mutually equivalent probability measures PTj ; j = 0; : : : ;M; and a family W Tj ; j = 0; : : : ;M �1of processes in such a way that: (i) for any j = 1; : : : ;M the process W Tj follows a d-dimensionalstandard Brownian motion under the probability measure PTj ; (ii) for any j = 1; : : : ;M � 1; theforward LIBOR rate L(�; Tj) satis�es the SDEdL(t; Tj) = L(t; Tj)�(t; Tj) � dW Tj+1t ; 8 t 2 [0; Tj ];with the initial condition L(0; Tj) = B(0; Tj)�B(0; Tj+1)�j+1B(0; Tj+1) :As already mentioned, the construction of the model is based on backward induction, therefore westart by de�ning the forward LIBOR rate with the longest maturity, i.e., TM�1: We postulate thatL(�; TM�1) = L(�; T �1 ) is governed under the underlying probability measure P by the following SDE(note that, for simplicity, we have chosen the underlying probability measure P to play the role ofthe forward LIBOR measure for the date T �)dL(t; T �1 ) = L(t; T �1 )�(t; T �1 ) � dWt;with the initial condition L(0; T �1 ) = B(0; T �1 )�B(0; T �)�MB(0; T �) :Put another way, we haveL(t; T �1 ) = B(0; T �1 )�B(0; T �)�MB(0; T �) exp�Z t0 �(u; T �1 ) � dWu � 12 Z t0 j�(u; T �1 )j2du�:Since B(0; T �1 ) > B(0; T �); it is clear that the L(�; T �1 ) follows a strictly positive martingale underPT� = P: The next step is to de�ne the forward LIBOR rate for the date T �2 : For this purpose,we need to introduce �rst the forward probability measure for the date T �1 : By de�nition, it is aprobability measure Q; equivalent to P; and such that processesU2(t; T �k ) = B(t; T �k )�M�1B(t; T �1 )are Q-local martingales. It is important to observe that the process U2(�; T �k ) admits the followingrepresentation U2(t; T �k ) = �M�1�MU1(t; T �k )�ML(t; T �1 ) + 1 :Let us formulate an auxiliary result, which is a straightforward consequence of Ito's rule.

4 Models of Forward LIBOR and Swap RatesLemma 3.1 Let G and H be real-valued adapted processes, such thatdGt = �t � dWt; dHt = �t � dWt:Assume, in addition, that Ht > �1 for every t and denote Yt = (1 +Ht)�1: Thend(YtGt) = Yt��t � YtGt�t� � �dWt � Yt�t dt�:It follows immediately from Lemma 3.1 thatdU2(t; T �k ) = �kt � �dWt � �ML(t; T �1 )1 + �ML(t; T �1 ) �(t; T �1 ) dt�for a certain process �k: Therefore it is enough to �nd a probability measure under which the processW T�1t def= Wt � Z t0 �ML(u; T �1 )1 + �ML(u; T �1 ) �(u; T �1 ) du =Wt � Z t0 (u; T �1 ) du; 8 t 2 [0; T �1 ];follows a standard Brownian motion (the de�nition of (�; T �1 ) is clear from the context). This canbe easily achieved using Girsanov's theorem, as we may putdPT�1dP = exp�Z T�10 (u; T �1 ) � dWu � 12 Z T�10 j (u; T �1 )j2du�; P-a.s.We are in a position to specify the dynamics of the forward LIBOR rate for the date T �2 under PT�1 ;namely we postulate that dL(t; T �2 ) = L(t; T �2 )�(t; T �2 ) � dW T�1t ;with the initial condition L(0; T �2 ) = B(0; T �2 )�B(0; T �1 )�M�1B(0; T �1 ) :Let us now assume that we have found processes L(�; T �1 ); : : : ; L(�; T �m): In particular, the forwardLIBOR measure PT�m�1 and the associated Brownian motion W T�m�1 are already speci�ed. Our aimis to determine the forward LIBOR measure PT�m : It is easy to check thatUm+1(t; T �k ) = �M�m�1�M�mUm(t; T �k )�M�mL(t; T �m) + 1 :Using Lemma 3.1, we obtain the following relationshipW T�mt =W T�m�1t � Z t0 �M�mL(u; T �m)1 + �M�mL(u; T �m) �(u; T �m) du; 8 t 2 [0; T �m]:The forward LIBOR measure PT�m can thus be easily found using Girsanov's theorem. Finally, wede�ne the process L(�; T �m+1) as the solution to the SDEdL(t; T �m+1) = L(t; T �m+1)�(t; T �m+1) � dW T�mt ;with the initial condition L(0; T �m+1) = B(0; T �m+1)�B(0; T �m)�M�mB(0; T �m) :Remarks. If the volatility coe�cient �(�; Tm) : [0; Tm]! Rd is deterministic, then, for each date t 2[0; Tm]; the random variable L(t; Tm) has a lognormal probability law under the forward probabilitymeasure PTm+1 : Such a model,2 commonly referred to as the lognormal model of LIBOR rates,was examined through di�erent means by Miltersen et al. (1997), Brace et al. (1997), Musiela andRutkowski (1997a), Jamshidian (1997). Let us mention that the futures Libor rate and its derivativesare studied in Section 5 below.2In fact, some authors examined a fully continuous-time version of the lognormal model of forward LIBOR rates,in which the rates L(�; T ) are speci�ed for all maturities.

Marek Rutkowski 53.1 Spot LIBOR MeasureThe backward induction approach to modelling of forward LIBOR rates was re-examined and essen-tially generalized by Jamshidian (1997). In this section, we present brie y his alternative approachto the modelling of forward LIBOR rates. As made apparent in the preceding section, in the directmodelling of LIBOR rates, no explicit reference is made to the bond price processes, which are usedto formally de�ne a forward LIBOR rate through equality (2). Nevertheless, to explain the ideathat underpins Jamshidian's approach, we shall temporarily assume that we are given a family ofbond prices B(t; Tj) for the future dates Tj ; j = 1; : : : ;M: By de�nition, the spot LIBOR measure isthat probability measure equivalent to P; under which all relative bond prices are local martingales,when the price process obtained by rolling over one-period bonds, is taken as a numeraire. Theexistence of such a measure can be either postulated, or derived from other conditions.3 Let us putGt = B(t; Tm(t))m(t)Yj=1 B�1(Tj�1; Tj); 8 t 2 [0; T �]; (3)where m(t) = inf fk 2 N j kXi=1 �i � tg = inf fk 2 N jTk � tg:It is easily seen that Gt represents the wealth at time t of a portfolio which starts at time 0 with oneunit of cash invested in a zero-coupon bond of maturity T1; and whose wealth is then reinvested ateach date Tj ; j = 1; : : : ;M � 1; in zero-coupon bonds which mature at the next date; that is, Tj+1:De�nition 3.2 A spot LIBOR measure PL is any probability measure on (;FT�); which is equiv-alent to P; and such that for any j = 1; : : : ;M the relative bond price B(t; Tj)=Gt follows a localmartingale under PL:Note thatB(t; Tk+1)=Gt = m(t)Yj=1 �1 + �jL(Tj�1; Tj�1)��1 kYj=m(t)+1 �1 + �jL(t; Tj�1)��1;so that all relative bond prices B(t; Tj)=Gt; j = 1; : : : ;M are uniquely determined by a collectionof forward LIBOR rates. In this sense, G is the correct choice of the reference price process in thepresent setting. We shall now concentrate on the derivation of the dynamics under PL of forwardLIBOR rates L(�; Tj); j = 1; : : : ;M: Our aim is to show that these dynamics involve only thevolatilities of forward LIBOR rates (as opposed to volatilities of bond prices or other processes).Therefore, it is possible to de�ne the whole family of forward LIBOR rates simultaneously under oneprobability measure (of course, this feature can also be deduced from the preceding construction).To facilitate the derivation of the dynamics of L(�; Tj); we postulate temporarily that bond pricesB(t; Tj) follow Ito processes under the underlying probability measure P; more explicitlydB(t; Tj) = B(t; Tj)�a(t; Tj) dt+ b(t; Tj) � dWt� (4)for every j = 1; : : : ;M; where, as before, W follows a d-dimensional standard Brownian motionunder an underlying probability measure P (it should be stressed, however, that we do not assumehere that P is a forward (or spot) martingale measure). Combining (3) with (4), we obtaindGt = Gt�a(t; Tm(t)) dt+ b(t; Tm(t)) � dWt�:3One may assume, e.g., that bond prices B(t; Tj) satisfy the weak no-arbitrage condition, meaning that there existsa probability measure ~P; equivalent to P; and such that all processes B(t; Tk)=B(t; T �) are ~P-local martingales.

6 Models of Forward LIBOR and Swap RatesFurthermore, by applying Ito's rule to equality1 + �j+1L(t; Tj) = B(t; Tj)B(t; Tj+1) ; (5)we �nd that dL(t; Tj) = �(t; Tj) dt+ �(t; Tj) � dWt;where �(t; Tj) = B(t; Tj)�j+1B(t; Tj+1)�a(t; Tj)� a(t; Tj+1)�� �(t; Tj)b(t; Tj+1)and �(t; Tj) = B(t; Tj)�j+1B(t; Tj+1) �b(t; Tj)� b(t; Tj+1)�: (6)Using (5) and the last formula, we arrive at the following relationshipb(t; Tm(t))� b(t; Tj+1) = jXk=m(t) �k+1�(t; Tk)1 + �k+1L(t; Tk) : (7)By de�nition of a spot LIBOR measure PL; each relative price process B(t; Tj)=Gt follows a localmartingale under PL: Since, in addition, PL is assumed to be equivalent to P; it is clear that it isgiven by the Dol�eans exponential, that isdPLdP = exp�Z T�0 hu � dWu � 12 Z T�0 jhuj2 du�; P-a.s.for some adapted process h: It it not hard to check, using Ito's rule, that h need to satisfya(t; Tj)� a(t; Tm(t)) = �b(t; Tm(t))� ht� � �b(t; Tj)� b(t; Tm(t))�; 8 t 2 [0; Tj ];for every j = 1; : : : ;M: Combining (6) with the last formula, we obtainB(t; Tj)�j+1B(t; Tj+1) �a(t; Tj)� a(t; Tj+1)� = �(t; Tj) � �b(t; Tm(t))� ht�;and this in turn yieldsdL(t; Tj) = �(t; Tj) � ��b(t; Tm(t))� b(t; Tj+1)� ht� dt+ dWt�:Using (7), we conclude that process L(�; Tj) satis�esdL(t; Tj) = jXk=m(t) �k+1�(t; Tk) � �(t; Tj)1 + �k+1L(t; Tk) dt+ �(t; Tj) � dWLt ;where the process WLt = Wt � R t0 hu du follows a d-dimensional standard Brownian motion underthe spot LIBOR measure PL: To further specify the model, we assume that processes �(t; Tj); j =1; : : : ;M; have the following form�(t; Tj) = �j�t; L(t; Tj); L(t; Tj+1); : : : ; L(t; TM )�; 8 t 2 [0; Tj ];where �j : [0; Tj]�RM�j+1 ! Rd are known functions. In this way, we obtain a system of SDEsdL(t; Tj) = jXk=m(t) �k+1�k(t; Lk(t)) � �j(t; Lj(t))1 + �k+1L(t; Tk) dt+ �j(t; Lj(t)) � dWLt ;where we write Lj(t) = (L(t; Tj); L(t; Tj+1); : : : ; L(t; TM )): Under mild regularity assumptions, thissystem can be solved recursively, starting from L(�; TM�1): The lognormal model of forward LIBORrates corresponds to the choice of �(t; Tj) = �(t; Tj)L(t; Tj); where �(�; Tj) : [0; Tj ] ! Rd is adeterministic function for every j:

Marek Rutkowski 74 Models of Forward Swap RatesWe assume, as before, that the tenor structure T0 = 0 < T1 < � � � < TM = T � is given. Recall that�j = Tj � Tj�1 for j = 1; : : : ;M; and thus Tj =Pji=1 �i for every j = 0; : : : ;M:4.1 Fixed-Maturity SwapsWe will �rst analyse the model of forward swap rates developed by Jamshidian (1996, 1997). Forany �xed j; we consider a �xed-for- oating forward (payer) swap which starts at time Tj and hasM � j accrual periods, whose consecutive lengths are �j+1; : : : ; �M : The �xed interest rate paid ateach of reset dates Tl for l = j + 1; : : : ;M equals �; and the corresponding oating rate, L(Tl); isfound using the formulaB(Tl; Tl+1)�1 = 1 + (Tl+1 � Tl)L(Tl) = 1 + �l+1L(Tl);i.e., it coincides with the LIBOR rate L(Tl; Tl): It is not di�cult to check, using no-arbitrage argu-ments, that the value of such a swap equals (by convention, the notional principal equals 1)FS t(�) = B(t; Tj)� MXl=j+1 clB(t; Tl); 8 t 2 [0; Tj ]; (8)where cl = ��l for l = j + 1; : : : ;M � 1; and cM = 1 + ��M : Consequently, the associated forwardswap rate, �M�j(t; Tj); that is, that value of a �xed rate � for which such a swap is worthless attime t; is given by the formula�M�j(t; Tj) = B(t; Tj)�B(t; TM )�j+1B(t; Tj+1) + � � �+ �MB(t; TM ) (9)for every t 2 [0; Tj ]; j = 1; : : : ;M � 1: In this section, we consider the family of forward swap rates~�(t; Tj) = �M�j(t; Tj) for j = 1; : : : ;M �1: Let us stress that the underlying swap agreements di�erin length, however, they all have a common expiration date, T � = TM :Suppose for the moment that we are given a family of bond prices B(t; Tm); m = 1; : : : ;M; ona �ltered probability space (;F;P) equipped with a Brownian motion W: As in Section 3, we �ndit convenient to postulate that P = PT� is the forward measure for the date T �; and the processW = W T� is the corresponding Brownian motion. For any m = 1; : : : ;M � 1; we introduce the�xed-maturity coupon process ~G(m) by setting (recall that T �l = TM�l; in particular, T �0 = TM )~Gt(m) = MXl=M�m+1 �lB(t; Tl) = m�1Xk=0 �M�kB(t; T �k ); 8 t 2 [0; TM�m+1]: (10)A forward swap measure is that probability measure equivalent to P; which corresponds to thechoice of the �xed-maturity coupon process as a numeraire asset. We have the following de�nition.De�nition 4.1 For a �xed j = 1; : : : ;M; a probability measure ~PTj on (;FTj ); equivalent to P;is said to be the �xed-maturity forward swap measure for the date Tj if, for every k = 1; : : : ;M; therelative bond priceZM�j+1(t; Tk) def= B(t; Tk)~Gt(M � j + 1) = B(t; Tk)�jB(t; Tj) + � � �+ �MB(t; TM ) ; 8 t 2 [0; Tk ^ Tj ];follows a local martingale under ~PTj :Put another way, for any �xed m = 0; : : : ;M � 1; the relative bond pricesZm(t; T �k ) = B(t; T �k )~Gt(m) = B(t; T �k )�M�m+1B(t; T �m�1) + � � �+ �MB(t; T �) ; 8 t 2 [0; T �k ^ T �m];

8 Models of Forward LIBOR and Swap Ratesare bound to follow local martingales under the forward swap measure ~PT�m�1 : It follows immediatelyfrom (9) that the forward swap rate for the date T �m equals~�(t; T �m) = B(t; T �m)�B(t; T �)�M�m+1B(t; T �m�1) + � � �+ �MB(t; T �) ; 8 t 2 [0; T �m];or equivalently, ~�(t; T �m) = Zm(t; T �m)� Zm(t; T �); 8 t 2 [0; T �m]:Therefore ~�(�; T �m) also follows a local martingale under the forward swap measure ~PT�m�1 : Moreover,since obviously ~Gt(1) = �MB(t; T �); it is evident that Z1(t; T �k ) = ��1M FB(t; T �k ; T �); and thus theprobability measure ~PT� can be chosen to coincide with the forward martingale measure PT� :Our aim is to construct a model of forward swap rates through backward induction. As onemight expect, the underlying bond price processes will not be explicitly speci�ed. We assume thatwe are given a family of bounded adapted processes ~�(�; Tj); j = 1; : : : ;M � 1; which represent thevolatilities of processes ~�(�; Tj): In addition, an initial term structure of interest rates, given by afamily B(0; Tj); j = 0; : : : ;M; of bond prices, is known. We wish to construct a family of forwardswap rates in such a way that d~�(t; Tj) = ~�(t; Tj)~�(t; Tj) � d ~W Tj+1t (11)for any j = 1; : : : ;M � 1; where each process ~W Tj+1 follows a standard Brownian motion under thecorresponding forward swap measure ~PTj+1 : The model should also be consistent with the initialterm structure of interest rates, meaning that~�(0; Tj) = B(0; Tj)�B(0; T �)�j+1B(0; Tj+1) + � � �+ �MB(0; TM ) : (12)We proceed by backward induction. The �rst step is to introduce the forward swap rate for the dateT �1 by postulating that ~�(�; T �1 ) solves the SDEd~�(t; T �1 ) = ~�(t; T �1 )~�(t; T �1 ) � d ~W T�t ; 8 t 2 [0; T �1 ]; (13)with the initial condition ~�(0; T �1 ) = B(0; T �1 )�B(0; T �)�MB(0; T �) :To specify the process ~�(�; T �2 ); we need �rst to introduce a forward swap measure ~PT�1 and an associ-ated Brownian motion ~W T�1 : To this end, notice that each process Z1(�; T �k ) = B(�; T �k )=�MB(�; T �)follows a strictly positive local martingale under ~PT� = PT� : More speci�cally, we havedZ1(t; T �k ) = Z1(t; T �k ) 1(t; T �k ) � dW T�t (14)for some adapted process 1(�; T �k ): According to the de�nition of a �xed-maturity forward swapmeasure, we postulate that for every k; the processZ2(t; T �k ) = B(t; T �k )�M�1B(t; T �1 ) + �MB(t; T �) = Z1(t; T �k )1 + �M�1Z1(t; T �1 )follows a local martingale under ~PT�1 : Applying Lemma 3.1 to processes G = Z1(�; T �k ) and H =�M�1Z1(�; T �1 ); it is easy to see that for this property to hold, it su�ces to assume that the processW T�1 ; which is given by the formula~W T�1t = ~W T�t � Z t0 �M�1Z1(u; T �1 )1 + �M�1Z1(u; T �1 ) 1(u; T �1 ) du; 8 t 2 [0; T �1 ];

Marek Rutkowski 9follows a Brownian motion under ~PT�1 (probability measure ~PT�1 is yet unspeci�ed, but will be soonfound through Girsanov's theorem). Note thatZ1(t; T �1 ) = B(t; T �1 )�MB(t; T �) = ~�(t; T �1 ) + Z1(t; T �) = ~�(t; T �1 ) + ��1M : (15)Di�erentiating both sides of the last equality, we get (cf. (13) and (14))Z1(t; T �1 ) 1(t; T �1 ) = ~�(t; T �1 )~�(t; T �1 ):Consequently, ~W T�1 is explicitly given by the formula~W T�1t = ~W T�t � Z t0 �M�1~�(u; T �1 )1 + �M�1��1M + �M�1~�(u; T �1 ) ~�(u; T �1 ) du; 8 t 2 [0; T �1 ]:We are in a position to de�ne, using Girsanov's theorem, the associated forward swap measure ~PT�1 :Subsequently, we introduce the process ~�(�; T �2 ); by postulating that it solves the SDEd~�(t; T �2 ) = ~�(t; T �2 )~�(t; T �2 ) � d ~W T�1twith the initial condition ~�(0; T �2 ) = B(0; T �2 )�B(0; T �)�M�1B(0; T �1 ) + �MB(0; T �) :For the reader's convenience, let us consider one more inductive step, in which we are looking for~�(t; T �3 ): We now consider processesZ3(t; T �k ) = B(t; T �k )�M�2B(t; T �2 ) + �M�1B(t; T �1 ) + �MB(t; T �) = Z2(t; T �k )1 + �M�2Z2(t; T �2 ) ;so that ~W T�2t = ~W T�1t � Z t0 �M�2Z2(u; T �2 )1 + �M�2Z2(u; T �2 ) 2(u; T �2 ) du 8 t 2 [0; T �2 ]:It is useful to note thatZ2(t; T �2 ) = B(t; T �2 )�M�1B(t; T �1 ) + �MB(t; T �) = ~�(t; T �2 ) + Z2(t; T �); (16)where in turn Z2(t; T �) = Z1(t; T �)1 + �M�1Z1(t; T �) + �M�1~�(t; T �1 )and the process Z1(�; T �) is already known from the previous step (clearly, Z1(�; T �) = 1=dM ). Di�er-entiating the last equality, we may thus �nd the volatility of the process Z2(�; T �); and consequently,to de�ne ~PT�2 :Let us now turn to the general case. We proceed by induction with respect to m: Suppose thatwe have already found forward swap rates ~�(�; T �1 ); : : : ; ~�(�; T �m); the forward swap measure ~PT�m�1 ;and the associated Brownian motion ~W T�m�1 : Our aim is to determine the forward swap measure~PT�m ; the associated Brownian motion ~W T�m ; and the forward swap rate ~�(�; T �m+1): To this end,we postulate that processesZm+1(t; T �k ) = B(t; T �k )~Gt(m+ 1) = B(t; T �k )�M�mB(t; T �m) + � � �+ �MB(t; T �) = Zm(t; T �k )1 + �M�mZm(t; T �m)follow local martingales under ~PT�m� : In view of Lemma 3.1, applied to processes G = Zm(�; T �k ) andH = Zm(�; T �m); it is clear that we may set~W T�m�t = ~W T�t � Z t0 �M�mZm(u; T �m)1 + �M�mZm(u; T �m) m(u; T �m) du; 8 t 2 [0; T �m]: (17)

10 Models of Forward LIBOR and Swap RatesTherefore it is su�cient to analyse the processZm(t; T �m) = B(t; T �m)�M�m+1B(t; T �m�1) + � � �+ �MB(t; T �) = ~�(t; T �m) + Zm(t; T �):To conclude, it is enough to notice thatZm(t; T �) = Zm�1(t; T �)1 + �M�m+1Zm�1(t; T �) + �M�m+1~�(t; T �m�1) :Indeed, from the preceding step, we know that the process Zm�1(�; T �) is a (rational) functionof forward swap rates ~�(�; T �1 ); : : : ; ~�(�; T �m�1): Consequently, the process under the integral signon the right-hand side of (17) can be expressed using the terms ~�(�; T �1 ); : : : ; ~�(�; T �m�1) and theirvolatilities (since the explicit formula is rather lengthy, it is not reported here). Having found theprocess ~W T�m and probability measure ~PT�m ; we introduce the forward swap rate ~�(�; T �m+1) through(11){(12), and so forth. If all volatilities are deterministic, the model is termed the lognormal modelof �xed-maturity forward swap rates.4.2 Amortising SwapsSo far, it was implicitly assumed that we deal with a �xed-for- oating payer swap, with constantnotional principal. Let us now consider the so-called amortising swap, in which the notional principalPj varies over time (we assume, however, that it is deterministic). We assume the cash ows at timeTj of the underlying swaps are Pj�jL(Tj�1; Tj�1) and ��Pj�j : Therefore the value at time t 2 [0; Tj ]of a swap which starts at time Tj equalsFS t(�) = MXi=j+1Pi�B(t; Ti�1)� (1 + ��i)B(t; Tj)�; 8 t 2 [0; Tj ]:The associated forward swap rate, �a(t; Tj); equals�a(t; Tj) = Pj+1�B(t; Tj)�B(t; Tj+1)�+ � � �+ PM�B(t; TM�1)�B(t; TM )�Pj+1�j+1B(t; Tj+1) + � � �+ PM�MB(t; TM ) ;or equivalently,�a(t; Tj) = Pj+1B(t; Tj) +Pm�1i=j+1(Pi+1 � Pi)B(t; Ti)� PMB(t; TM )PMi=j+1 ~�iB(t; Ti) ;where ~�i = Pi�i: Assume that processes ~G(m) are given by formula (10), with �'s substituted with~�'s. We may apply the recursive procedure based on the same set of general ideas as before. Let usfocus, for simplicity, on the di�erences which occur in the �rst two steps. Instead of (15), we nowhave Z1(t; T �1 ) = B(t; T �1 )~�MB(t; T �) = P�1M �a(t; T �1 ) + ��1M : (18)This allows us to �nd ~PT�1 ; and thus also to introduce the swap rate �a(�; T �2 ): Similarly, equality(16) now takes the following formZ2(t; T �2 ) = B(t; T �2 )~�M�1B(t; T �1 ) + ~�MB(t; T �) = P�1M�1��a(t; T �2 )+(PM�1�PM )Z2(t; T �1 )+PMZ2(t; T �)�;where in turn Z2(t; T �i ) = Z1(t; T �i )1 + ~�M�1Z1(t; T �1 ) ; i = 0; 1:

Marek Rutkowski 11The auxiliary process Z2(�; T �2 ) may thus be expressed in terms of processes which have been foundin the �rst step (Z1(�; T �) = 1=~�M and Z1(�; T �1 ) is given by (18)). More generally, we haveZm(t; T �m) = P�1M�m+1��a(t; T �m) + m�1Xi=1 (PM�i � PM�i+1)Zm(t; T �i ) + PMZm(t; T �)�;where Zm(t; T �i ) = Zm�1(t; T �i )1 + ~�M�m+1Zm�1(t; T �m�1) ; i = 0; : : : ;m� 1:Since processes Zm�1(�; T �i ); i = 0; : : : ;m�1 are already known from the preceding step, we concludethat the backward induction is still feasible, subject to suitable minor modi�cations.4.3 Fixed-maturity SwaptionsFor a �xed, but otherwise arbitrary, date Tj ; j = 1; : : : ;M�1; we consider a swaption with expirationdate Tj ; written on a payer swap settled in arrears, with �xed rate �; which starts at date Tj andhas M � j accrual periods (it is referred to as the jth swaption in what follows). Notice that the jthswaption can be seen as a contract which pays to its owner the amount �lP (�M�j(Tj ; Tj)� �)+ ateach settlement date Tl; where l = j+1; : : : ;M (we write P to denote a constant notional principal).Equivalently, the jth swaption pays an amount~Y = MXl=j+1 �lPB(Tj ; Tl)�~�(Tj ; Tj)� ��+at maturity date Tj : It is useful to observe that ~Y admits the following representation~Y = ~GTj (M � j)P �~�(Tj ; Tj)� ��+:Recall that the model of �xed-maturity forward swap rates speci�es the dynamics of the process~�(�; Tj) through the following SDEd~�(t; Tj) = ~�(t; Tj)~�(t; Tj) � d ~W Tj+1t ;where ~W Tj+1 follows a standard d-dimensional Brownian motion under the corresponding forwardswap measure ~PTj+1 : The de�nition of the forward swap measure ~PTj+1 implies that any processof the form B(t; Tk)= ~Gt(M � j) is a local martingale under ~PTj+1 : Furthermore, from the generalconsiderations concerning the choice of a numeraire (see, e.g. Geman et al. (1995) or Musiela andRutkowski (1997b)) it is easy to see that, the arbitrage price �t(X) of an attainable claim X =g(B(Tj ; Tj+1); : : : ; B(Tj ; TM )); which settles at time Tj ; equals�t(X) = ~Gt(M � j)E ~PTj+1 � ~G�1Tj (M � j)X j Ft�; 8 t 2 [0; Tj ]:Applying the last formula to the swaption's payo� ~Y ; we obtainfPS jt = �t( ~Y ) = ~Gt(M � j)P E ~PTj+1 �(~�(Tj ; Tj)� �)+ j Ft�; 8 t 2 [0; Tj ];where fPS jt stands for the price at time t of the jth swaption. We assume from now on that~�(�; Tj) : [0; Tj ] ! Rd is a bounded deterministic function; that is, we place ourselves within theframework of the lognormal model of �xed-maturity forward swap rates. We have the followingresult, due to Jamshidian (1996).

12 Models of Forward LIBOR and Swap RatesProposition 4.1 For any j = 1; : : : ;M�1; the arbitrage price at time t 2 [0; Tj ] of the jth swaptionequals fPS jt = MXl=j+1 �lPB(t; Tl)�~�(t; Tj)N�~h1(t; Tj)�� �N�~h2(t; T )��;where N denotes the standard Gaussian probability distribution function, and~h1;2(t; Tj) = ln(~�(t; Tj)=�)� 12 ~v2(t; Tj)~v(t; Tj) ;with ~v2(t; Tj) = R Tjt j~�(u; Tj)j2 du:To obtain a closed-from solution for the price of a swaption written on a forward swap rate asso-ciated with a given family of amortising swaps, it is convenient to assume the framework introducedin Section 4.2. In this setting, it is enough to replace in the valuation formula above the constantprincipal P by the variable principal Pl; the swap rate �(t; Tj) by �a(t; Tj); and the volatility function�(�; Tj) by the (deterministic) volatility �a(�; Tj) of the process �a(�; Tj):4.4 Fixed-length SwapsIn this section, we no longer assume that the underlying swap agreements have di�erent lengths butthe same maturity date. On the contrary, the length, K; of a swap will now be �xed, however, itsmaturity date will vary. For instance, for a forward swap which starts at time Tj ; j = 1; : : : ;M�K;the �rst settlement date is Tj+1; and its maturity date equals TK+j : The value at time t 2 [0; Tj ] ofsuch a swap equals (cf. (8)) FS jt (�) = B(t; Tj)� j+KXl=j+1 clB(t; Tl);where cl = ��l for l = j+1; : : : ; j+K � 1; and cj+K = 1+��j+K : The forward swap rate of lengthK; for the date Tj ; is that value of the �xed rate which makes the underlying forward swap startingat Tj worthless at time t: Using the last formula, we �nd easily that�K(t; Tj) = B(t; Tj)�B(t; Tj+K)�j+1B(t; Tj+1) + � � �+ �j+KB(t; Tj+K) :For convenience, we will write �(t; Tj) instead of �K(t; Tj): Also, we denote M� =M �K + 1:Remarks. It is worthwhile to note that the forward LIBOR rate L(t; Tj) coincides with the one-period forward swap rate over the time period [Tj ; Tj+1]: Formally, the model of forward LIBORrates presented in Section 3 may be seen as a special case of the model of �xed-length forward swaprates. However, since in this case K = 1; the construction is simpler, and requires less assumptions.For any m = 1; : : : ;M�; we introduce the �xed-length coupon process G(j) by settingGt(m) def= M�m+1Xl=M��m+1 �lB(t; Tl) = m+K�2Xk=m�1 �M�kB(t; T �k ); 8 t 2 [0; Tj ]:As one might expect, a modi�ed forward swap measure which we are now going to introduce corre-sponds to the choice of a �xed-length coupon process as a numeraire asset.De�nition 4.2 For a �xed j = 1; : : : ;M�; a probability measure PTj on (;FTj ); equivalent to theunderlying probability measure P; is called the �xed-length forward swap measure for the date Tj if,for every k = 1; : : : ;M; the relative bond priceUM��j+1(t; Tk) def= B(t; Tk)Gt(M�� j + 1) = B(t; Tk)�jB(t; Tj) + � � �+ �j+K�1B(t; Tj+K�1) ; 8 t 2 [0; Tk ^ Tj ];follows a local martingale under PTj :

Marek Rutkowski 13It is clear that each forward swap rate �(�; Tj); j � M�� 1; follows a local martingale underthe forward swap measure for the date Tj+1; since �(�; Tj) = UM��j+1(t; Tj) � UM��j+1(t; Tj+K):If we assume, in addition, that �(�; Tj) is a strictly positive process, it is natural to postulate that itsatis�es d�(t; Tj) = ~�(t; Tj)�(t; Tj) � dW Tj+1t ;where W Tj+1t follows a standard Brownian motion under PTj+1 : The initial condition is�(0; Tj) = B(0; Tj)�B(0; Tj+K)�j+1B(0; Tj+1) + � � �+ �j+KB(0; Tj+K) :As expected, the construction of �xed-length forward swap rates model also relies on the backwardinduction. In contrast to both previously studied models, the terminal forward swap measure PTM�does not coincide with the \usual" forward measure for the date TM� ; however. Nevertheless, we still�nd it convenient to proceed backwards in time { that is, to start by postulating that the underlyingprobability measure P represents the forward swap measure for the date TM� ; and then to de�nerecursively the forward swap measures corresponding to the preceding dates, TM��1; TM��2; : : : ; T1:To proceed further, we need to assume that we are given a family �(�; Tj); j = 1; : : : ;M � K;of volatilities of forward swap rates �(�; Tj); as well as the initial term structure of bond pricesB(0; Tj); j = 1; : : : ;M: As one might easily guess, the �rst step is to postulate that �(�; TM��1);which equals�(0; TM��1) = B(0; TM��1)�B(0; TM )�M�B(0; TM�) + � � �+ �MB(0; TM ) = B(0; TM��1)�B(0; TM )Gt(1) ;satis�es d�(t; TM��1) = �(t; TM��1)�(t; TM��1) � dWt;with obvious initial condition. This means, in particular, that we have chosen PTM� = P andW TM� =W: Our next goal is to introduce the forward swap measure for the date TM��1:Case of equal accrual periods. Assume �rst, for simplicity, that �j = � is constant. In this casethe backward induction goes through as in previous cases. Recall that�(t; TM��2) = B(t; TM��2)�B(t; TM�1)�B(t; TM��1) + � � �+ �B(t; TM�1) = B(t; TM��2)�B(t; TM�1)Gt(2) :On the other hand, we haveU2(t; Tk) = B(t; Tk)Gt(2) = B(t; Tk)�B(t; TM��1) + � � �+ �B(t; TM�1) ;so that the relative bond price U2(�; Tk) admits the following representationU2(t; Tk) = U1(t; Tk)1 + ��(t; TM��1) :In view of Lemma 3.1, it is obvious that the process U2(�; Tk) follows a local martingale under PTM��1whenever the processW TM��1t = W TM�t � Z t0 ��(u; TM��1)1 + ��(u; TM��1) �(u; TM��1) dufollows a standard Brownian motion under this measure. To �nd PTM��1 , it is enough to applyGirsanov's theorem. A general induction step relies on the following relationshipUM��m+1(t; Tk) = B(t; Tk)Gt(M��m+ 1) = UM��m(t; Tk)1 + ��(t; Tm) ;

14 Models of Forward LIBOR and Swap Rateswhich allows us to de�ne PTM��m and W TM��m provided that the measure PTm+1 and processes�(�; Tm); W Tm+1 are already known. We setW Tmt = W Tm+1t � Z t0 ��(u; Tm)1 + ��(u; Tm) �(u; Tm) du; 8 t 2 [0; Tm];and we determine PTm through Girsanov's theorem. Given W Tm and PTm ; we postulate that theforward swap rate �(�; Tm�1) is governed by the SDEd�(t; Tm�1) = ~�(t; Tm�1)�(t; Tm�1) � dW Tmt :This shows that the knowledge of all swap rate volatilities (and, of course, the initial term structure)is su�cient if the purpose is to construct an arbitrage-free family of forward swap rates. Somewhatsurprisingly, this is no longer the case if the length of the accrual period varies over time.General case. We �rst deal with the date TM��2: Note that�(t; TM��2) = B(t; TM��2)�B(t; TM�1)�M��1B(t; TM��1) + � � �+ �M�1B(t; TM�1) = B(t; TM��2)�B(t; TM�1)Gt(2) :Furthermore, for any k we haveU2(t; Tk) = B(t; Tk)Gt(2) = B(t; Tk)�M��1B(t; TM��1) + � � �+ �M�1B(t; TM�1) :The last equality may be given the following equivalent formU2(t; Tk) = U1(t; Tk)1 +�1U1(t; TM ) + �M��1�(t; TM��1) = U1(t; Tk)1 + V1(t) ;where we write �1 = �M � �M��1; and the de�nition of the process V1 is clear from the context.To proceed further, we need to assume that the dynamics of the process U1(�; TM ) = B(�; TM )=G(1)under PTM� is known; more precisely, it is enough to specify its volatility, since it is clear thatU1(�; TM ) is governed by the SDE of the formdU1(t; TM ) = U1(t; TM )�1(t; TM ) � dW TM�tfor some process �1(�; TM ): Suppose that �1(�; TM ); and thus also U1(�; TM ); are known. Let usdenote v1(t) = �1U1(t; TM )�1(t; TM ) + �M��1�(t; TM��1)�(t; TM��1):Then W TM��1t = W TM�t � Z t0 v1(u)1 + V1(u) du; 8 t 2 [0; TM��1];and thus the probability measure PTM��1 can be found explicitly. Next, the swap rate �(�; TM��2)is de�ned by setting d�(t; TM��2) = �(t; TM��2)�(t; TM��2) � dW TM��1t :Let us now consider a general induction step. We assume that we have already found PTm+1and W Tm+1 ; and thus also �(�; Tm): In order to de�ne �(�; Tm�1); we need �rst to specify PTm andW Tm : We have the following relationshipUM��m+1(t; Tk) = B(t; Tk)Gt(M��m+ 1) = UM��m(t; Tk)1 + VM��m(t) ;

Marek Rutkowski 15where VM��m(t) = �M��mUM��m(t; Tm+K) + �m�(t; Tm)and �M��m = �m+K � �m: We have, as expectedW Tmt = W Tm+1t � Z t0 vM��m(u)1 + VM��m(u) du; 8 t 2 [0; Tm];where vM��m(t) = UM��m(t; Tm+K)�M��m(t; Tm+K) + �m�(t; Tm)�(t; Tm)and �M��m(t; TM�m+1) represents the volatility of the processUM��m(�; Tm+K) = G�1(M��m+ 1)B(�; Tm+K):Concluding, in order to construct a model of �xed-length forward swap rates through backwardinduction, we �nd it convenient to assume, in addition, that the volatilities of relative bond pricesUm(�; TM�m+1) = G�1(m)B(�; TM�m+1); m = 1; : : : ;M�� 2;are exogenously given.4.5 Fixed-length SwaptionsFor any date Tj ; j = 1; : : : ;M��1; let us consider a payer swaption with expiration date Tj ; writtenon a payer swap, with �xed rate �; which starts at date Tj and has K accrual periods. The jthswaption may now be seen as a contract that paysY = j+KXl=j+1 �lPB(Tj ; Tl)��(Tj ; Tj)� ��+at time Tj : It is useful to notice thatY = GTj (M�� j)P ��(Tj ; Tj)� ��+:In the framework of the �xed-length forward swap rates model, the forward swap rate �(�; Tj) satis�esd�(t; Tj) = �(t; Tj)�(t; Tj) � dW Tj+1t ;where W Tj+1 follows a standard d-dimensional Brownian motion under PTj+1 : In view of the de�-nition of the forward swap measure ~PTj+1 ; any process of the form B(t; Tk)=Gt(M�� j) is a localmartingale under PTj+1 : Therefore the price of an attainable claim X; which settles at time Tj anddepends on the bond prices, equals�t(X) = Gt(M�� j)E PTj+1 �G�1Tj (M�� j)X j Ft�; 8 t 2 [0; Tj ]:In particular, for the jth �xed-length swaption we getcPS jt = �t(Y ) = Gt(M�� j)P E PTj+1 �(�(Tj ; Tj)� �)+ j Ft�; 8 t 2 [0; Tj ]:Let us place ourselves within the framework of the lognormal model of �xed-length forward swaprates, in which the volatilities �(�; Tj) of �xed-length forward swap rates are assumed to followbounded deterministic functions. Then we have the following result, which is a counterpart ofProposition 4.1. It is useful to note that if K = 1; the jth swaption may be identi�ed with the jthcaplet. Therefore Proposition 4.2 covers also the valuation of caplets (and thus also caps) in thelognormal model of forward LIBOR rates presented in Section 3.

16 Models of Forward LIBOR and Swap RatesProposition 4.2 For any j = 1; : : : ;M�� 1; the arbitrage price at time t 2 [0; T � j] of the jthswaption equals cPS jt = l=j+KXl=j+1 �lPB(t; Tl)��(t; Tj)N�h1(t; Tj)�� �N�h2(t; T )��;where N denotes the standard Gaussian probability distribution function, andh1;2(t; Tj) = ln(�(t; Tj)=�)� 12 v2(t; Tj)v(t; Tj) ;with v2(t; Tj) = R Tjt j�(u; Tj)j2 du:It is worthwhile to observe the swaption's price at time 0 depends only on the initial termstructure and the volatility of the underlying swap. In particular, the volatilities �M��m(�; Tm+K) ofrelative bond prices, which were used as auxiliary inputs in model's construction, do not enter thevaluation formula. For this reason, to get the valuation formula above, we may equally well takethese volatilities equal identically to zero. On the other hand, it is reasonable to expect that thechoice of these volatilities would in uence the price processes of underlying bonds, and thus also theswaption's price at time t > 0: The choice of the underlying bond price structure is, of course, evenmore important in valuation { also at time 0 { of derivative securities with American features, suchas Bermudan swaptions.4.6 Bond PricesAll models of forward rates examined in this note are arbitrage-free, in the following sense: it ispossible to �nd a family B(�; Tj); j = 1; : : : ; n of bond prices, which is compatible with forwardrates speci�ed by a model, and which possesses the arbitrage-free features. The last property meansthat all bond prices discounted by the price of the bond with the longest maturity follow localmartingales under some probability measure equivalent to the original probability measure. Theactual procedure of �nding such a family depends on a particular model and requires further studies(of practical, rather than theoretical, nature). In the case of the lognormal model of forward LIBORrates, an interesting method of a direct construction of underlying bond prices { not only for thedates Tj ; but indeed for any maturity { was proposed recently by Sawa (1997). Essentially, hismethod combines the HJM methodology with the (linear or non-linear) interpolation. It shouldbe stressed, however, that his approach takes also into account the properties of a �nite family offorward LIBOR rates predicted by the lognormal model.Since various models of forward market rates are inconsistent with each other, in general (thisproperty is obvious for lognormal models), another important issue which arises in this context isthe model choice. We refer to Jamshidian (1997) for an interesting discussion of this point. Letus only mention that, essentially, he advocates the use of di�erent models for di�erent classes ofinterest-rate derivatives. When combined with each other, these models may give rise to a structurewhich is not arbitrage-free { this feature should not be seen as their major de�ciency, however.Indeed, this is a typical situation in most real-life applications of the arbitrage pricing theory. Onthe positive side, since the focus is on the modelling of interest rates which are directly observedin the market, the model calibration should hopefully prove easier than in the case of traditionalmodels of instantaneous rates.Still, if for whatever reasons we insist on the absence of arbitrage in the combined model as acompulsory requirement, a judiciously chosen arbitrage-free combination of various models seems tobe a plausible solution. At �rst glance, it would be reasonable to focus on a combined model, whichwould encompass the forward swap rates model for longer maturities, the forward LIBOR ratesmodel for shorter maturities, and the classical by now Heath-Jarrow-Morton (1992) bond pricingmethodology for maturities from outside a given tenor structure. Details of such a constructionwould depend on the features of a particular market and the purpose of the modelling.

Marek Rutkowski 175 Futures Rates and PricesLet us introduce the discrete-time savings account B by the formulaBTj = jYk=1�1 + �kL(Tk�1)� = jYk=1B�1(Tk�1; Tk):Notice that BT1 = 1 + �1L(0) = 1=B(0; T1) is deterministic (consequently, the \spot" martingalemeasure will coincide with the forward Libor measure for the date T1). It is clear that BTj = GTjfor any j = 0; : : : ;M (the process G is given by (3)).Remarks. It is worthwhile to point out that B is the unique discrete-time savings account implied bythe model of forward LIBOR rates, a continuous-time savings account is not uniquely determined.The continuous-time process G; which was introduced by Jamshidian (1997) (see Section 3.1 of thisnote), is merely an example of a continuous-time savings account implied by the family of forwardLIBOR rates. In general, there is no reason to expect that a continuous-time savings account ~Bcompatible with the family L(�; Tj) of forward LIBOR rates would necessarily satisfy ~BTj = BTjfor j = 1; : : : ;M: In particular, it seems plausible that a realistic model of bond prices speci�esBT1 as a random variable rather than as a deterministic constant. It should thus be stressed thatin what follows we work as if the process G was indeed chosen as the continuous-time savingsaccount implied by the family of forward LIBOR rates (which are lognormally distributed under theforward measures). Such a choice implies that prices of all bond with maturities less than T1 areknown and they follow necessarily deterministic functions. The bottom line is that the results ofthis section concerning Eurodollar futures and Eurodollar futures options are valid only under ourspeci�c assumptions.5.1 Spot Martingale MeasureFor our further purposes, it is essential to investigate the concept of a spot martingale measurewithin the framework of the model of forward LIBOR rate. We postulate that the spot martingalemeasure P� is de�ned through the Radon-Nikod�ym derivativedPT�dP� = 1c�BT� ; P�-a.s.,where c� is a constant. Equivalently dP�dPT� = c�BT� ; PT�-a.s., (19)where c� = 1=EPT� (BT�) = B(0; T �): Of course, we need now to justify this convention.Remarks. In the standard HJM methodology, the modelling is done under the spot measure, P� say.The forward measure PT for any date T is then introduced by settingdPTdP� = 1B(0; T )BT ; P�-a.s.In the present context, the order is reversed; that is, forward measures for discrete set of datesare known by construction, and \spot" measure is de�ned with reference to the forward measures.Let us emphasize once again that we did not specify a family of bond prices, except for bondswith maturities in the �rst interval and bond prices B(Tj ; Tj+1): This makes the concept of a spotprobability measure more ambiguous than in the traditional HJM framework, in which the savingsaccount process is uniquely speci�ed through the short-term rate rt = f(t; t):As one might expect, the probability measure P� given by (19) is closely related to spot LIBORmeasure PL; introduced in Section 3.1. To be more speci�c, we have the following result.

18 Models of Forward LIBOR and Swap RatesLemma 5.1 Let P� be a probability measure such thatdP�dPT� = B(0; T �)GT� ; PT�-a.s. (20)Then the process B(�; Tk)=G follows a martingale under P� for k = 1; : : : ;M:Proof. For simplicity, we assume that all processes B(�; Tk)=B(�; T �) follow a martingale under PT�(this allows us to work directly with conditional expectations rather than to apply Ito's rule). Weneed to show that EP��B(Tk; Tk)GTk ���Ft� = B(t; Tk)Gt ; 8 t 2 [0; Tk]:The left-hand-side in the last formula equalsEP��B(Tk; Tk)GTk ���Ft� = EPT� (GT�G�1Tk j Ft)EPT� (GT� j Ft) :From the properties of the forward LIBOR measure, we obtainEPT� (GT� j Ft) = GtB�1(t; T �); 8 t 2 [0; T �]:Therefore, it is enough to show thatJ def= EPT� (GT�G�1Tk j Ft) = B(t; Tk)B�1(t; T �):If k =M � 1; the last equality is obvious. If k < M � 1; then for every t � Tk < T �1 ; we haveJ = EPT�� 1B(Tk; Tk+1) � � � 1B(T �1 ; T �) ���Ft�= EPT�� 1B(Tk; Tk+1) � � � 1B(T �2 ; T �1 ) EPT��B(T �1 ; T �1 )B(T �1 ; T �) ���FT�2 � ���Ft�= EPT�� 1B(Tk; Tk+1) � � � 1B(T �3 ; T �2 ) 1B(T �2 ; T �) ���Ft�:Repeating, if necessary, this procedure we �nd easily thatEPT� (GT�G�1Tk j Ft) = EPT�� 1B(Tk; T �) ���Ft� = B(t; Tk)B(t; T �) ;as expected. 2In view of Lemma 5.1, probability measure P� given by (20) (or equivalently, by (19)) satis�esthe de�nition of the spot LIBOR measure PL: If it was known, in addition, that the spot LIBORmeasure is uniquely determined, this would mean, of course, that P� = PL: Since the uniquenessof PL is unclear, we prefer to work in what follows with the measure P� (which is, by de�nition,unique). It should be made clear that a spot martingale measure is not well-de�ned in the modelof forward LIBOR rates, since the model does not uniquely specify the implied (continuous-time)savings account. It seems that this can be done in various way (for instance, by specifying all bondprices). Notice that PL corresponds essentially to the choice of the process G as a savings account.By choosing G as a savings account, one implicitly de�nes also all bond prices through the standardformula B(t; T ) = Gt EPL(G�1T j Ft); 8 t � T � T �:For convenience, we shall frequently refer to P� as the spot probability measure. Let us nowconsider the dates Tj j = 1; : : : ;M � 1: We denote by P�j the restriction of P� to the �-�eld FTj {that is, P�j = P�jFTj ; 8 j = 1; : : : ;M � 1:

Marek Rutkowski 19Our aim is to show that the probability measure P�j can be introduced without making reference toP�; we shall use instead the forward LIBOR measure PTj for a given date as the reference probabilitymeasure.Remarks. Let us place ourselves for the moment within the standard HJM setup. For any maturitiesU � T; we have dPUdPT = B(0; T )BTB(0; U)BU ; PT -a.s.Consequently dPUdPT jFU = EPT�B(0; T )BTB(0; U)BU ���FU� = B(0; T )B(0; U) EPT�BTBU ���FU�:Furthermore, EPT�BTBU ���FU� = EP�(B�1U j FU )EP�(B�1T j FU ) = 1EP�(BUB�1T j FU ) = 1B(U; T ) :Concluding, we have dPUdPT = dPUdPT jFU = cB(U; T ) ; (21)where c is the normalizing constantc = hEPT �B�1(U; T )�i�1 = EPU (B(U; T )) = B(0; T )B(0; U) :It seems reasonable to expect that relationship (21) is universal, in particular, that it also holdsin the framework of the forward LIBOR rates model. This is indeed true, as the following resultshows.4Lemma 5.2 For any j = 1; : : : ;M � 1 we havedPTjdPTj+1 = B(0; Tj+1)B(0; Tj) 1B(Tj ; Tj+1) = 1 + �j+1L(Tj ; Tj)1 + �j+1L(0; Tj) (22)and dPTjdPT� = B(0; T �)B(0; Tj) EPT��M�1Yk=j 1B(Tk; Tk+1) ���FTj�; (23)or equivalently dPTjdPT� = B(0; T �)B(0; Tj) EPT��M�1Yk=j �1 + �k+1L(Tk; Tk)� ���FTj�: (24)Proof. Recall that we have (see Section 3)dPTjdPTj+1 = exp�Z Tj0 (u; Tj) � dW Tj+1u � 12 Z Tj0 j (u; Tj)j2du�; PTj+1 -a.s.,where (u; Tj) = �j+1L(u; Tj)1 + �j+1L(u; Tj) �(u; Tj); 8u 2 [0; Tj ]:4Notice that formula (22) agrees with what we already know, namely, that it is possible to determine the forwardLIBOR measure PTj�1 if the forward LIBOR measure PTj and the process L(�; Tj�1) are speci�ed.

20 Models of Forward LIBOR and Swap RatesSince Tj is �xed, we �nd it convenient to write L(t) = L(t; Tj); �(t) = �(t; Tj); etc. in what follows.Let us de�ne the process J by settingJt def= exp�Z t0 �j+1L(u)1 + �j+1L(u) �(u) � dW Tj+1u � 12 Z t0 ��� �j+1L(u)1 + �j+1L(u) �(u)���2du�for t 2 [0; Tj]: It is clear that J0 = 0 anddJt = Jt �j+1L(t)1 + �j+1L(t) �(t) � dW Tj+1t :On the other hand, the process It def= 1 + �j+1L(t)1 + �j+1L(0)satis�es: I0 = 1 anddIt = �j+1dL(t)1 + �j+1L(0) = �j+1�(t)L(t)1 + �j+1L(0) � dW Tj+1t = It �j+1L(t)1 + �j+1L(t) �(t) � dW Tj+1t :This shows that It = Jt for every t 2 [0; Tj ]: By setting t = Tj ; we obtain equality (22). To provethe second formula, we proceed by induction. For j =M � 1 the formula follows immediately from(22). Let us consider j =M � 2: For any A 2 FT�2 we haveB(0; T �)B(0; T �2 ) ZA 1B(T �2 ; T �1 ) 1B(T �1 ; T �) dPT� = B(0; T �1 )B(0; T �2 ) ZA 1B(T �2 ; T �1 ) dPT�1 = PT�2 (A):More generally, for any j and arbitrary event A 2 FTj ; we haveB(0; T �)B(0; Tj) ZA M�1Yk=j 1B(Tk; Tk+1) dPT�= B(0; T �)B(0; Tj) ZA 1B(Tj ; Tj+1) EPT�� M�1Yk=j+1 1B(Tk; Tk+1) ���FTj+1� dPT�= B(0; Tj+1)B(0; Tj) ZA 1B(Tj ; Tj+1) dPTj+1 = PTj (A);where the second equality follows from the induction step, and the last one is an immediate conse-quence of (22). This completes the proof of the lemma 2We are in a position to prove the following auxiliary result.Lemma 5.3 For any j = 1; : : : ;M; we havedP�j = cjBTjdPTj ; PTj -a.s., (25)where cj = 1=EPTj (BTj ) = B(0; Tj):Proof. We have (cf. (20))dP�dPT� jFTj = EPT� (B(0; T �)GTM j FTj ) = B(0; T �)GTjEPT��M�1Yk=j 1B(Tk; Tk+1) ���FTj�:Using Lemma 5.2, we �nd easily thatdP�dPT� jFTj = B(0; Tj)GTj dPTjdPT� :This proves (25) upon simpli�cation. 2In view of the last result, when dealing with futures contracts with maturity Tj ; we may introducethe spot martingale measure on (;FTj ) using equality (25) rather than (20). It appears that sucha choice considerably simpli�es the calculations.

Marek Rutkowski 215.2 Futures LIBOR Rate and Eurodollar Futures PriceBy the market convention, the settlement price of the Eurodollar futures contract equals (cf. Aminand Ng (1997)) E(T1; T1) = 1� �2L(T1; T1)at the contract's expiration date T1: Consequently, at any date t 2 [0; T1] the Eurodollar futuresprice (or index) equalsE(t; T1) = EP��1� �2L(T1; T1) j Ft� = 1� �2EP��L(T1; T1) j Ft�:Let us formally de�ne the futures LIBOR rate Lf (�; T1) for the date T1 by settingLf (t; T1) = EP�(L(T1; T1) j Ft) = EP�2 (L(T1; T1) j Ft); 8 t 2 [0; T1];so that E(t; T1) = 1� �2Lf (t; T1): Recall thatBT2 = �1 + �1L(0; 0)��1 + �2L(T1; T1)�and dP�2 = B(0; T2)BT2dPT2 :Therefore, using the abstract Bayes rule, we obtainLf (t; T1) = EPT2 �(1 + �2L(T1; T1))L(T1; T1) j Ft�EPT2 �1 + �2L(T1; T1) j Ft� = ItJt :It is evident that Jt = EPT2 �1 + �2L(T1; T1) ��Ft� = 1 + �2L(t; T1):Furthermore,It = EPT2 �(1 + �2L(T1; T1))L(T1; T1) ��Ft� = EPT2��1 + �2L(t; T1)e�t� 12 v2t �L(t; T1)e�t� 12 v2t ���Ft�;where �t = Z T1t �(u; T1) � dW T2u ; v2t = Var (�t) = Z T1t j�(u; T1)j2 du: (26)Notice that the random variable �t is independent of the �-�eld Ft and it has the Gaussian lawwith zero mean and the variance v2t : Consequently, we haveIt = L(t; T1) + �2L2(t; T1)EPT2 �e2�t�v2t � = L(t; T1) + �2L2(t; T1)ev2t : (27)Combining the formulae above, we conclude thatLf (t; T1) = L(t; T1)1 + �2L(t; T1) �1 + �2L(t; T1)eR T1t j�(u;T1)j2 du�for every t 2 [0; T1]: We are in a position to formulate the following result.Proposition 5.1 The futures LIBOR rate for the date T1 equalsLf (t; T1) = L(t; T1)�1 + �2g1(t)L(t; T1)�1 + �2L(t; T1) (28)and the Eurodollar futures price, for the futures contract that settles at time T1; equalsE(t; T1) = 1� �22g1(t)L2(t; T1)1 + �2L(t; T1) ; (29)where g1(t) = exp�Z T1t j�(u; T1)j2 du�; 8 t 2 [0; T1]: (30)

22 Models of Forward LIBOR and Swap RatesThe next result deals with the dynamics of the futures LIBOR rate and Eurodollar futures priceunder the spot measure P�1: It is useful to observe that, under our present assumptions, equalityP�1 = PT1 is satis�ed. For this reason, we do not need to introduce the \spot" Brownian motionW �; since we may and do assume that W � =W T1 :Proposition 5.2 The futures LIBOR rate satis�esdLf (t; T1) = Lf (t; T1)Zt�(t; T1) � dW �t ; (31)where W � =W T1 ; and the process Z is given by the formulaZt = 11 + �2L(t; T1) + �2g1(t)L(t; T1)1 + �2g1(t)L(t; T1) ; 8 t 2 [0; T1]: (32)The Eurodollar futures price satis�esdE(t; T1) = �E(t; T1)� 1�Zt�(t; T1) � dW �t : (33)Proof. Since Lf (t; T1) is given by an explicit formula (29), it is enough to apply the Ito formula.Let us denote Lft = Lf (t; T1); L(t) = L(t; T1); �(t) = �(t; T1) and g(t) = g1(t): It is clear thatLft = XtYt; where Xt = L(t)�1 + �2L(t)��1; Yt = 1 + �2L(t)g(t):It is useful to observe thatdL(t) = L(t)�(t) � dW T2t ; dg(t) = �g(t)j�(t)j2dt:Using Ito's rule, we �nd thatdXt = �1� �2Xt��Xt�(t) � dW T2t � �2X2t j�(t)j2dt�and dYt = �2g(t)L(t)��(t) � dW T2t � j�(t)j2dt�:This in turn implies thatdLft = �2g(t)L(t)Xt��(t) � dW T2t � j�(t)j2dt�+ �1� �2Xt�Yt�Xt�(t) � dW T2t � �2X2t j�(t)j2dt�+ �1� �2Xt��2Xtg(t)L(t)j�(t)j2dt:After rearranging, we obtaindLft = �(1� �2Xt)Yt + �2g(t)L(t)�Xt�t � �dW T2t � �2Xt�(t) dt�:Observe that dW T2t � �2Xt�(t) dt = dW T2t � �2L(t)1 + �2L(t) �(t) dt = dW T1t :Furthermore, by simple algebra we obtain��1� �2Xt�Yt + �2g(t)L(t)�Xt = Lft �2� �2Xt � Y �1t �:Upon substitution, this yields the dynamics (31). To derive (33), it is enough to observe thatdE(t; T1) = ��2dLf (t; T1) = ��2Lf (t; T1)Zt�(t; T1) � dW T1tand ��2Lf (t; T1) = E(t; T1)� 1:This completes the proof of the proposition. 2

Marek Rutkowski 235.3 Eurodollar Futures OptionsThe Eurodollar futures call option's payo� equals (at its expiry date T ) equalsEFC T = �E(T; T1)�K�+: (34)Equivalently, in terms of the futures LIBOR rates Lf ; we haveEFC T = �2��� Lf (T; T1)�+; (35)where the strike rate � satis�esK = 1��2�: We shall examine Eurodollar futures call option which issettled at the margin (that is, the option is itself a futures contract). From the general considerationsconcerning futures contracts, it follows that we need to evaluate the conditional expectationEFC t = EP���E(T; T1)�K�+ j Ft� = �2EP����� Lf (T; T1)�+ j Ft�: (36)For simplicity, unless explicitly stated otherwise, we assume that T = T1: i.e., the option's expirydate and the settlement date of the underlying Eurodollar futures contract coincide. Similarly asin the preceding section, we �nd it convenient to make the calculations under the forward LIBORmeasure PT2 : From the Bayes rule, we obtainEP����� Lf (T1; T1)�+ j Ft� = EPT2 ��1 + �2L(T1; T1)���� L(T1; T1)�+ j Ft�EPT2 �1 + �2L(T1; T1) j Ft� = ~It~Jt :It is clear that it is enough to evaluate ~It: For this purpose, note that~It = EPT2��1 + �2L(t; T1)e�t� 12 v2t ���� L(t; T1)e�t� 12 v2t �+ ���Ft�;where �t and vt are given by (26). Put another way, ~It = h(L(t; T1)); where the function h : R! Requals h(x) = EPT2��1 + �2xe�t� 12 v2t ���� xe�t� 12 v2t �+�:Let us write h(x) = h1(x) + h2(x); withh1(x) = EPT2 ��� xe�t� 12 v2t �+ = EPT2 ��� xe�t� 12Var (�t)�+and h2(x) = �2xEPT2 ��e�t� 12 v2t � ye2�t�2v2t �+ = �2xEPT2 ��e�t� 12Var (�t) � ye�t� 12Var (�t)�+;where y = xev2t and �t = 2�t: To end the calculations, it is enough to employ the following standardlemma.Lemma 5.4 Let (�; �) be a zero-mean, jointly Gaussian, two-dimensional random variable, de�nedon a probability space (;F ;P): For any positive real numbers a and b; we haveEP�ae �� 12 Var (�) � be�� 12 Var (�)�+ = aN(d1)� bN(d2);where d1;2 = ln(a=b)� 12Var (� � �)pVar (� � �) :

24 Models of Forward LIBOR and Swap RatesFor h1; Lemma 5.4 gives (notice that �t � �t = �t in our case)h1(x) = �N(e1)� xN(e2);where e1;2 = e1;2(t; T1) = ln(�=x)� 12 v2tvt :For h2; it yields h2(x) = �2x��N(e1)� xev2tN(e2)�;where e1;2 = e1;2(t) = ln(�=x)� v2t � 12 v2tvt :We have thus proved the following valuation result.Proposition 5.3 Consider a Eurodollar futures call option with expiry date T1; settled at the marginand written on a futures contract that also matures at time T1: Its price at time t 2 [0; T1] equalsEFC t = �21 + �2L(t; T1) ��N(e1)� L(t; T1)N(e2) + �2L(t; T1)��N(e1)� L(t; T1)ev2tN(e2)��;where e1;2 = e1;2(t; T1) = ln(�=L(t; T1))� 12 v2tvtand e1;2 = e1;2(t; T1) = ln(�=L(t; T1))� v2t � 12 v2tvt :Recall that � is the strike LIBOR rate { that is, the option's exercise price equals K = 1� �2�:Observe also that �1 + �2L(t; T1)��1 = B(t; T2)B(t; T1)is the forward price at time t of T2-maturity zero-coupon bond for the settlement date T1:Remarks. It is worthwhile to recall that the arbitrage price at time t 2 [0; T1] of a caplet (i.e., a calloption on the LIBOR rate) with strike rate �; maturing at T1; equals5Cpl t = �2B(t; T2)�L(t; T1)N(~e1)� �N(~e2)�; (37)where ~e1;2 = ~e1;2(t; T1) = ln(L(t; T1)=�)� 12 v2tvtand, as before, v2t = R T1t j�(u; T1)j2 du: From the put-call parity relationship, we deduce easily thatthe price of a corresponding oorlet (i.e., put option on the LIBOR rate) equalsFrl t = �2B(t; T2)��N(e1)� L(t; T1)N(e2)�; (38)In this regard, it is interesting to note that if N(e1) � 0 (and thus also N(e1) � 0) thenEFC t � �2 B(t; T2)B(t; T1) ��N(e1)� L(t; T1)N(e2)� = Frl tB(t; T1) :It seems reasonable to expect that the last formula should give a good approximation of EFC t; atleast for at-the-money or out-of-the-money Eurodollar futures call options with long time to expirydate (note that the condition � � L(t; T1) is equivalent to E(t; T1) � K).5We assume, of course, the framework of the lognormal model of forward LIBOR rates. Formula (37) is thus aspecial case of the valuation result established in Proposition 4.2.

Marek Rutkowski 255.4 Long-term Eurodollar Futures ContractsSo far, we have considered only futures contracts that settle at time T1; short-term Eurodollarfutures contract, say. Our aim is now to examine a long-term Eurodollar futures contract { thatis, a contract that settles at time Tj for some j � 2: For the next reset date, T2; we introduce thefutures LIBOR rate Lf (�; T2) by the formulaLf (t; T2) = EP�(L(T2; T2) j Ft) = EP�3 (L(T2; T2) j Ft); 8 t 2 [0; T2]:Since BT3 = �1 + �1L(0; 0)��1 + �2L(T1; T1)��1 + �3L(T2; T2)�and dP�3 = B(0; T3)BT3dPT3 ;we have Lf (t; T2) = EPT3 �(1 + �2L(T1; T1))(1 + �3L(T2; T2))L(T2; T2)� j Ft)EPT3 �(1 + �2L(T1; T1))(1 + �3L(T2; T2) j Ft� = ItJt :Suppose �rst that t belongs to the interval [T1; T2]: Then L(T1; T1) is measurable with respect tothe �-�eld Ft; and thus we obtainLf (t; T2) = EPT3 �(1 + �3L(T2; T2))L(T2; T2) j Ft�EPT3 �1 + �3L(T2; T2) j Ft� ; 8t 2 [T1; T2]:Consequently, reasoning along the similar lines as in Section 5.2, one �nds easily that (cf. (28))Lf (t; T2) = L(t; T2)�1 + �3g2(t)L(t; T2)�1 + �3L(t; T2) ; 8 t 2 [T1; T2];where g2(t) = exp�Z T2t j�(u; T2)j2 du�:The Eurodollar futures price for the settlement date T2 satis�es E(t; T2) = 1 � �3Lf (t; T2) so that(cf. (29)) E(t; T2) = 1� �32g2(t)L2(t; T2)1 + �3L(t; T2) ; 8 t 2 [T1; T2];Assume now that t 2 [0; T1): Observe �rst thatJt = EPT3 �(1 + �2L(T1; T1))EPT3 �1 + �3L(T2; T2) j FT1� j Ft�= EPT3 �(1 + �2L(T1; T1))(1 + �3L(T1; T2)) j Ft�since L(�; T2) follows PT3 -martingale. On the other hand, for It we haveIt = EPT3 �(1 + �2L(T1; T1))EPT3 �(1 + �3L(T2; T2))L(T2; T2) j FT1� j Ft�;where in turn (cf. (27))EPT3 �(1 + �3L(T2; T2))L(T2; T2) j FT1� = L(T1; T2)�1 + �3g2(T1)L(T1; T2)�:We conclude that for any t 2 [0; T1) we haveIt = EPT3 �(1 + �2L(T1; T1))(1 + �3g2(T1)L(T1; T2))L(T1; T2) j Ft�Recall that L(T1; T2) = L(t; T2) exp�Z T1t �(u; T2) � dW T3u � 12 Z T1t j�(u; T2)j2du�;

26 Models of Forward LIBOR and Swap Ratesand L(T1; T1) = L(t; T1) exp�Z T1t �(u; T1) � dW T2u � 12 Z T1t j�(u; T1)j2du�;where dW T2u = dW T3u � �3L(u; T2)1 + �3L(u; T2) �(u; T2) du:The last equality makes it obvious that a closed-form solution for It and Jt is not easily available,since the joint conditional law with respect to Ft of (L(T1; T1); L(T1; T2)) is not explicitly known.Let us �x t 2 [0; T1): To proceed further, we shall substituteW T2 with the process ~W T2 that satis�esd ~W T2u = dW T3u � �3L(t; T2)1 + �3L(t; T2) �(u; T2) du:Put more explicitly, the random variable L(T1; T1) is replaced by the random variable ~L(T1; T1);which is given by the formula~L(T1; T1) = wtL(t; T1) exp�Z T1t �(u; T1) � dW T3u � 12 Z T1t j�(u; T1)j2du�;where wt = exp�� �3L(t; T2)1 + �3L(t; T2) Z T1t �(u; T2) du�: (39)The joint conditional law underPT3 of the two-dimensional random variable (ln ~L(T1; T1); lnL(T1; T2))is Gaussian, with explicitly known parameters. For i = 1; 2 we write�t(i) = Z T1t �(u; Ti) � dW T3u ; v2t (i) = Var ��t(i)� = Z T1t j�(u; Ti)j2 du:Then It � ��13 EPT3��1 + xe�t(1)� 12 v2t (1)��1 + g2(T1)ye�t(2)� 12 v2t (2)�ye�t(2)� 12 v2t (2)�and Jt � EPT3��1 + xe�t(1)� 12 v2t (1)��1 + ye�t(2)� 12 v2t (2)��;where x = �2wtL(t; T1) and y = �3L(t; T2): Standard calculations now lead to the following result.Proposition 5.4 The futures LIBOR rate Lf (�; T2) satis�esLf (t; T2) = L(t; T2)�1 + �3g2(t)L(t; T2)�1 + �3L(t; T2) ; 8 t 2 [T1; T2];and Lf (t; T2) � L(t; T2)�1 +K(t; T1)�12t + �3g2(t)L(t; T2)�1 +K(t; T1)�12t ��1 +K(t; T1) + �3L(t; T2)�1 +K(t; T1)�12t � ; 8 t 2 [0; T1);where �12t equals �12t = exp�Z T1t �(u; T1) � �(u; T2) du�;K(t; T1) = �2wtL(t; T1) and wt is given by (39). In particular, if t 2 [0; T1) and �12t = 1; thenLf (t; T2) � L(t; T2)�1 + �3g2(t)L(t; T2)�1 + �3L(t; T2) :Finally, if t 2 [0; T1) and �(u; T1) = 0 for every u 2 [t; T1]; thenLf (t; T2) = L(t; T2)�1 + �3g2(t)L(t; T2)�1 + �3L(t; T2) :

Marek Rutkowski 27The statement of Proposition 5.4 may be reformulated as follows: if the conditional covarianceCovFt� lnL(T1; T1); lnL(T1; T2)� = ln�12t = Z T1t �(u; T1) � �(u; T2) du = 0; 8 t 2 [0; T1);then Lf (t; T2) � L(t; T2)�1 + �3g2(t)L(t; T2)�1 + �3L(t; T2) ; 8 t 2 [0; T2]:Let us stress that the right-hand side in the last formula always gives the exact value of Lf (t; T2)for any t 2 [T1; T2]: Note also that in the last statement of the proposition we deal with the ratherunappealing case when L(T1; T1) = L(t; T1) (this means, informally, that there is no incrementalrandomness in the dynamics of L(�; T1) over the time interval [t; T1]). Finally, let us observe that inorder to determine the Eurodollar futures price for the date T2 it is enough to use the relationshipE(t; T2) = 1� �3Lf (t; T2); 8 t 2 [0; T2]:Remarks. The case of a Eurodollar futures contract that settles at time Tj for j � 3 can be treatedalong the same lines. It seems natural to evaluate the conditional expectations under the forwardLIBOR measure PTj+1 ; working backwards in time from the date Tj :AcknowledgementsThis work was carried out during the author's stay at the University of New South Wales in Sydney.The author would like to thank Professor Marek Musiela from the University of New South Walesfor his hospitality and for stimulating discussions. The author expresses also his gratitude for the�nancial support from the Australian Research Council.References[1] Amin, K. and Ng, V.K.: Inferring future volatility from the information in implied volatility inEurodollar options: a new approach. Rev. Finan. Stud. 10 (1997), 333{367.[2] Ball, C.A. and Torous, W.N.: Bond price dynamics and options. J. Finan. Quant. Anal. 18(1983), 517{531.[3] Brace, A.: Arbitrage free lognormal interest rate models. Working paper, Risk Seminars, HongKong Singapore Tokyo, 1997.[4] Brace, A., G�atarek, D. and Musiela, M.: The market model of interest rate dynamics. Math.Finance 7 (1997), 127{154.[5] Flesaker, B.: Arbitrage free pricing of interest rate futures and forward contracts. J. FuturesMarkets 13 (1993), 77{91.[6] Geman, H., ElKaroui, N. and Rochet, J.C.: Changes of numeraire, changes of probabilitymeasures and pricing of options. J. Appl. Probab. 32 (1995) 443{458.[7] Hansen, O.K.: Theory of an arbitrage-free term structure. Econom. Finan. Computing, Summer(1994), 67{85.[8] Heath, D., Jarrow, R. and Morton, A.: Bond pricing and the term structure of interest rates:A new methodology for contingent claim valuation. Econometrica 60 (1992), 77{105.[9] Jamshidian, F.: Pricing and hedging European swaptions with deterministic (lognormal) for-ward swap rate volatility. Preprint, Sakura Global Capital, 1996.

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